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# Introductory Physics I PHY 231

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This 249 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 231 at Michigan State University taught by Staff in Fall. Since its upload, it has received 398 views. For similar materials see /class/207627/phy-231-michigan-state-university in Physics 2 at Michigan State University.

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Date Created: 09/19/15

Chapter 12 The Laws of Thermodynamics Principles of Thermodynamics CI Energy is conserved 0 FIRST LAW OF THERMODYNAMICS 0 Examples Engines Infernal gt Mechanical FricTion Mechanical gt In rernal CI All processes mus r increase enfropy o SECOND LAW OF THERMODYNAMICS o En rropy is measure of disorder 0 Engines can noT be 100 efficienT Converting Internal Energy to Mechanical 3 Work done by expansion WzFAx FPA szAVA Example A cylinder of radius 5 cm is kepT aT pressure wiTh a pisTon of mass 75 kg a WhaT is The pressure inside The cylinder b If The gas expands such ThaT The cylinder rises 120 cm whaT work was done by The gas c WhaT amounT of The work wenT inTo changing The graviTaTionaI PE of The pisTon d Where did The resT of The work go Solution Given M 75 A 750052 Ax012 Pam 1013x105 Pa 1 Find P903 P M8 P 195Ox105 Pa gas atm b Find W905 WP AAx 1838J gas C Wgr39avify W 883 J d Where did The ofher39 Compressing The ou rside air39 work go Quiz Review A massive copper pisTon Traps an ideal gas as shown To The righT The pisTon is allowed To freely slide up and down and equilibraTe wiTh The ouTside air Pick The mosT correcT sTaTemenT A The pressure is higher inside The cylinder Than ouTside B The TemperaTure inside The cylinder is higher Than ouTside The cylinder C If The gas is heaTed by applying a l flame To cylinder and The pisTon comes To a new equilibrium The inside pressure will have increased D All of The above E A and C 1sz M l v Some Vocabulary p 1 El Isobaric l o P cons ran r 4 A o P EIIsovolume rr39Ic 0 V cons ran r Y A r EIIso rher39mal p K o T cons ran r V El Adiaba ric 0 Q O P vlt Example Outside Air Room T Atm P A massive pis ron rraps an amoun r of Helium gas as shown The pis ron freely slides up and down The sys rem i ni rially equilibra res a r room rempera rure a Weigh r is slowly added To The pis ron isofhermaIy compressing The gas ro half i rs original volume b APb9ltPCl BTbgtltTa CWabgt0 E Qab gtO Vocabulary Wab is work done by gas be rween a and b Qab is hea r added To gas be rween a and b Example Outside Air Room T Atm P A massive pis ron rraps an amoun r of Helium gas as shown The pis ron freely slides up and down The sys rem i ni rially equilibra res a r room rempera rure a Weigh r is slowly added To The pis ron adiaba caIy compressing The gas ro half i rs original volume b Vocabulary Wab is work done by gas be rween a and b Qab is hea r added To gas be rween a and b Example Outside Air Room T Atm P A massive pis ron rraps an amoun r of Helium gas as shown The pis ron freely slides up and down The sys rem i ni rially equilibra res a r room rempera rure a The gas is cooled isobarically compressing The gas ro half i rs original volume b A Pbgtlt Pa BWa gt O CTbgt TCl D Ubgt Ua EQab gtO Vocabulary Wab is work done by gas be rween a and b Qab is hea r added To gas be rween a and b Work from closed cycles Consider cycle A gt B gt A x P Work from closed cycles Consider cycle A gt B gt A A P WABA Area vlt Work from closed cycles Reverse The cycle make if coun rer39 clockwise P Example Pam 4 3 v 2 B 1 4 I IV o 1 2 3 Vm3 m ZINE Thomson Emoks cm a WhaT amounT of work is performed by The gas in The cycle IAFI WIAFI 3P0Tm J b How much heaT was inser39Ted i nTo The gas in The cycle IA FI AU O Q 304x105 J c WhaT amounT of work is performed by The gas in The cycle IBFI W 3O4x105 J One More Example Consider a mono ronic ideal gas which expands according 751 To The PV diagram EP kPa A a Wha r work was done by 50 The gas from A To B b Wha r hea r was added To B The gas be rween A and B 25 quot C c Wha r work was done by V m3 The gas from B To C l l l d Wha r hea r was added To 02 04 O 6 The gas beween B and C e Wha r work was done by The gas from C To A f Wha r hea r was added To The gas from C To A Solution A 75 0 Find WAB WAB Area 20000 J 50 39 b Find QAB 25 02 04 06 Fir39s r find UA and UB UA 22500 J UB 22500 J AU 0 Finay solve for Q Solution A 7539 c Find W3C WAB Area 10000 J 50 d QBC 25quot 02 04 06 Fir39s r find uB and uc uB 22500 J UC 7500 J AU 15000 Finay solve for Q Q 25000 J Solution 5 P kPa A 7539 2 Find WCA WAB Area 0 J 50 f QCA 25quot B C V m3 I I I L I I I V 02 04 06 4am find uc and L1 UC 7500 J UA 22500 J AU 15000 Finay solve for Q Q 15000 J Continued Example Take solu rions from Ias r problem and find a Ne r work done by gas in The cycle b Amoun r 0f hea r added To gas WAB WBC WCA 10000 J QAB quot QBC QCA 10000 J This does NOT mean Tha r The engine is 100 efficien r ReVieW QUiZ Outside Air Room T Atm P A massive pis ron Traps an amoun r of Helium gas as shown The pis ron freely slides up and down The sys rem i ni rially equilibra res a r room rempera rure T TCl 20 C a The gas is Then slowly hea red un ril T Tb 100 C b A PCl Pb B Wab gt O 639 Qab gt Wab D All of The above E A and C Vocabulary Wab is work done by gas be rween a and b Qab is hea r added To gas be rween a and b Entropy CI Measure of Disorder of The sys rem randomness ignorance CI 5 kglogN N of possible arrangemen rs for fixed E and Q py Why do we use IogN Consider rwo sys rems A and B NA NB of ways To arrange en rir39e sys rem 10g 10gNA10gNB S A S B En rr opy of bo rh sys rems Entropy CI To ral En rropy always rises 2nd Law of Thermodynamics CI Adding hea r raises en rropy Defines Temperature in Kelvin Why does Q ow from hot to cold EIConsider39 rwo sys rems one wi rh TA and one wi rh TB CIAIIow Q gt 0 To flow from TA ro TB EIEn rr39opy changed by AS QTB QTA EIIf TA gt T3 rhen AS gt O EISys rem will achieve more randomness by exchanging hea r un ril TB TA Ef ciencies of Engines CI Consider a cycle described by W work done by engine QhoT Qhof hea r Thai flows lh I39O engine from source a r Thor Qcold hea r exhaus red from engine a r lower Temperature Tcold CI Efficiency is defined Qhot Qcold 1 Qcold Qhot Qhot Qcold gt Qhot gt Qcold gt Tycold gt Tycold 1201 Qhot not lst and 2nd Laws of Thermodynamics 1 Energy is conserved YOU CAN T WIN 2 En rr39opy always increases YOU CAN T BREAK EVEN Carnot Engines CI Idealized engine CI Mos r efficient possible Carnot Cycle Energy resermir 3 Th a D gt A B gt C Adiabatic C I Adiabalic compression c I y expansinn Q0 Q0 d I gt I Isothermal cnmpm inn Energy reservoir at T 2003 Thomson BrnoksICole C 1004 omson BrooksCole Example An ideal engine CarnoT is raTed aT 50 efficiency when iT is able To exhausT heaT aT a TemperaTure of 20 C If The exhausT TemperaTure is lowered To 30 C whaT is The new efficiency Solution Given e05 when Tcold2293 K Find e when Tcoldz 243 K FirsT find ThoT 1 T T01d 586 K 1 6 hot 0 Now find e given Thof586 K and Tcold243 K e 0586 Refrigerators Given Refrigera red region is a r Tcold Hea r exhaus red ro region wi rh ThoT Qho Find Efficiency v Qcold 1 Qhot Qcold Qhot Qcold 1 Qhot gt Qcold Qhot gt not 1110f Tycold Qcold Tycold No re Highes r efficiency for39 small T differences I Heat Pumps Given Inside is a r ThoT OUTSide is 0139 Tcold Qhof Find Efficiency Qhot 1 Qhot Qcold 1 Qcold Qhot Qhot gt Qcold Qhot gt not 1110f Tycold Qcold Tycold Like Refrigerator Highes r efficiency for small AT Example A modern gas furnace can work aT pr39acTically 100 efficiency ie 100 of The heaT from burning The gas is conver39Ted i nTo heaT for39 The home Assume ThaT a heaT pump wor39ks aT 50 of The efficiency of an ideal heaT pump If elecTr39iciTy cosTs 3 Times as much per39 kwhr39 as gas for39 whaT range of ouTside Temper39aTur39es is iT advanTageous To use a heaT pump Assume Timide 295 K Solution Find T for39 which e 3 for39 hea r pump Above This T use a hea r pump Below This T use gas T295 50 of ideal hea r pump T 295 Z4SSOK 27 C Real engines emit more than heat Lecture 32 Transverse and longitudinal Waves Waves on a Wire Guitar strings Waves travelling down a wire or a rope are transverse waves and their wavespeed increases the greater the tension in the wire and reduces as the mass of the string or wire increases The precise formula is 7 F 12 v M 1 Travelling waves are like water waves that move in a particular direction A different sort of wave occurs on guitar string because both ends of the string are xed The vibrations of the string do not seem to move with the wavespeed 1 Why not These waves seem to stay in one place and for this reason are called standing waves They can be thought of as a sum of travelling waves where each of the travelling waves in the string moves with the wavespeed v For standing waves to occur there needs to be special relationships between the wavelength of the wave and the length ofthe string In the case of a guitar the wave amplitude must be zero at each end of the string so that an integral number of half wavelengths must t in the string length L We must then have 1 2L iAnn L or An Y 2 The vibrations of the string cause a pressure wave in the air and this pressure wave is the sound which we hear as we will return to later Notice that waves on a guitar string are transverse standing waves The fundamental wavelength is A1 2L The frequency associated with this fundamental f1 vAl The frequencies associated with the other wavelengths are given W U n fnn vnf1 3 Sound Waves Applications Sound waves are longitudinal waves and have different names depending on the frequency of the waves Human Audible frequencies lie in the range 20H2 lt f lt 20000H2 lnfrasonic waves lie below the audible frequency range while ultrasonic waves have frequencies above the audible Ultrasonic waves are used extensively in biology ranging from low intensity applica tions in imaging eg ultrasound to high intensity applications in surgery eg CUSA A piezoelectric device can be used to generate sound waves as the piezo material oscillates at the same frequency as an applied oscillatory voltage The percentage of the sound wave that is re ected is given by PR ph MYHOO 4 h M Contrast is stongest when the differences in density are largest The speed of sound The speed of sound in a material in a liquid or a gas is given by B U if2 5 where B is the bulk modulus and p is the density In a solid the sound velocity is given by Y 1 2 v i 6 where Y is the Young7s modulus Sound travels faster in water or a solid than in air The speed of sound is temperature dependent and in air the dependence is approximated byin meterss T 331 7 12 7 v 273K Intensity I lntensity is equal to power per unit area and has the units Wmz We therefore have 7 Power 7 8 7 Area 7 A At The limits of human hearing at 1kH2 are 1042mm2 lt 1 lt 1Wm2 9 where the lower limit is the faintest sounds that can be heard while the upper limit is the pain threshold Acoustic surgery is possible at very high intensities which requires careful focusing of the acoustic waves Typical Lecture 18 Principles of Thermodynamics Energy is conserved FIRST LAW OF THERMODYNAMICS Examples Engines Heat gt Mechanical Energy Friction Mechanical Energy gt Heat All processes must increase entropy SECOND LAW OF THERMODYNAMICS Entropy is measure of disorder Engines can not be 100 efficient Work done on a gas Adding hea r Q can Change Temperature Change of Change state of matter Imema39 Energy AU Can also change AU by doing work on The gas W i A Flt Aygt FAX Aw First Law of Thermodynamics Conserva rion of Energy Can change internal energy AU by Adding hea r ro gas Q Doing work on gas W PAV No re Work done by The gas Work done on The gas W PAV by the gas Add hea r gt Increase In r Energy amp Gas does work Example 121 A cylinder of radius 5 cm is kept at pressure with a piston of mass 75 kg a What is The pressure inside The cylinder 1950x105 Pa b If The gas expands such Thai The cylinder rises 120 cm who work was d by39l39h g one e as1838J c What amount of The work went into changing The gravi39l39a39l39ional PE of The piston 883 J d Where did The rest of The work go Compressing The outside air Example 122a A massive copper piston traps an ideal gas as shown to the right The piston is allowed to freely slide up and down and equilibrate with the outside air The pressure inside the cylinder is pressure outside the a Greater than b Less than c Equal to Example 122b A massive copper piston traps an ideal gas as shown to the right The piston is allowed to freely slide up and down and equilibrate with the outside air The temperature inside the cylinder is the temperature T gt outside lfi a Greater than b Less than c Equal to Example 122c A massive copper piston traps an ideal gas as shown to the right The piston is allowed to freely slide up and down and equilibrate with the outside air If the gas is heated by a steady flame and the piston rises to a new equilibrium position the new pressure will be than the previous pressure a Greater than b Less than c Equal to Some Vocabulary Isobaric P constant Isovolumetr39ic V constant WO Isothermal T constant AU O ideal gas Adiabatic 0 Q O vlt vlt vlt vlt Example 123a A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Weight is slowly added to the piston isothermaly compressing the gas to half its original volume b Pb is Pa a Greater than b Less than c Equal to Outside Air Room T Atm P Example 123b Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Wei ht is sow added to the iston 9 Y P 1 T isothermaly compressing the gas to am half its original volume b A Tb is To a Greater than b Less than c Equal to Example 123c Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Weight is slowly added to the piston L T isothermaly compressing the gas to gt half its original volume b Wab is 0 quot a Greater than A b Less than c Equal to Vocabulary Wab is work done by gas betwen a an Example 123d Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Wei ht is sow added to the iston 9 Y P 1 T isothermaly compressing the gas to am half its original volume b A Ub is Ua a Greater than b Less than c Equal to Example 123e Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Wei ht is sow added to the iston 9 Y P 1 T isothermaly compressing the gas to am half its original volume b A Qab is o a Greater than b Less than c Equal to Vocabulary Qab is heat added to gasetween a and b Example 12394a Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Wei ht is slowl added to the iston 9 Y P 1 T adiabaticaly compressing the gas to am half its original volume b A Pb l5 Pa a Greater than b Less than c Equal to Example 12394b Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Wei ht is slowl added to the iston 9 Y P 1 T adiabaticaly compressing the gas to am half its original volume b Web is 0 a Greater than b Less than c Equal to Example 12394c Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Wei ht is slowl added to the iston 9 Y P 1 T adiabaticaly compressing the gas to am half its original volume b A Qab is o a Greater than b Less than c Equal to Example 12394d Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Wei ht is slowl added to the iston 9 Y P 1 T adiabaticaly compressing the gas to am half its original volume b A Ub l5 Ua a Greater than b Less than c Equal to Example 12394e Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a Wei ht is slowl added to the iston 9 Y P 1 T adiabaticaly compressing the gas to am half its original volume b A Tb l5 1l a Greater than b Less than c Equal to Example 12395a Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a The gas is cooled isobaricaly compressing the gas to half its original volume b Pb is Pa a Greater than b Less than c Equal to Example 12395b Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a The gas is cooled isobaricaly compressing the gas to half its original volume b Wab is 0 a Greater than b Less than c Equal to Example 12395c Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a The gas is cooled isobaricaly compressing the gas to half its original volume b Tb is To a Greater than b Less than c Equal to Example 12395d Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a The gas is cooled isobaricaly compressing the gas to half its original volume b Ub is U a a Greater than b Less than c Equal to Example 12395e Outside Air Room T Atm P A massive piston traps an amount of Helium gas as shown The piston freely slides up and down The system initially equilibrates at room temperature a The gas is cooled isobaricaly compressing the gas to half its original volume b Qab is o a Greater than b Less than c Equal to PV Diagrams Pa rh moves ro right Wby The gas Area under curve Pa rh moves ro left Wby m gas Area under curve V Won The gas Why The gas Work from closed cycles Consider cycle A gt B gt A AP Area Area work done by gas Work from closed cycles Consider cycle A gt B gt A AP WABA Area work vlt done by gas Work from closed cycles Reverse The cycle make if counter clockwise A P A Area Area gtB work done by gas Work from closed cycles Reverse The cycle make if counter clockwise AP WAgtBgtA Are vlt work BO done by gas Internal Energy in closed cycles in closed cycles Example 126 a What amount of work is performed by the gas in the cycle IAFI Harm 4 A W304x105 J 3 b How much heat was inserted into the gas in the cycle IAFI Q 304x105 J c What amount of work is 4llll 0 1 2 3 4 performed by the gas In the cycle IBFI V 3 w 304x105 J a mu mumquot Brookscal Consider a monoTonic ideal gas Example 127 a WhaT work was done by P kPa The gas from A To B 75 55 A 20000 J b WhaT heaT was added To The gas beTween A and B 50 20000 c WhaT work was done by B The gas from B To C 25quot 10000 J C d WhaT heaT was added To V ms The gas beween B anczisC39SOO J l l l e WhaT work was done39 by 02 04 06 The gas from C To A f WhaT heaT was added To The gas from C To A 15000 J Example Continued Take solutions from last problem and find a Net work done by gas in The cycle b Amount of heat added To gas WAB WBC WCA J QAB QBc QCA 10000 J This does NOT mean That The engine is 100 efficient Example 128a Consider an ideal gas undergoing The Trajectory Through The PV diagram In going from A To B To C The work done BY The gas is O agt blt c A l P W V Example 128b In going from A To B To C The change of The inTernal energy of The gas is O agt blt c A l P W V Example 128c A C P D In going from A To B To C The amount of heat added To The gas is O W V agt blt c Example 128d A C P D In going from A To B To C To D To A The work done BY The gas is O W V agt blt c Example 128e In going from A To B To C To D To A The change of The inTernal energy of The gas is O agt blt c A l P W V Announcements 0 ExTr39a sTudy session SaTur39day December 7 100 PM Room 1415 BPS 1 THE FINAL EXAM WILL TAKE PLACE AT 800 PM 1000 PM MONDAY DEC 9 A misprinT in The syllabus lisTed The Time as 900 PM 2 DifferenT secTions will Take The final aT differenT locaTions SECTION I C102 Wilson Wilson is a large residence hall on Wilson Ave on The souThwesT side of campus SECTION II 138 CEM This is in The ChemisTr39y building which is jusT nor39Th of The BPS building where we have lecTur39e SECTION III N130 BCC This is in The Business Complex CenTer39 aT The nor39ThwesT corner39 of Bogue and Shaw Announcements 3 The makeup exam will be given aT 545 PM Tuesday Dec 10 in room 1415 BPS The room adjacenT To where we have lecTure If you need To Take The makeup exam please email Dr Tung or Dr PraTT and include a brief explanaTion as To why you are unable To Take The exam aT The regular Time 4 The same rules which applied To The midTerms will apply To The final In parTicular please bring your picTure ID IDs will be checked carefully aT The end of The exam one page of noTes you can use boTh sides a calculaTor a 2 pencil Where they come from Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 1 problem 1 problem 1 problem 2 problems 2 problems 2 problems 1 problem 1 problem 3 problems 1 problem Chapter 11 2 problems Chapter 12 2 problems Chapter 13 3 problems Chapter 14 3 problems Where they come from 9 of the problems are new meaning that they were written from scratch 3 are from the previous midterms 5 are from HW These are mostly from the last 4 problem sets 5 are from inclass quizzes 2 are from inclass examples 1 is from the practice exam e frequency of a tunin fork is 468 Hz Find the length of e none f the above 2 An elevator with mass m 1000 kg is going up with acceleration a 20 ms2 Find the tension T in the supporting cable a 5000 kg ms2 b 8000 N c 10000 kg ms2 11800N e 15000 kg ms2 3 In a ballistic pendulum experiment two processes occur in suscession l A bullet at high initial velocity v0 is stopped in a wooden block which is suspended from the ceiling by a string and the combined system acquires a velocity of V and 2 the system executes a pendulum motion in Which the velocity decreases as the pendulum rises I until it reaches a maXimum height and turns back Linear momentum along the direction of motion is conserved in a process 2 but not 1 roc l but not 2 c both 1 and 2 d neither 1 nor 2 e none of the above l39N l l In the same ballistic pendulum problem as above mechanical energy kinetic plus potential energies is conserved in rocess 2 but not 1 b process 1 but not 2 c both 1 and 2 d neither 1 nor 2 e none of the above 5 If you increase the power level to your speakers by 27 dB by What factor is the amplitude of the pressure wave increase a 501 24 c 149 d 386 e none of the above 6 Waves propagate at 8 ms along a stretched string The end of the string is vibrated up and down once every 25 s What is the wavelength of the waves that travel along the string a 10m Om c 50m d 15m e none of the above 7 Which one of the following quantities is at a maximum when an object in simple harmonic motion is at zero displacement a potential energy b frequency c acceleration inetic energy e force 8 One person hits the rail with a hammer while another one is some distance away with one ear on the rail Suppose that the time difference between the arrivals of the two sounds traveling through air and through steel is 398 s If the speed of sound in air was 340 ms how far in km are the two persons apart For steel Young39s modulus is 201011Nm2 and the density is 7800 kgm3 145 km b 234 km 0 568 km 1 355 km e none of the above 9 A force of magnitude 30 N acts on a 12 kg block If the initial velocity of the block is 4 ms and the force acts in the direction of this velocity over What distance must this force act to change the block s velocity from 4 to 6 ms a 20 m V m l h 5139 I b 02 In 1 3 N d 40 m Lquotquot39I e none of the above 10 A gas is at an initial pressure of 15 104 Pa and occupies a volume of 025 m3 The slow addition of 25000 cal of heat to the system causes it to expand isobarically to a volume of 040 m3 What is the increase in the internal energy of the system a 2250 J b 25000 cal c 5972 J 1 107 X 103 J 102 x 103J 11 A 05 kg mass oscillates in simple harmonic motion on a spring which has a spring constant of 2500 Nm The spring was initially compressed 12 cm a negative displacement What is the velocity in ms When the mass is at a positive displacement of 5 cm a 35 ms b 58 ms 77 ms d 151 ms e 225 ms 12 An ideal Carnot engine takes in 20 kcal of heat at 80 CC and exhausts some of this at 10 CC How much work in J must have been done by the engine a 422 gtlt105 b 1009 c 100 900 d 212 gtlt104 e 2120 No right answer should be 16600 J 13 A 100 kg mass is pushed up a frictionless inclined plane How much work has to be done against gravity to move the block from the bottom to the top of a plane Whose height is 5 m and which makes an angle 9 37 with the horizontal a 3913 J b 500 J 4900 J 1 1960 J e none of the above T T m a J 14 A heat engine exhausts heat of amount Qc 3000 J While performing useful work of the amount W15 00 J What is the efficiency of the engine a 15 b 50 c 60 3 e 100 15 Heat is added to 2 kg ofice at 18 CC HOW many kilocalories are required to change the ice to steam at 110 CC a 14580 b 14496 0 3600 14676 e none of the above 16 A ball is thrown horizontally from the top of a 15 m high cliff If the initial speed of the ball is 10 ms What is the speed of the ball when it hits the ground a 10 ms b 298 ms 0 15 ms 198 ms nk l l 111F5 e none of the above 5 m 17 A block of aluminum with a mass of 38 kg is at 8 CC and comes in contact With a hot block of copper With a mass of 8 kg at 56 CC What is the nal temperature a 215 0C 06 0C c 353 0C d 427 0C e none of the above 18 Find the amount of heat Q transferred per square meter in 1 hr by conduction through a composite wall of thickness 036 m if the inside temperature is Ti and the outside temperature is To Assume the thermal conductivity of this wall to be 10 Js m C C Let T1 30 CC and TO 10 C C then the amount of heat is a 2000 J b 10000 J c 36000 J 00000 J e none of the above 19 A gas cylinder of volume 30 liters contains an ideal gas at temperature 27 degrees C and pressure 1870 kPa Some of the gas leaks until the pressure falls to 1330 kPa How many moles of gas leaked assuming that the temperature remains constant during this process a 25 b 47 65 d 96 e none of the above 20 A wooden beam of length L is hinged at one end A force F acts on the other end at an angle as shown in the gure The resulting torque which determines the rotational motion of the beam around the pivot point is F cosG b L F cosG c L F d L F sine e L F sine 21 A piece of moon rock reads 282 grams on a scale when in air but 14 grams in alcohol speci c gravity 079 What is its density in kgm3 a 567 b 1268 c 1453 1569 e none of the above 22 A steel Wire 100 m long at 10 CC has a coefficient of linear expansion of l 10395C Give its change in length as the temperature changes from 10 CC to 40 CC a 10 cm 0 cm c 40 cm d 10 m e 03 m 23 A puck of mass m is attached to a string and is Whirled around in a circular motion Initially the radius of the circle is R and the angular speed is 0 The string is gradually extended outward until the radius becomes 2R What is the angular speed at the new radius 4 b032 cc d20 e40 24 Consider a child playing on a swing As she reaches the lowest point in her swing which of the following statements is true a her acceleration is downward and equal to g 98 ms2 b the tension in the rope is equal to her weight mg c the tension in the rope is equal to her mass times her acceleration ma he tension is equal to mg mv2L where L is the gth of the swing e none of the above Example 133 PHY2317 Spring 2008 A 36 kg block is attached to a spring of constant k 600 Nm The block is pulled 35 cm away from its equilibrium position and is pushed so that it has an initial velocity of 50 cms at t 0 What is the position of the block at t 075 seconds Answer The formula for position as a function of time is zt Acoswt 7 b We will also need velocity as a function of time 0t 701A sinwt 7 b We are given m 36 kg7 k 600 Nm7 00 35 cm7 00 50 cms Note that the amplitude is not 35 cm7 because it does not start from rest The initial push will make the amplitude larger than 35 cm First7 we can nd the angular frequency from w ikm 40825 rads Next7 we must nd the amplitude and the phase In general7 A and b are always determined by the initial conditions7 00 and 00 We plug t 0 into the equations for position and velocity and set them equal to the initial conditions 00 Acos7 Acos 35 cm 00 701A sin7 wA sin 50 cms Note that I used cos7 cos and sin7 7 sin We now have 2 equations for 2 unknowns7 A and b You can solve for b by taking the ratio of the second equation over the rst equation on each side wAsin i 50 cms ACOS 35 cm tanltltz5034993 Our calculator gives 0 03366 We can then plug this back into either of the initial condition equations to nd A 37081 cm Now that we know A7 157 and 017 we can get the nal answer by plugging t 075 s into the formula for The nal answer is 0075 s 7339 cm One nal comment Make sure your calculator is set for angles in radians ap er Solids and Fluids Elasticity Archimedes Principle Bernoulli39s Equation States of Matter Solid Liquid Gas 0 Plasmas Solids Stress and Strain Stress Measure of force felt by material Force Stress 7 Area SI units are Pascals 1 Pa 1 NM2 same as pressure Solids Stress and Strain Strain Measure of deformation F AL St 39 7 mm L IAL dimensionless Young s Modulus Tension Example 91 King Kong a 80x104 kg monkey swings from a 320 m cable from the Empire State building If the 30 cm diameter cable is made of steel Y18x1011 Pa Tensile STF BSS by how much will the cable stretch tensile strain 197 m Measure of stiffness Tensile refers to tension Shear Modulus Fixed face Bulk Modulus quotquot Change in Pressure h jf Volume STrain Pascals as units for Pressure 1Pa1Nm2 Example 92 A large solid sTeel Y18x1011 Pa block L 5 m W4 m H3 m is submerged in The Mariana Trench where The pressure is 75x107 Pa a B whaT ercenTa e does The len Th chan e y O841 g g g b WhaT are The changes in The lengTh widTh and heighT 208 mm 167 mm 125 mm c By whaT percenTage does The volume change O125 Solids and Liquids Solids have Young39s Bulk and Shear moduli Liquids have only bulk moduli Ultimate Strength Maximum FA before fracTure or crumbling DifferenT for compression and Tension Densities TABLE 93 Density of Some Common Substances Substance pkgm3 a Ice 0917 X 103 Aluminum 270 X 103 Iron 786 X 103 Copper 892 x 103 Silver 105 x 103 Lead 113 x 103 Gold 193 gtlt 103 Platinum 214 X 103 Uranium 187 X 103 Substance pkgm3a Water 100 X 103 Glycerin 126 X 103 Ethyl alcohol 0806 X 103 Benzene 0879 X 103 Mercury 136 X 103 Air 129 Oxygen 143 Hydrogen 899 X 10 2 Helium 179 X 10 1 All values are at standard atmospheric temperature and pressure STP de ned as 0 C 273 K and l aLm 1013 X 10 Pa To convert to grams per cubic centimeter multiply by 10 3 2003 Thomson BrooksCole Density and Specific Gravity Densities depend on temperature pressure Specific gravity ratio of density to density of H20 at 4 oc Example 93 The specific gravity of gold is 193 What is the mass in kg and weight in lbs of 1 cubic meter of gold 19300 kg 42549lbs Pressure amp Pascal39s Principle Pressure applied to any part of an enclosed fluid is transmitted undimished to every point of the fluid and to the walls of the containerquot Each face feels same force Transmitting force An applied force F1 can be amplified Hydraulic press Examples hydraulic brakes forklifts car lifts etc Pressure and Depth aloe Purl w is weight w M8 pVg pAhg Sum forces to zero PA POA wzo Factor A Mgul w m A Example 95 skip Find fhe pressure af 10000 m of wafer DATA Afmospheric pressure 1015x10s Pa 982x107 Pa Example 96 Assume fhe ufimafe sfrengfh of legos is 40x104 Pa If fhe densify of Iegos is 150 kgms whaf is fhe maximum possible heighf for a lego fower 272 m Example 97 Esfimafe fhe mass of fhe Earfh39s afmosphere given fhaf afmospheric pressure is 1015x10s Pa Dafa Rh636x10 m 526x10 kg Archimedes Principle Any objecf complefely or parfially submerged in a fluid is buoyed up by a force whose magnifude is equal fo fhe weighf of fhe fluid displaced by fhe objecf Proving Archimedes Principle Hop Ing Rap Fnat Boothom 791 h Fbuttom pgD Example 98 A helicopfer lowers a probe info Lake Michigan which is suspended on a cable The probe has a mass of 500 kg and ifs average densify is 1400 kgm Whaf is fhe fension in fhe cable 1401 N Example 99a A wooden ball of mass M and volume V floats on a swimming pool The density of the wood is pwood ltpH20 The buoyant force acting on the ball is a Mg upward b PHonV upward C pHZOpwoodgv upward Example 99b A steel ball of mass M and volume V rests on the bottom of a swimming pool The density of the steel is psml gtszo The buoyant force acting on the ball is a Mg upward b PHonV upward C psteelpH20gv upward Example 910 A small swimming pool has an area of 10 square meters A wooden 4000kg statue of density 500 kgm3 is then floated on top of the pool How far does the water rise Data Density of water 1000 kgm3 40 cm Floating Coke Demo SKIP The can will a Float b Sink Paint Thinner Demo SKIP When I pour in the paint thinner the cylinder will a Rise b Fall Equation of Continuity What goes in must come out mass density Ax pAvAt A Mass that passes a point in pipe during time At t i tummy n in u Example 911 Water flows through a 40 cm diameter pipe at 5 cms The pipe then narrows downstream and has a diameter of of 20 cm What is the velocity of the water through the smaller pipe 20 cms Laminar or Streamline Flow Fluid elements move along smooth paths Friction in laminar flow is called viscosity 2003 Thomson BrooksCole Turbulence Fluid elements move along irregular paths Sets in for high velocity gradients small pipes or instabilities Ideal Fluids Laminar Flow Bernoulli s Equation Sum of P KEV and PEV is constant How can we derive this gt No turbulence Nonviscous gt No friction between fluid layers Incompressible gt Density is same everywhere 1 1 AKE Mv va 2 2 1 1 APE ngz ng1 pAngz pAng1 PIAIAxl P2142sz PlAV PZAV 2003 Thomson Bro Point 1 W 2 Flel Fzsz 1 v Bernoulli s Equation derivation inksCole t 2 Consider a volume AV of mass AM of incompressible fluid AXQ P2A2 Example 912 A very large pipe carries water with a very slow P1 velocity and empties into a small pipe with a high velocity If P2 is 7000 Pa lower than P1 what is the velocity of the water in A2 the small pipe A1 Venturi Meter 374 ms Applications of Bernoulli s Equation Venturi meter Curve balls Airplanes Beach Ball amp Straws Demos Example 913a Consider an ideal incompressible fluid choose gt lt or Example 913b xvii A1 Consider an ideal incompressible fluid choose gt lt or Mass that passes 1quot in one second mass that passes 2quot in one second a blt cgt P1 P2 a b lt c gt Example 913c Consider an ideal incompressible fluid choose gt lt or V1 V2 a blt cgt Example 913d A1 Consider an ideal incompressible fluid choose gt lt or P1 P2 a blt cgt Lecture 8 Last Lecture Work for noncons ran r force Spring force Power Chapter 6 Momentum and Collisions Momentum De nition Newton39s 2quotd Law IE m Conservation of Momentum True for isolafed particles no external forces Proof Recall F12F21 Newton s 3rd Law gt E2 210 gt0 Al Al 213 for isolated particles never changes Momentum is a Vector quantity Bo rh 2px and Zpy are conserved px mvx py mvy Example 61 An astronaut of mass 80 kg pushes away from a space station by throwing a 075 kg wrench which moves with a velocity of 24 ms relative to the original frame of the astronaut What is the astronaut39s recoil speed 0225 ms Center of mass does not accelerate mle1 quot12sz m3Ax3 cm quot11 m2 m3 Atm1Ax1At quot12sz At m3Ax3 At quot11 m2 m3 m1m2 m3 0 if total P is zero Example 62 Ted and his iceboaT combined mass 240 kg resT on The frictionless surface of a frozen lake A heavy rope mass of 80 kg and lengTh of 100 m is laid ouT in a line along The Top of The lake IniTially Ted and The rope are aT resT AT Time 0 Ted Turns on a wench which winds 05 m of rope onTo The boaT every second a WhaT is Ted39s velociTy jusT afTer The wench Turns on 012 ms b WhaT IS The velocITy of The rope aT The same TI 7e c WhaT is The Ted39s speed jusT as The rope finishes m S d How far did The cenTerofmass of TedboaTrope moae e How far did Ted move 1 f How far did The cenTerofmass of The rope movgg 2 Example 63 A 1967 Corvette of mass 1450 kg moving with a velocity of 100 mph 447 ms slides on a slick street and collides with a Hummer of mass 3250 kg which is parked on the side of the street The two vehicles interlock and slide off together What is the speed of the two vehicles immediately 133 ms 3o9 mph after they join Impulse Useful For sudden changes where the exact details of the Force are dif cult to determine I aaaaaaaaaaaaaaaaaaaaa an For nonconstant F Impulse Area under F vs t curve Bungee Jumper Demo Example 64 a 1305 kgms A pitcher throws a 0145kg baseball so that it crosses home plate horizontally with a speed of 40 ms It is hit straight back at the pitcher with a final speed of 50 ms a What is the impulse delivered to the ball b Find the average force exerted by the bat on the ball if the two are in contact for 20 x 10393 s c What is the acceleration experienced by the ball b 6525 N c 45000 ms2 Collisions Momentum is always conserved in a collision Classification of collisions ELASTIC Both energy amp momentum are conserved INELASTIC Momentum conserved not energy Perfectly inelastic gt objects stick Lost energy goes to heat Examples of Perfectly Inelastic Collisions o Catching a baseball 0 Football tackle 0 Cars colliding and sticking o Bat eating an insect Examples of Perfectly Elastic Collisions o Superball bouncing 0 Electron scattering Ball Bounce Demo Example 65a A superball bounces off the floor A The net momentum of the earthsuperball is conserved B The net energy of the earthsuperball is conserved C Both the net energy and the net momentum are conserve D Neither are conserved Example 65b A astronaut floating in space catches a baseball A Momentum of the astronautbaseball is conserved B Mechanical energy of the astronautbasebal is conserve C Both mechanical energy and momentum are conserved D Neither are conserved Example 65c A proton scatters off another proton No new particles are created A Net momentum of two protons is conserved B Net kinetic energy of two protons is conserved C Both kinetic energy and momentum are conserved D Neither are conserved Perfectly Inelastic collision in 1dimension 0 Final velocities are the same Example 66 A 5879lb 2665 kg Cadillac Escalade going 35 mph smashes into a 2342lb 1061 kg Honda Civic also moving at 35 mph1564 ms in the opposite directionThe car39s collide and stick a What is the final velocity of the two vehicles b What are the equivalent brickwallquot speeds for each vehicle a 673 ms 151 mph b 199 mph for Cadillac 501 mph for Civic Example 67 A proton mp167x1027 kg elastically collides with a target proton which then moves straight forward If the initial velocity of the projectile proton is 30x106 ms and the target proton bounces forward what are a the final velocity of the projectile proton b the final velocity of the target proton Elastic collision in 1dimension 1 Conservation of Energy 2 Conservation of Momentum 0 Rearrange both equations and divide Elastic collision in 1dimension Final equations for headon elastic collision 0 Relative velocity changes sign 0 Equivalent to Conservation of Energy Example 68 An proton mp167x1027 kg elastically collides with a target deuteron mD2mp which then moves straight forward If the initial velocity of the projectile proton is 30x106 ms and the target deuteron bounces forward what are a the final velocity of the projectile proton b the final velocity of the target deuteron vp 1Ox106 ms vd 20x106 ms Headon collisions with heavier objects always lead to reflections vo k Example 69a M The mass M1 enters from the left with velocity v0 and strikes the mass M2M1 which is initially at rest The collision is perfectly elastic 0 Just after the collision v2 v0 Agt Blt c vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2M1 which is initially at rest The collision is perfectly elastic Just after the collision v1 0 Agt Blt c vo k Example 69c M The mass M1 enters from the left with velocity v0 and strikes the mass M2M1 which is initially at rest The collision is perfectly elastic Just after the collision P2 Mlvo 3 llA vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2M1 which is initially at rest The collision is perfectly elastic At maximum compression the energy stored in the spring is 12M1v02 Agt Blt c vo k Example 69e M The mass M1 enters from the left with velocity v0 and strikes the mass M2lt M1 which is initially at rest The collision is perfectly elastic Just after the collision v2 v0 Agt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2lt M1 which is initially at rest The collision is perfectly elastic Just after the collision v1 0 Agt Blt c vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2lt M1 which is initially at rest The collision is perfectly elastic Just after the collision P2 Mlvo gt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2lt M1 which is initially at rest The collision is perfectly elastic At maximum compression the energy stored in the spring is 12M1v02 A gt B lt c vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2gtM1 which is initially at rest The collision is perfectly elastic Just after the collision v2 v0 gt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2gtM1 which is initially at rest The collision is perfectly elastic Just after the collision v1 0 gt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2gtM1 which is initially at rest The collision is perfectly elastic Just after the collision P2 Mlvo gt Blt Chapter 2 OneDimensional Motion Motion at fixed velocity Definition of average velocity Motion with fixed acceleration Graphical representations Displacement vs position Position x relative to origin Displacement Ax xfxi 2002 Brooks Cole Publishing a division ofThomson Learning Average velocity quot i7W110i Average velocity Can be positive or negative Depends only on initialfinal positions eg if you return to original position average velocity is zero Instantaneous velocity Let time interval approach zero Defined for every instance in time Equals average velocity if v constant 5PEED is absolute value of velocity Graphical Representation of Average Velocity Position xm 100 2002 Brooks Cole Publishing a division ofThomson Learning gtQAx40m 80 6039 A At 30 s 40 20 Time 155 0 10 20 30 40 50 Between and T is slope of blue line Graphical Representation of Instantaneous Velocity Position xln 2002 Brooks Cole Publishing a division otThomson Learning 100 80 60 40 20 Time ts O 10 20 30 40 50 vt30 is slope of tangent green line Example 21 Carol starts at a position xt0 15 m At t20 s Carol39s position is xt2 s45 m At t40 s Carol39s position is xt4 s25 m a What is Carol39s average velocity between tO and t2 s b What is Carol39s average velocity between t2 and t4 s c What is Carol39s average velocity between tO and t4 s a 15 ms b 35 ms c 10 ms Example 22 On a mission to rid Spartan Stadium of vermin an archer shoots an arrow across the stadium at an unlucky rat 200 meters away The archer hears the squeal 22 seconds later What was the velocity of the arrow The speed of sound is 330 ms Visualize the problem Example 22 On a mission to rid Spartan Stadium of vermin an archer shoots an arrow across the stadium at an unlucky rat 200 meters away The archer hears the squeal 22 seconds later What was the velocity of the arrow The speed of sound is 330 ms V 125 ms X m Example 23a a C US d The instantaneous velocity A a is zero at Bbampd C c amp e Example 23b The instantaneous velocity is A a negative at B b C c D d Ee Example 23c The average velocity is zero in V the Interval B bd C cd D ce E de SPEED Speed is M and is always positive Average speed is sum over IAxI elements divided by elapsed time Example 23d The average velocity is A 43 negative in the intervas 3 0C C ce D de Example 24 x a What is the average 8 velocity between B and E 6 b What is the average speed between B and E 2 0 2 4 6 8 10 12 t s a 02 ms b 12 ms Acceleration The rate of change of the velocity Average acceleration measured over finite time interval Instantaneous acceleration measured over infinitesimal interval At gt O Accelerometer Demo Graphical Description of Acceleration Acceleration is slope of tangent line in v vs t graph Graphical Description of Acceleration a is positivenegative when v vs t is 15 Example 20 I l I I l I I I I II E 10 gtlt Z O al I I I I I I I I I I I 39 O 5 IO 15 t 0 At which points does the position equal zero A a only B a and d C b only D b amp cl Example 25c 20 l I I I I I I I I I I 1 2 e I E 10 gtlt Z O I l I I I I I I I I I I O 5 IO 15 t t A a At which point is the velocity negative 3 b C c D d Ee risingfalling or when x vs t curves upwardsdownwards Example 20 I I I I I I I I I l I e I 3 1o gtlt Z O al I I I I I I I I I I I I O 5 10 15 t 0 At which points does the velocity equal zero A a B b only C c only D b amp d E a amp d Example 20 E 10 x Z Z O I I I I I I I I I I I I O 5 IO 15 t t A ac At which segments is the 3 cd acceleration negative C ce D de 23 24 Example 25e 20 I I I I I I I I I I I a O G 5 1 t t At which points does the gone of The below 39 9 acceleration equal zero C C D d E e a vs t is a constant v vs t is a straight line x vs t is a parabola Constant Acceleration Eqs of Motion 25 26 Solving Problems with Eqs of Motion 5 variables Ax t v0 vf a 2 equations 3 variables must be given so that 2 equations can solve for 2 unknowns Example 26 Crash Houlihan speeds down the intersate when she slams on the brakes and slides into a concrete barrier The police measure skid marks to be 60 m long and from a tape recording know that she was breaking from 35 seconds Furthermore they know that her Mercedes would decelerate at 55 msz while skidding What was Crash39s speed when she hit the barrier 7 52 ms 28 Other Forms of Eqs of Motion Substitute to eliminate vf V V at sz O 0 t Other Forms of Eqs of Motion Substitute to eliminate v0 Ax vf atvft 29 30 Other Forms of Eqs of Motion Substitute to eliminate t v v v v Ax 0 f f 0 Final List of 1d Equations Which one should I use Each Eq has 4 of the 5 variables Ax t v0 v amp a Ask yourself Which variable am I not given and not interested inquot If that variable is t use Eq 5 Example 26 Revisited Crash Houlihan speeds down the intersate when she slams on the brakes and slides into a concrete barrier The police measure skid marks to be 60 m long and from a tape recording know that she was breaking from 35 seconds Furthermore they know that her Mercedes would decelerate at 55 ms2 while skidding What was Crash39s speed when she hit the barrier 7 52 ms 32 Exam le 27a A drag racer starts her p car from rest and accelerates at 100 ms2 for the entire distance of a 400 m 14 mi race How much time was required to finish the race Example 27b A drag racer starts her car from rest and accelerates at 100 ms2 for the entire distance of a 400 m 14 mi race What was her final speed Example 27c A drag racer starts her car from rest and finishes a race in 35 seconds with a constant acceleration for the entire distance of a 400 m 14 mi race What was her final speed V l Free Fall Galileo Objects under the influence of gravity no resismnce faquot wifh consul downward Father was a musician experimented with music acceleration if near Earth39s surface Initiay was a professor Teaching premeds g 981 ms2 Developed telescope 1610 Milky Way stars quot Use the usual equations with a gt g MOONS Of JUPi l39er Phases of Venus Measured g Quantified mechanics In 1632 published Dialogue concerning the two greatest quot world systems AA nmul nullIn AF L Example 28a 33932 Example 28b A man throws a brick upward from the top of a 50 m building The brick has an initial 35 upward velocity of 20 ms a How high above the building does the brick get before it falls A man drops a brick off the top of a 50m building The brick has zero initial velocity s u s v u t 3002 a How much time is required for fhe brick 10 h fhe ground b How much time does the brick spend going upwards c What is the velocity of the brick when it passes the man going downwards d What is the velocity of the brick when JOU 5IP b What is the velocity of the brIck when Ibhgtslghg ground uguizaj uosmouuo uogsgngp 2 Eugqsglqnd aloa sxomg zouz b 313 ms it hits the ground E 25805 e At what time does the brick hit the fi Example 28b Example 29a A man throws a brick upward from the top A man throws a brick upward from the of a 50 m building The brick has an initial top of a building Assume the coordinate 5 upward velocity of 20 ms system is defined with positve defined as i39Ef quot39 a How high above the building does the Upward briCk gef before I quot5 At 39A39 the acceleration is positive I r J1 iw v b How much time does the brick spend 3 m a True 39 d s b False 9 goIng upwar s c 20 ms 1 c What is the velocity of the brick when d 372 ms it passes the man going downwards e 533 s a Ii d What is the velocity of the brick when it hits the ground e At what time does the brick hit the Eugumaj uoswauua noisyup z Eugusuqnd awn 5110019 0 i i Example 29b Example 29c A man Throws a brick upward from The Top of a building Assume The coordinaTe A man Throws a brick upward from The Top of a building Assume The coordinaTe 10v1 4 sysTem is defined wiTh posiTve defined as 2H sysTem is defined wiTh posiTve defined as 44 upward S upward 3 AT 339 The velociTy is zero AT 339 The acceleraTion is zero a True a True 7 V b False b False r W quot 333323 43 44 Example 29d Example 29e A man Throws a brick upward from The if m Top of a building Assume The coordinaTe quotFi W sysTem is defined wiTh posiTve defined as mew EMAquot A man Throws a brick upward from The Top of a building Assume The coordinaTe i am sysTem is defined wiTh posiTve defined as g 91 N upward E upward E AT 639 The velociTy is negaTive AT 639 The acceleraTion is negaTive a True g a True g b False b False m 45 m 46 Example 29f Example 299 A man Throws a brick upward from The USN Top of a building TRUE OR FALSE jEZ JMmA Assume The coordinaTe sysTem is defined wiTh posiTve defined as upward A man Throws a brick upward from The Top of a building Assume The coordinaTe sysTem is defined wiTh posiTve defined as 2 quot1 quot upward E 39 C I 03 A5 n u 7200 mV ZOO The speed aT C39 and aT A39 are equal a True b False Th v ociTy aT C39 and aT A39 are equal a rue b False JO noisinip JO noisinip Chapter 2 OneDimensional Motion Motion at fixed velocity Definition of average velocity Motion with fixed acceleration Graphical representations Displacement vs position Position x relative to origin Displacement Ax xfxi 2002 Brooks Cole Publishing a division ofThomson Learning Average velocity quot i7W110i Average velocity Can be positive or negative Depends only on initialfinal positions eg if you return to original position average velocity is zero Instantaneous velocity Let time interval approach zero Defined for every instance in time Equals average velocity if v constant 5PEED is absolute value of velocity Graphical Representation of Average Velocity Position xm 100 2002 Brooks Cole Publishing a division ofThomson Learning gtQAx40m 80 6039 A At 30 s 40 20 Time 155 0 10 20 30 40 50 Between and T is slope of blue line Graphical Representation of Instantaneous Velocity Position xln 2002 Brooks Cole Publishing a division otThomson Learning 100 80 60 40 20 Time ts O 10 20 30 40 50 vt30 is slope of tangent green line Example 21 Carol starts at a position xt0 15 m At t20 s Carol39s position is xt2 s45 m At t40 s Carol39s position is xt4 s25 m a What is Carol39s average velocity between tO and t2 s b What is Carol39s average velocity between t2 and t4 s c What is Carol39s average velocity between tO and t4 s a 15 ms b 35 ms c 10 ms Example 22 On a mission to rid Spartan Stadium of vermin an archer shoots an arrow across the stadium at an unlucky rat 200 meters away The archer hears the squeal 22 seconds later What was the velocity of the arrow The speed of sound is 330 ms Visualize the problem Example 22 On a mission to rid Spartan Stadium of vermin an archer shoots an arrow across the stadium at an unlucky rat 200 meters away The archer hears the squeal 22 seconds later What was the velocity of the arrow The speed of sound is 330 ms V 125 ms X m Example 23a a C US d The instantaneous velocity A a is zero at Bbampd C c amp e Example 23b The instantaneous velocity is A a negative at B b C c D d Ee Example 23c The average velocity is zero in V the Interval B bd C cd D ce E de SPEED Speed is M and is always positive Average speed is sum over IAxI elements divided by elapsed time Example 23d The average velocity is A 43 negative in the intervas 3 0C C ce D de Example 24 x a What is the average 8 velocity between B and E 6 b What is the average speed between B and E 2 0 2 4 6 8 10 12 t s a 02 ms b 12 ms Acceleration The rate of change of the velocity Average acceleration measured over finite time interval Instantaneous acceleration measured over infinitesimal interval At gt O Accelerometer Demo Graphical Description of Acceleration Acceleration is slope of tangent line in v vs t graph Graphical Description of Acceleration a is positivenegative when v vs t is 15 Example 20 I l I I l I I I I II E 10 gtlt Z O al I I I I I I I I I I I 39 O 5 IO 15 t 0 At which points does the position equal zero A a only B a and d C b only D b amp cl Example 25c 20 l I I I I I I I I I I 1 2 e I E 10 gtlt Z O I l I I I I I I I I I I O 5 IO 15 t t A a At which point is the velocity negative 3 b C c D d Ee risingfalling or when x vs t curves upwardsdownwards Example 20 I I I I I I I I I l I e I 3 1o gtlt Z O al I I I I I I I I I I I I O 5 10 15 t 0 At which points does the velocity equal zero A a B b only C c only D b amp d E a amp d Example 20 E 10 x Z Z O I I I I I I I I I I I I O 5 IO 15 t t A ac At which segments is the 3 cd acceleration negative C ce D de 23 24 Example 25e 20 I I I I I I I I I I I a O G 5 1 t t At which points does the gone of The below 39 9 acceleration equal zero C C D d E e a vs t is a constant v vs t is a straight line x vs t is a parabola Constant Acceleration Eqs of Motion 25 26 Solving Problems with Eqs of Motion 5 variables Ax t v0 vf a 2 equations 3 variables must be given so that 2 equations can solve for 2 unknowns Example 26 Crash Houlihan speeds down the intersate when she slams on the brakes and slides into a concrete barrier The police measure skid marks to be 60 m long and from a tape recording know that she was breaking from 35 seconds Furthermore they know that her Mercedes would decelerate at 55 msz while skidding What was Crash39s speed when she hit the barrier 7 52 ms 28 Other Forms of Eqs of Motion Substitute to eliminate vf V V at sz O 0 t Other Forms of Eqs of Motion Substitute to eliminate v0 Ax vf atvft 29 30 Other Forms of Eqs of Motion Substitute to eliminate t v v v v Ax 0 f f 0 Final List of 1d Equations Which one should I use Each Eq has 4 of the 5 variables Ax t v0 v amp a Ask yourself Which variable am I not given and not interested inquot If that variable is t use Eq 5 Example 26 Revisited Crash Houlihan speeds down the intersate when she slams on the brakes and slides into a concrete barrier The police measure skid marks to be 60 m long and from a tape recording know that she was breaking from 35 seconds Furthermore they know that her Mercedes would decelerate at 55 ms2 while skidding What was Crash39s speed when she hit the barrier 7 52 ms 32 Exam le 27a A drag racer starts her p car from rest and accelerates at 100 ms2 for the entire distance of a 400 m 14 mi race How much time was required to finish the race Example 27b A drag racer starts her car from rest and accelerates at 100 ms2 for the entire distance of a 400 m 14 mi race What was her final speed Example 27c A drag racer starts her car from rest and finishes a race in 35 seconds with a constant acceleration for the entire distance of a 400 m 14 mi race What was her final speed V l Free Fall Galileo Objects under the influence of gravity no resismnce faquot wifh consul downward Father was a musician experimented with music acceleration if near Earth39s surface Initiay was a professor Teaching premeds g 981 ms2 Developed telescope 1610 Milky Way stars quot Use the usual equations with a gt g MOONS Of JUPi l39er Phases of Venus Measured g Quantified mechanics In 1632 published Dialogue concerning the two greatest quot world systems AA nmul nullIn AF L Example 28a 33932 Example 28b A man throws a brick upward from the top of a 50 m building The brick has an initial 35 upward velocity of 20 ms a How high above the building does the brick get before it falls A man drops a brick off the top of a 50m building The brick has zero initial velocity s u s v u t 3002 a How much time is required for fhe brick 10 h fhe ground b How much time does the brick spend going upwards c What is the velocity of the brick when it passes the man going downwards d What is the velocity of the brick when JOU 5IP b What is the velocity of the brIck when Ibhgtslghg ground uguizaj uosmouuo uogsgngp 2 Eugqsglqnd aloa sxomg zouz b 313 ms it hits the ground E 25805 e At what time does the brick hit the fi Example 28b Example 29a A man throws a brick upward from the top A man throws a brick upward from the of a 50 m building The brick has an initial top of a building Assume the coordinate 5 upward velocity of 20 ms system is defined with positve defined as i39Ef quot39 a How high above the building does the Upward briCk gef before I quot5 At 39A39 the acceleration is positive I r J1 iw v b How much time does the brick spend 3 m a True 39 d s b False 9 goIng upwar s c 20 ms 1 c What is the velocity of the brick when d 372 ms it passes the man going downwards e 533 s a Ii d What is the velocity of the brick when it hits the ground e At what time does the brick hit the Eugumaj uoswauua noisyup z Eugusuqnd awn 5110019 0 i i Example 29b Example 29c A man Throws a brick upward from The Top of a building Assume The coordinaTe A man Throws a brick upward from The Top of a building Assume The coordinaTe 10v1 4 sysTem is defined wiTh posiTve defined as 2H sysTem is defined wiTh posiTve defined as 44 upward S upward 3 AT 339 The velociTy is zero AT 339 The acceleraTion is zero a True a True 7 V b False b False r W quot 333323 43 44 Example 29d Example 29e A man Throws a brick upward from The if m Top of a building Assume The coordinaTe quotFi W sysTem is defined wiTh posiTve defined as mew EMAquot A man Throws a brick upward from The Top of a building Assume The coordinaTe i am sysTem is defined wiTh posiTve defined as g 91 N upward E upward E AT 639 The velociTy is negaTive AT 639 The acceleraTion is negaTive a True g a True g b False b False m 45 m 46 Example 29f Example 299 A man Throws a brick upward from The USN Top of a building TRUE OR FALSE jEZ JMmA Assume The coordinaTe sysTem is defined wiTh posiTve defined as upward A man Throws a brick upward from The Top of a building Assume The coordinaTe sysTem is defined wiTh posiTve defined as 2 quot1 quot upward E 39 C I 03 A5 n u 7200 mV ZOO The speed aT C39 and aT A39 are equal a True b False Th v ociTy aT C39 and aT A39 are equal a rue b False JO noisinip JO noisinip Chapter 11 Energy in Thermal Processes Vocabulary 3 Kinds of Energy 0 Internal Energy U Energy of microscopic motion and inter molucular forces 0 Work W FAx PAV is work done by compression next chapter 0 Heat Q Energy transfer from microscopic contact next chapter Temperature and Specific Heat m Speci c Heats of Some Materials at Atmospheric Pressure 0 Add energy gt T rises Substance Jkg C calg C Aluminum Belyllium Cadmium Copper Mass Germanium Glass Property of material Gold Ice Iron Lead Mercury Silicon Silve r 1 calorie 4186 J 900 0215 1820 0436 230 0055 387 00924 322 0077 837 0200 129 00308 2090 0500 448 0107 g 128 00305 138 0033 3 703 0168 2 234 0056 in t l g 57 Water 4 186 100 g Example 111 Bobby J oe drinks a 130 calorie can of soda If the efficiency for turning energy into work is 20 how many 4 meter floors must Bobby J oe ascend in order to work off the soda and maintain her 55 kg mass Nfloors 39 4 Example 112 Aluminum has a specific heat of 0924 calg C If 110 g of hot water at 90 C is added to an aluminum cup of mass 50 g which is originally at a temperature of 23 C what is the final temperature of the equilibrated watercup combo T 873 C Phase Changes and Latent Heat 0 T does not rise when phases change at constant P 0 Examples solid gt liquid fusion liquid gt vapor vaporization 0 eat energy required to change phases Property of substance transition TABLE 39I 12 Latent Heals of Fusion and Vaporization Latent Heal Latent Heat of Melting of Fusion Boiling Vaporization Substance Point C Jkg calg Point C J kg calg Helium 26065 5221 X 1095 125 26893 209 x 10quot 199 Nitrogen 997 255 X 10quot 609 l958l 201 X 105 480 Oxygen 21879 138 x 10 l 330 18207 213 x 10 509 a I 114 101 x10quot M t 78 85lX105 v 000 gtlt 10 quot 10000 220 x 10 i 110 5181 X 10 I 44400 320 x 10quot a 3273 245 x 10 585 1 750 870 x 10quot 208 Aluminum 000 307 x 10quot 948 2 450 114 x 107 2 720 an Simr 90080 882 x 104 211 2 191 233 x 101 558 5 Gold 1 00300 044 x 1039 154 2 000 158 x 10 377 E Copper 1 083 134 X 10 quot 320 1 187 506 x 10 I 210 g Example 113 10 li139ers of wafer is hea139ed from 12 C 1390 100 C 139hen boiled away a How much energy is required 1390 bring 139he wa139er 1390 boiling b How much ex139ra energy is required 1390 vaporize 139he wa139er c If elec139rici1y cos139s 75 per MWhr wha139 was 139he cos139 of hea139ing and boiling 139he wa139er a Q 88x104 cal 368x105 J b Q 54x105 cal 226x106 J c 55 Example 114 Consider Bobby Joe from 139he previous example If 139he 80 of 139he 130 kcals from her soda wen139 in139o hea139 which was 139aken from her body from radia139ion how much wafer was perspired 1390 maintain her normal body 139empera1ure Assume a la139en139 hea139 of vaporiza139ion of 540 calg even 139hough T 37 C 193g A can of soda has 325 g of H20 Some fluid drips away Three Kinds of Heat Transfer Conduc139ion 0 Shake your neighbor pass if down 0 Examples Hea139ing a skillet losing hea139 139hrough 139he walls Convec139ion 0 Move ho139 region 1390 a differen139 loca139ion 0 Examples Ho139wa139er hea139ing for buildings Circula139ing air Uns139able a139mospheres 0 Radiation 0 Ligh139 is emitted from ho139 objec139 0 Examples 139ars Incandescen139 bulbs Conduction 0 Power depends on area A 139hickness Ax 139empera139ure difference AT and conduc139ivi1y of ma139erial 2003 Thomson Bro ksICDIe Conduc139ivi1y is properfy Energy ow of ma139erial fOY Thgt Tc 1 Example 115 A copper p0139 of radius 12 cm and 139hickness 5 mm si139s on a burner and boils wafer The 139empera139ure of 139he burner is 115 C while 139he 139empera139ure of 139he inside of 139he p0139 is 100 C Wha139 mass of wafer is boiled away every minu139e DATA kcu 397 Wm C m143 kg Conductivities and Rvalues 0 Conduc139ivi1y k 0 Properfy of Ma139erial 0 SI uni139s are Wm C RValue Properfy of material and 139hickness Ax Measures resis139ance 1390 hea139 Useful for comparing insula139ion produc139s Quo139ed values are in AWFUL uni139s Thermal Conductivities Conducitivities Thermal and Rvalues Conductivity Substance Js In C Metals at 25 C 2 TABLE 39I 14 RVaIues for Common Aluminum 238 Building Materials Copper 397 5 Gold 314 5 C Iron 795 g Lend 547 7 Mammal i r m 51039 41 g Hardwood Siding 10 in thick Gases at 20ac E Wood shingles lapped 087 A L023 4 83 Brick 40 in thick 400 Helium 0138 a Concrete block lled cores 193 llydmgcn 0172 Styrofoam 10 in thick 50 Nin ngcn 0023 4 Fiber glass batting 35 in thick 1090 Oxygen 0023 8 Fiber glass batting 60 in thick 1880 Nonmemls Fiber glass board 10111t111Ck 435 Asbmms 021 Cellulose ber 10 in thick 370 lunamc L3 Flat glass 0125 in thick 089 Glass L84 Insulating glass 025in space 154 CE 16 Vertical air space 35 in thick 101 Rubber 02 Air lm 017 luci L60 Diy wall 050 in thick 045 Wood 010 Sheathing 050 in thick What makes a good heat conductor Free electrons metals Easy transport of sound lattice vibrations tiff is good Low Density is good Pure crystal structure Diamond is perfect Example 116a An large pipe carries steam at 224 C across a large industrial plant The outside of the pipe is at room temperature 24 C The pipe is 120 m long and has a diameter of 70 cm The pipe is constructed of an insulating material of conductivity k 262 Wm C In order to reduce the rate of heat loss through the pipe by a factor of 12 an engineer could a Reduce the length of the pipe by a factor of 12 b Reduce the diameter of the pipe by a factor of 12 c Increase the thickness of the pipe by a factor of 2 d All of the above e None of the above Example 116b An large pipe carries steam at 224 C across a large industrial plant The outside of the pipe is at room temperature 24 C The pipe is 120 m long and has a diameter of 70 cm The pipe is constructed of an insulating material of conductivity k 262 Wm C In order to reduce the rate of heat loss through the pipe by a factor of 12 an engineer could a Make the pipe using a new material with twice the conductivity 5 24 Wm C b Redesign the pipe to double the Rvalue c All of the above d None of the above Example 116c An large pipe carries steam at 224 C across a large industrial plant The outside of the pipe is at room temperature 24 C The pipe is 120 m long and has a diameter of 70 cm The pipe is constructed of an insulating material of conductivity k 262 Wm C In order to reduce the rate of heat loss through the pipe by a factor of 12 an engineer could a Reduce the density of steam in the pipe by a factor of 12 b Reduce the temperature of the steam to 124 C c Reduce the velocity of the steam through the pipe by a factor of 12 d All of the above e None of the above Consider a la Rvalues for layers ered system e lassair lass AT AT1 HP39 vv AT1 AT AT3 AT4 AT2 AT3 PR2PR3 A A A P all ri ri ZR1R2 R3 Example 117 Consider Three panes of glass each of Thickness 5 mm The panes Trap Two 25 cm layers of air in a large glass door How much power leaks Through a 20 rn2 glass door if The TemperaTure ouTside is 40 C and The TemperaTure inside is 20 C DATA kg 084 Wm C km 00234 Wrn C lass P 558W Convection 0 If warm air blows across The room iT is convecTion 0 If There is no wind iT is conducTion 0 Can be insTigaTed by Turbulence or insTabiliTies Why are windows triple paned To sTop convecTion Transfer of heat by radiation 0 All objecTs emiT lighT if T gt 0 0 Colder objecTs emiT longer wavelengThs red or infrared 0 HoTTer objecTs emiT shorTer wavelengThs blue or ulTravioleT 0 Tefan39s Law give power of emiTTed radiaTion EmissiviTy o 56696x10398 Wm2 K4 0ltelt1 usually near 1 is The TefanBoszmann consTanT Example 118 If The TemperaTure of The Sun fell 5 and The radius shrank 10 whaT would be The percenTage change of The Sun39s power oquuT 34 Example 119 DATA The sun radiaTes 374x1026 W DisTance from Sun To EarTh 15x1011 m Radius of EarTh 636x106 m a WhaT is The inTensiTy powerm2 of sunlighT when iT reaches EarTh a 1323 wm2 b How much power is absorbed by EarTh in sunlighT assume ThaT none of The sunlighT is reflected b 1 68x1017 W c WhaT average TemperaTure would allow EarTh To radiaTe an amounT of power equal To The amounT of sun power absorbed cT276K3 C37 F What is neglected in estimate Earth is not at one single temperature Some of Sun39s energy is reflected Emissivity lower at Earth39s thermal wavelengths than at Sun39s wavelengths Radioactive decays inside Earth Hot underground less so in Canada Most of Jupiter39s radiation Example 1110a Two Asteroids A and B orbit the Sun at the same radius R Asteroid B has twice the surface area of A Assume both asteroids absorb 100 of the sunlight and have emissivities of 10 The average temperature of B T a 14TA b 12TA 5 TA d ZTA e 4TA Example 1110b Two identical asteroids A and B orbit the sun Asteroid B is located twice as far from sun as Asteroid A RB2RA Assume both asteroids absorb 100 of the sunlight and have emissivities of 10 The average temperature of B TB a 14TA b 12T c 2 12TA d 2 14TA 3 TA Example 1110c Two Asteroids A and B orbit the Sun at the same radius R Asteroid B is painted with reflective paint which reflects 34 of the sunlight while asteroid A absorbs 100 of the sunlight Both asteroids have emissivities of 10 The average temperature of B T a 14TA b 12T c 2 12TA d 2 14TA 2 24013 Example 1110d Two Asteroids A and B orbit the Sun at the same radius R Asteroid B has an emissivity of 025 while the emissivity of asteroid A is 10 Both asteroids absorb 100 of the sunlight The average temperature of B T a 4TA b ZTA c 21 ZTA d 214TA e 234TA Greenhouse Gases Sun is much hotter than Earth so sunlight has much shorter wavelengths than light radiated by Earth infrared Emissivity of Earth depends on wavelength CO2 in Earth39s atmosphere reflects in the infrared Barely affects incoming sunlight Reduces emissivity e of reradiated heat 002 Calcm39m39a rm in ppmv sume Enquelecmmxssmn mu Angular Displacement Chapter 7 Circular motion about AXIS Three measures of angles 1 Degrees 0 Reference 2 Revolutions 1 rev 360 deg 1 6 Rotational Motion 3 Radians 2n rads 360 deg 21 Universal Law of Gravitation Kepler39s Laws w R 5 19 0 Reference line 2003 Thomson BrooksCole An ular Dis lacement cont 9 p Example 71 Change in An automobile wheel has a radius of 42 cm If a distance of a poim car drives 10 km through what angle has the J39 Reference wheel rotated S 2751 N N counts revolutions line 2 1 9 0 is in radians a In reVOIUTions a N a k b Iquot radians b e 238x104 radians C Iquot degrees c G 136x106 degrees Reference line 2003 Thomson BrooksCole Angular Speed can be given in Example 72 Revolutionss y o A race car engine can turn at a maximum rate of 12000 Rad39 quot5s quotgt called 00 rpm revolutions per minute 1 x a What is the angular velocity in radians per second 7 0f 39Linea39quot Speed 139 39quot b If helipcopter blades were attached to the 6i V 27 N revolutions crankshaft while it turns with this angular velocity what 0 x t is the maximum radius of a blade such that the speed of My 6f 6i in rads the blade tips stays below the speed of sound DATA The speed of sound IS 343 msa 1256 rads b 27 cm Angular Acceleration 0 03 must be in radians per sec Denoted by on Units are rads2 Every point on rigid object has same mand 0c RotationalLinear Equivalence AeeAx WOHVO X90 191 Linear and Rotational Motion Analogies Rotational Motion Linear Motion to a7 v v A97 0 t AFL 2 wfw0oct vv0at 1 2 1 2 A9w0t706t Ax V0tilll 2 2 A9 tuft loth Ax VII latz 2 2 2 2 2 2 a7 a V i7oo 9 ivioan 2 2 2 2 Example 73 A pottery wheel is accelerated uniformly from rest to a rate of 10 rpm in 30 seconds a What was the angular acceleration in rads2 b How many revolutions did the wheel undergo during that time a 00349 radsz b 250 revolutions Linear movement of a rotating point Distance x mg Different points have 0 d39 t 39 d Speed V rm Ifferen Inear spee s Acceleration 1 r06 Only works for angles in radians Example 74 A coin of radius 15 cm is initially rolling with a rotational speed of 30 radians per second and comes to a rest after experiencing a slowing down of on 005 rads2 a Over what angle in radians did the coin rotate b What linear distance did the coin move a 90 rad b 135 cm Example 78 A race car speeds around a circular track a If the coefficient of friction with the tires is 11 what is the maximum centripetal acceleration in g s that the race car can experience b What is the minimum circumference of the track that would permit the race car to travel at 300 km hr a 11 gquots b 404 km in real life curves are banked Example 79 A curve with a radius of curvature of 05 km on a highway is banked at an angle of 20 If the highway were frictionless at what speed could a car drive without sliding off the road 423 ms 945 mph Skip Example 710 A yo yo is spun in a circle as shown If the length of the 3 string is L 35 cm and the i circular path is repeated 15 1 times per second at what 1 angle 0 with respect to the 1 vertical does the string bend 6 716 degrees quot axonsnows v uosmoux coo Example 711a Jout of page I Iinto page Which vector represents acceleration a A bE cF dB eJ Example 711b Jout of page J I into page Which vector represents net force acting on car a A b E c F d B e J Example 711c Jout of page I Iinto page If car moves at IIdesignII speed which vector represents the force acting on car from contact with road a D bE cG dI eJ Example Jout of page A Iinto page If car moves slower than quotdesignquot 39speed which vector represents frictional force acting on car from contact with road neglect air resistance a B bC cE dF eI Example 712 skip A roller coaster goes upside down performing a circular loop of radius 15 m What speed does the roller coaster need at the top of the loop so that it does not need to be held onto the track 121 ms Accelerating Reference Frames Fma F ma af Fictitious force Looks like quotgravitationalquot force Example 713 Which of these astronauts experiences quotweightlessnessquot BOB who is stationary and located billions of light years from any star or planet TED who is falling freely in a broken elevator CAROL who is orbiting Earth in a low orbit ALICE who is far from any significant stellar object in a rapidly rotating space station A BOB amp TED B TED C BOB TED amp CAROL D BOB CAROL amp ALICE E BOB TED CAROL amp ALICE Newton s Law of Universal Gravitation Always attractive Proportional to both masses Inversely proportional to separation squared Gravitation Constant Determined experimentally Henry Cavendish 1798 Light beam mirror g Light amplify motion SOUI C6 Example 714 Given In SI units 6 667x1039 g981 and the radius of Earth is 638 x106 Find Earth39s mass Example 715 Given The mass of Jupiter is 173x1027 kg and Period of 1039s orbit is 17 days Find Radius of 1039s orbit r 185x109 m 599x1024 kg TYChO Brahe 1545391601 Uraniborg on an island near Copenhagen K39 Lost part of nose in a duel EXTREMELY ACCURATE astronomical observations nearly 10X improvement corrected for atmosphere 0 Believed in Retrograde Motion lMSS 39Firen39zg Hired Kepler to work as mathematician First to Johannes Kepler Johannes Kepler 15711630 Explain planetary motion 1571391630 Investigate the formation of pictures with a pin hole camera First to explain the principles of how a telescope works discover and describe total internal reflection explain that tides are caused by the Moon Explain the process of vision suggest that the Sun rotates about Its axns by refraction within the eye derive the birth year of Christ that is now Formulate eyeglass designed universauy accepted for nearsightedness and farsighfednesy derive logarithms purely based on mathematics He tried to use stellar parallax caused by the Earth39s orbit to measure the distance to the stars the same principle as depth perception Today this branch of research is called astrometry Explain the use of both eyes for depth perception Isaac Newton 1642 1727 Invented Calculus Formulated the universal law of gravitation Showed how Kepler39s laws could be derived from an inverse square law force Invented Wave Mechanics Numerous advances to mathematics and geometry Example 716a Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Astronaut Ted whizzes around in a circular orbit cg the same radius a Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward and is at a radius R but hasn39t hit the ground yet Which astronauts experience weightlessness A All 4 B Ted and Carol C Ted Carol and Aliced rol Alice Example 716b Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Astronaut Ted whizzes around in a circular orbit at the same radius cam39 Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward and is at a radius R but hasn39t hit the ground yet Assume each astronaut weighs w180 lbs on Earth Alice The gravitational force acting on is A w Ted B ZERO C Example 716c Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Astronaut Ted whizzes around in a circular orbit at the same radius camquot Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward and is at a radius R but hasn39t hit the ground yet Assume each astronaut weighs w180 lbs on Earth Alice The gravitational force acting 0 Alice is A w B ZERO Ted Example 716d Astronaut Bob stands atop the highest mountain of planet Earth which has radius R Astronaut Ted whizzes around in a circular orbit at the same radius C 0 Astronaut Carol whizzes around in a circular orbit a radius 3R Astronaut Alice is simply falling straight downward and is at a radius R but hasn39t hit the ground yet Assume each astronaut weighs w180 lbs on Earth at The gravitational force acting on quot 52 Carol is A w B w3 C w9 Ted D ZERO Example 716e Astronaut Bob stands atop the highest mountain of planet Earth which has radius Astronaut Ted whizzes around in a circular orbit at the same radius carol Astronaut Carol whizzes around in a circular orbit of radius 3R Astronaut Alice is simply falling straight downward and is at a radius R but hasn39t hit the ground yet Which astronauts undergo an acceleration g98 msz A Alice A B Bob and Alice C Alice and Ted D Bob Ted and Alice Ted E All four lice Kepler s Laws 1 Planets move in elliptical orbits with Sun at one of the focal points 2 Line drawn from Sun to planet sweeps out equal areas in equal times 3 The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet Kepler s First Law 0 Planets move in elliptical orbits with the Sun at one focus Any object bound to another by an inverse square law a will move in an elliptical path 0 Second focus is Sun empty Planet 2003 Thomson v BrooksCole Kepler s Second Law 0 Line drawn from Sun to planet will sweep out equal areas in equal times Area from A to B and C to D are the same A True for any central force due to angular momentum conservation next chapter Kepler s Third Law The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to te lanet For orbit around the Sun K5 297x103919 szm3 K is independent of the mass of the planet Derivation of Kepler s Third Law G 2 ma 2 msz mlt2T7r2gt2 R GM R3 272 2 F Example 717 Data Radius of Earth s orbit 10 AU Period of Jupiter39s orbit 119 years Period of Earth39s orbit 10 years Find Radius of Jupiter39s orbit 52 AU Gravitational Potential Energy PE mgh valid only near Earth39s surface 39 For arbitrary altitude h r RE h JL PILl t u 39 Zero reference level IS M 325 at rltgto quotWt 39 V s r At 7 r a 2003 Thomson BrooksCole Graphing PE vs position A o E 3 1O O7 x 20 H I O 3O LIJ D 4O 0 20 4O 60 r million meters 80 Example 718 You wish to hurl a projectile from the surface of the Earth Re 638x106 m to an altitude of 20x106 m above the surface of the Earth Ignore rotation of the Earth and air resistance a What initial velocity is required a 9736 ms b What velocity would be required in order for the projectile to reach infinitely high Ie what is the escape velocity 5 11181 ms c skip How does the escape velocity compare to the velocity required for a low earth orbit c 7906 ms Chapter 14 Sound Sound Waves ISound is longitudinal pressure oompression waves I Range of hearing 20 Hz to 20000 Hz FREQUENCY DEMO Speed of Sound Liquids and Gases B is bulk modulus p is massvolume Solids Y is Young39s modulus 331 ms is v at 0 C T is the absolute temperature Example 141 John Brown hits a steel railroad rail with a hammer Betsy Brown standing one mile down the track hears the bang through the cool 32 F air while her twin sister Boopsie is lying next to her and hears the bang through the steel by placing her ear on the track DATA Ysml20x10 Pa psw7850 kgm3 What is the time difference between the moments when Betsy and Boopsie hear the bang 454 s Intensity of Sound Waves Power 7 AE if 7 A A G Area SI units are Wm2 Intensity is proportional to square of amplitude pressure modulation Intensity Range for Human Hearing Threshold of Hearing 103912 Wm2 AP 103910 atm Threshold of Pain 10 Wm2 Decibel Scale TABLE 142 Intensity Levels in Decibels for Different Sources Sensation is logarithmic Source of Sound 3 dB Nearby jet airplane 150 Jackhammer machine gun 130 Siren rock concert 120 Subway power mower 100 Busy traf c 80 Vacuum cleaner 70 IO is ThreShOId of hearing Normal conversation o Mosquito buzzing 4O Nhisper 30 Threshold of PaIn Is Rustlng leaves 10 Therefore 1 Threshold of hearing 0 Intensity vs Intensity Level INTENSITY is PA Wmz INTENSITY LEVEL is in decibels dimensionless Sound Level Demo Example 142 A noisy machine in a factory produces a sound with a level of 80 dB How many machines can the factory house without exceeding the 100dB limit a 125 machines b 20 machines c 100 machines Spherical Waves Spherical wave from Energy propagates equally in all directions 2003 Thomson BreaksCole Example 143 skip A train sounds its horn as it approaches an intersection The horn can just be heard at a level of 50 dB by an observer 10 km away Treating the horn as a point source and neglect any absorption of sound by the air or ground a What is the average power generated by the horn a 126 W b What intensity level of the horn39s sound is observed by someone waiting at an intersection 50 m from the train b 96 dB Example 144 Bozo Bob buys a 20W train whistle and figures out that he won39t have any trouble standing 2 meters from the whistle since his stereo speakers are rated at 100 W and he has little trouble with the speakers turned all the way up What is the intensity level of the whistle 116 dB Doppler Effect A change in the frequency experienced by an observer due to motion of either the observer or the source DOPPLER DEMO 13 14 Doppler Effect Moving Observer When not moving fwa When moving f39 V VOI9SA O If observer moves away Us 0 Source S o lt7 A Observer Cole 0 15 16 Example 145 Mary is riding a roller coaster Her mother who is standing on the ground behind her yells out to her at a frequency of 1000 Hz but it sounds like 920 Hz v343 ms What is Mary39s speed 274 ms Doppler Effect Source in Motion 139 l VST l VSamp V 11 vSv f39 Vlv 18 Doppler Effect Source in Motion Approaching source Source leaving a Example 146 An Train has a brass band playing a song on a flaTcar As The Train approaches The sTaTion aT 214 ms a person on The planorm hears a TrumpeT play a noTe aT 3520 Hz DATA v 343 ms sound a WhaT is The True frequency of The TrumpeT a 3300 Hz b WhaT is The wavelengTh of The sound b 974 cm c If The TrumpeT plays The same noTe afTer passing The planorm whaT frequency would The person on The planorm hear C 3106 H2 20 Shock Waves Sonic Booms When The source velociTy approaches The speed of sound Conical wave from 2003 Thomson BrooksCole a Fig 14 1p439 Slide 15 Application speed radar Re ected wave gt Radar g un Transmitted wave moving car 23214411 Ii139il I ifrl 7630 53985 55mm 69 21 22 Application weather radar mate m amn ea Ier ervme 39 39 39 f l 1 r 739 u eon Ba quot7 1 x a y I f Cadillac quot 4 mama39th quot n Big Rapids M ma doppler effecT are measured Doppler Effect Both Obsewer and Source Moving wiTch appropriaTe signs if observer or source moves away 23 24 Example 147 At rest a car39s horn sounds the note A 440 Hz The horn is sounded while the car moves down the street A bicyclist moving in the same direction at 10 ms hears a frequency of 415 Hz DATA vsound 343 ms What is the speed of the car Assume the cyclist IS behind the car 313 ms Example 148a A train has a whistle with a frequency of a 1000 Hz as measured when the both the train and observer are stationary For a train moving in the positive x direction which observer hears the highest frequency when the train is at position x0 Observer A has velocity VAgt0 and has position XAgt0 Observer B has velocity VBgt0 and has position XBlt0 Observer C has velocity VClt0 and has position XCgt0 Observer D has velocity VDlt0 and has position XDlt0 Example 148b A train has a whistle with a frequency of a 1000 Hz as measured when the both the train and observer are stationary For a train moving in the positive x direction which observer hears the highest frequency when the train is at position x0 An observer with Vgt0 and position Xgt0 hears a frequency a gt 1000 Hz b lt 1000 Hz c Can not be determined Example 148c A train has a whistle with a frequency of a 1000 Hz as measured when the both the train and observer are stationary For a train moving in the positive x direction which observer hears the highest frequency when the train is at position x0 An observer with Vgt0 and position Xlt0 hears a frequency a gt 1000 Hz b lt 1000 Hz c Can not be determined Example 148d A train has a whistle with a frequency of a 1000 Hz as measured when the both the train and observer are stationary For a train moving in the positive x direction which observer hears the highest frequency when the train is at position x0 An observer with Vlt0 and position Xlt0 hears a frequency a gt 1000 Hz b lt 1000 Hz c Can not be determined Standing Waves Consider a wave and its reflection yngh Asin2 ift A sin2n0052nft 7 cos2 51n2 ft yk Asm2n Asin2ncosZ7rft cos2 sin2 ft yngh ym 2Asin27r00527rft Standing Waves x yright yleft 2A 311127TZJC0527T Factorizes into xpiece and tpiece Always ZERO at x0 or xmM2 31 Resonances Integral number of half wavele ths in length L t T8 N N 1 t 0 a m T Vibrating t T4 23778 bl 6 2003 Thomson BrooksICole Fig 1416p442 39 Slide18 C 9 OISgt10018 39 U 5w lll 8003 Resonance in String Demo 32 33 Nodes and antinodes 0A node is a minimum in the pattern OAn antinode is a maximum Fundamental 2nd 3rd Harmonics 2nd harmonic 2003 Thomson Brookleole 99 f1 b u n Fig 1418p443 Slide 25 34 35 Example 149 A cello string vibrates in its fundamental mode with a frequency of 220 vibrationss The vibrating segment is 700 cm long and has a mass of 120 g a Find the tension in the string a 163 N b Determine the frequency of the string when it vibrates in three segments b 660 Hz 36 Loose End Organ pipes open aT one end Example 1410 An organ pipe of lengTh 15 m is open aT one end and closed aT The oTher WhaT are The lowesT Two harmonic frequencies DATA Speed of sound 343 ms 572 Hz 1715 Hz Beats InTerference from Two waves wiTh sligthy differenT frequency hAlll I nm Qd v t VUUUUV UUUUVM 2003 Thamsan BrooksCole A Beat Frequency Derivation AfTer Time TbeaT Two sounds will differ by one compleTe cycle flTbeat fZTbeat 1 1 Tb eat fl f2 1 fbeat Them Beats Demo Standing Waves in Air Columns Chapter 11 Energy in Thermal Processes Vocabulary 3 Kinds of Energy CI In rernal Energy U Energy of a sys rem due ro microscopic mo rion and i n rermolucular forces CI Work W FAx PAV is work done by expansion nex r chap rer CI Hea r Q Energy Transfer from microscopic con rac r nex r chap rer Temperature and Speci c Heat m Speci c Heats of CI Add energy gt T rises Some Materials at Atmospheric Pressure Substance J kg C cal g C Aluminum 900 0215 Beryllium l 820 0436 mass of ma rer39lal Properly of Th Cadmium 230 0055 Copper 387 0092 4 Germanium 322 0077 Glass 837 0200 Gold 129 0030 8 Ice 2 090 0500 o CHZO 39 10 cal9 C Iron 448 0107 g 1 calor39le 4186 J Lead 128 00305 a Mercury 138 0033 g Silicon 703 0168 9 Silver 234 0056 Steam 1010 0480 a 4 186 100 U Example Converting calories Bobby Joe drinks a 130 calorie can of soda If The efficiency for39 Tur39ning energy inTo work is 10 how many 4 meTer39 floor39s musT Bobby Joe ascend in order39 To work off The soda and moi nTai n her39 55 kg mass Solution Fir39sT noTe ThaT The quotcaloriesquot isTed in food are acTually kilocalor ies J Q 130 kcal 13gtlt105 cal4186 l 544gtlt105 J ca 10 o of Q geTs conver39Ted To PE QlO mgh h 2 101m Nfloor39s Example Calorimetry Aluminum has a specific heaT of 0924 cal9 C If 110 g of hoT waTer aT 90 C is added To an aluminum cup of mass 50 g which is originally aT a TemperaTure of 23 C whaT is The final TemperaTure of The equilibraTed waTercup combo Solution EquaTe heaT loss of waTer wiTh heaT gain of cup mcupCAl 0C mHZOCHZO 0C Solve for T T 873 C Phase Changes and Latent Heat El T does noT r39ise when phases change aT consTanT P El Examples solid gt liquid fusion liquid gt vapor39 vaporiza rion El LaTenT heaT energy required To change phases Latent Heats of Fusion and Vaporization Pr39oper39l Latent Heat Latent Heat of Of Melting of Fusion Boiling Vaporization Substance Point C Jkg cal g Point C Jkg cal g Helium 26965 523 x 103 125 26893 209 x 104 499 Nitrogen 20997 255 x 104 609 19581 201 x 105 480 Oxygen 21879 138 x 10 l 330 18297 213 x 105 509 Ethyl alcohol 114 104 x 105 249 78 854 x 105 204 39 Water 000 333 gtlt 105 797 10000 226 x 10G 540 5 Sulfur 119 381 x 104 910 44460 326 x 105 779 3 Lead 3273 245 x 104 585 1 750 870 x 105 208 Aluminum 660 397 x 105 948 2 450 114 x 107 2 720 in Silver 96080 882 x 10 l 211 2193 233 x 10 558 g Gold 1 06300 644 x 10 1 154 2 660 158 x 10 377 E Copper 1083 134 x 105 320 1 187 506 x 106 1210 g Example Boiling water 10 iTers of waTer is heaTed from 12 C To 100 C Then boiled away a How much energy is required To bring The waTer To boiling b How much exTra energy is required To vaporize The waTer c If elecTriciTy cosTs 75 per 1000 kWhrs whaT was The cosT of boiling The waTer Solution 1 Given m1000 g c10 calg AT88 Find Q Q 88x104 cal 368x105 J Solution continued b Given L54O calg mIOOOg Find Q Q 54x10395 col 226x106 J c Given Q 226x106368x105 J Ra re 751000 kWhr39 Find cos r Fir39s r find rate in dollar39sJ 75 8 Rate dollarsJ 208 x 10 dollarsJ 100010003600 Then find ne r cos r Q mul riplied by rate 55 Announcements Mid rer39ms graded on scale of 11 Who wan rs ex rr39a r39eviewr39eci ra rion sessions You can pick up your exams no r bubble shee r in Friday helpr39oom Example Body cooling Consider Bobby Joe from The previous example If The 90 of The 130 kcols from her39 soda wenT inTo heoT which was Taken from her39 body from r39odioTion how much oner39 was per39spir39ed To moinToi n her39 normal body Temper39oTur39e Assume a IoTenT heoT of vaporizaTion of 540 calg even Though T 37 C Solution Given Q O9x13x105 col L 540 colg Find mmp mevapQL g A can of soda has 350 g of H20 Some fluid dr39ips away Three Kinds of Heat Transer CI Conduction 0 Shake your neighbor39 pass it down 0 Examples Heating a skillet Losing eat through the walls of a house CI Convection 0 Move hot region to a different location 0 Examples Hotwater39 heating for buildings Circulating air39 Unstable atmospher39es CI Radiation 0 Light is emitted from hot object 0 Examples Star39s Incandescent bulbs Conduction CI Power depends on area leng rh Temperature difference and conduc vifyof ma rerial 2003 Thomson BrooksCole Conduc rivi ry is proper ry Energy ow of ma rerial for Thgt 1 V Example A copper poT of radius 12 cm and Thickness 5 mm siTs on a burner and boils waTer The TemperaTure of The burner is 115 C while The TemperaTure of The inside of The poT is 100 C WhaT mass of waTer is boiled away every mi nuTe DATA kCu 397 Wm C Solution Given Ax 0005 m A TEr39Z r12 m Th 115 C TC 100 C Time 60 sec kCu 397 Wm C L 540 calg FirsT find The power P 539x104 WaTTs NexT find Q PTime 323x106 J Finally find m of vaporized waTer Remember m Q Lf L4186540 Jg m 13943 k9 Conductivities and Rvalues CI Conduc rivi ry 0 Property of Material 0 SI uni rs ar39e Wm C CI RValue 0 Property of ma rer39ial and Thickness Ax 0 Measures r39esis rance ro hea r 0 Useful for comparing insula rion produc rs o Quo red values are in AWFUL uni rs Thermal Conductivities Substance Thermal Conductivity Js m C Metals at 25 C Aluminum Copper Gold Iron Lead Silver Gases at 20 C Air Helium Hydrogen Nitrogen Oxygen N onmetals Asbestos Concre 6 Glass Ice Rubber Vv39ater Wood 238 397 314 795 347 427 0023 4 0138 0172 0023 4 0023 8 910315310019 uoswouleoaz 025 13 084 16 02 060 010 Conducitivities and Rvalues TABLE 39I 14 RValues for Common Building Materials Material Rvalue ft2 FhBtu Hardwood siding 10 in thick Wood shingles lapped Brick 40 in thick Concrete block lled cores Styrofoam 10 in thick Fiber glass batting 35 in thick Fiber glass batting 60 in thick Fiber glass board 10 in thick Cellulose ber 10 in thick Flat glass 0125 in thick Insulating glass 025in space Vertical air space 35 in thick Air lm Drywall 050 in thick Sheathing 050 in thick 435 370 089 154 101 017 045 132 2003 Thomson BrooksCole ARGH What makes a good heat conductor Fr39eequot elec rr39ons me rals Easy Transpor r of sound Ia r rice vibr39a rions STiff is good Low Densi ry is good Pur39e crystal structure Diamond is perfect Rvalues for layers Consider a layered sys rem eg glassair glass HH H AT I ATZ 3T3 AT4 ATAEAEAEW PR1 PR2 PR3 A A A P 22R1R2R3 Example Glass Door Consider Three panes of glass each of Thickness 5 mm The panes Trap Two 25 cm layers of air in a large glass door How much power leaks Through a 20 m2 glass door if The TemperaTure ouTside is 40 C and The TemperaTure inside is 20 C DATA kglass 084 Wm C kw 00234 Wm C Solution Known kglass084 Ax 0005 k A20 AT6O Find P Firs r find Rglass and R Rglasso00595 Rair1068 Nexf find R for all The layers Finay find The power glass airO0234 Axair0025 for one layer of each air Convection CI If warm air39 blows across The room if is convec rion El If There is no wind if is conduction El Can be ins riga red by Turbulence or39 ins rabili ries Why are windows triple paned To s rop convec rion Transfer of heat by radiation CI All objects emi r ligh r if T gt 0 CI Colder39 objects emi r longer wavelengths red or39 i nfr39ar39ed CI Ho r rer objects emi r shor rer39 waveleng rhs blue or39 ul rr39aviole r CI Sfefan s Law give power39 of emi r red radiation Emissivify G 56696x108 Wm20K4 O lt e lt 1 usually near39 1 is The S refanBol rzmann cons ran r Example If The TemperaTur39e of The Sun fell 5 and The radius shrank 10 o whaT would be The per39cenTage change of The Sun39s power39 oquuT Solution P 0A6T4 047Z39R28T4 F0 0A06T04 047tR eT04 092 O954 0660 34 o Example Power of the Sun DATA The sun radiaTes 374x1026 W DisTance from Sun To EarTh 15x1011 m Radius of EarTh 636x106 m a WhaT is The inTensiTy powerm2 of sunlighT when iT reaches EarTh b How much power is absorbed by EarTh in sunlighT assume ThaT none of The sunlighT is reflecTed c WhaT average TemperaTure would allow EarTh To radiaTe an amounT of power equal To The amounT of sun power absorbed Solution Given P3unl I2ear1 h3unl R a Find IPA of sunligh r a r The Ear rh39s orbi r earTh P sun 2 1320 Wm2 47rR earthsun3 b Find power absorbed by ear rh P 2 la a 7122 167x1017 w earth c Find average T of ear rh 2 WW 6 1 T42 T276 K3 C37 F O39Ae Why is the Earth warmer EarTh is noT aT one single TemperaTure EmissiviTy lower aT EarTh39s Thermal wavelengThs Than aT Sun39s wavelengThs RadioacTive decays inside EarTh HoT underground less so in Canada MosT of JupiTer39s radiaTion Greenhouse Gases EISun is much ho r rer rhan Ear rh so sunligh r has much shor rer waveleng rhs rhan ligh r radia red by Ear rh infrared EIEmissivi ry of Ear rh depends on waveleng rh CICOZ in Ear rh39s a rmosphere reflec rs in The infrared oBarer affec rs incoming sunligh r oReduces emissivi ry e of reradia red hea r 032 Cmmnlra tm in ppmv 360 350 340 330 320 310 300 290 230 270 l700 I50 l300 1350 l900 1950 2000 Year uxce En que te Comm ission 1990 Lecture 9 Last Lecture Momentum New ron39s 2nd Useful for short rimes Collisions p always conserved Elastic E conserved roo Inelastic E not conserved Example 67 A proton mp167x1027 kg elastically collides with a target proton which then moves straight forward If the initial velocity of the projectile proton is 30x106 ms and the target proton bounces forward what are a the final velocity of the projectile proton b the final velocity of the target proton Elastic collision in 1dimension 1 Conservation of Energy 2 Conservation of Momentum 0 Rearrange both equations and divide Elastic collision in 1dimension Final equations for headon elastic collision 0 Relative velocity changes sign 0 Equivalent to Conservation of Energy Example 68 An proton mp167x1027 kg elastically collides with a target deuteron mD2mp which then moves straight forward If the initial velocity of the projectile proton is 30x106 ms and the target deuteron bounces forward what are a the final velocity of the projectile proton b the final velocity of the target deuteron vp 1Ox106 ms vd 20x106 ms Headon collisions with heavier objects always lead to reflections vo k Example 69a M The mass M1 enters from the left with velocity v0 and strikes the mass M2M1 which is initially at rest The collision is perfectly elastic 0 Just after the collision v2 v0 Agt Blt c vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2M1 which is initially at rest The collision is perfectly elastic Just after the collision v1 0 Agt Blt c vo k Example 69c M The mass M1 enters from the left with velocity v0 and strikes the mass M2M1 which is initially at rest The collision is perfectly elastic Just after the collision P2 Mlvo 3 llA vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2M1 which is initially at rest The collision is perfectly elastic At maximum compression the energy stored in the spring is 12M1v02 Agt Blt c vo k Example 69e M The mass M1 enters from the left with velocity v0 and strikes the mass M2lt M1 which is initially at rest The collision is perfectly elastic Just after the collision v2 v0 Agt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2lt M1 which is initially at rest The collision is perfectly elastic Just after the collision v1 0 Agt Blt c vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2lt M1 which is initially at rest The collision is perfectly elastic Just after the collision P2 Mlvo gt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2lt M1 which is initially at rest The collision is perfectly elastic At maximum compression the energy stored in the spring is 12M1v02 A gt B lt c vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2gtM1 which is initially at rest The collision is perfectly elastic Just after the collision v2 v0 gt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2gtM1 which is initially at rest The collision is perfectly elastic Just after the collision v1 0 gt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2gtM1 which is initially at rest The collision is perfectly elastic Just after the collision P2 Mlvo gt Blt vo k Example M The mass M1 enters from the left with velocity v0 and strikes the mass M2gtM1 which is initially at rest The collision is perfectly elastic At maximum compression the energy stored in the spring is 12M1v02 gt Blt Example 610 772 2003 Thomsan BreaksCale Ballistic Pendulum used to measure speed of bullet 05kg block of wood swings up by height h 65 cm after stopping 809 bullet 227 ms What was bullet39s velocity Example 611 A 5g bulleT Traveling GT 500 ms embeds in a 1495 kg block of wood r39esTing on The edge of a 09m high Table How far does The block land from The edge of The Table 71 4 oursnmomowcu 3 Example 612 Tarzan M80 kg swings on a 12 m vine by letting go from an angle of 60 degrees from the vertical At the bottom of his swing he picks up Jane m50 kg To what angle do Tarzan and Jane swing 358 degrees indepedent of L or g Example 612a Tarzan M80 kg swings on a 12 m vine by letting go from an angle of 60 degrees from the vertical At the bottom of his swing he picks up Jane m50 kg To what angle do Tarzan and Jane swing To calculate Tarzan39s speed just before he picks up Jane you should apply A Conservation of Energy B Conservation of Momentum Example 612b Tarzan M80 kg swings on a 12 m vine by letting go from an angle of 60 degrees from the vertical At the bottom of his swing he picks up Jane m50 kg To what angle do Tarzan and Jane swing To calculate TarzanampJane39s speed just after their collision given the previous answer you should apply A Conservation of Energy B Conservation of Momentum Example 612c Tarzan M80 kg swings on a 12 m vine by letting go from an angle of 60 degrees from the vertical At the bottom of his swing he picks up Jane m50 kg To what angle do Tarzan and Jane swing To calculate TarzanampJane39s final height given the previous answer you should apply A Conservation of Energy B Conservation of Momentum Chapter 7 Rotational Motion Universal Law of Gravitation Kepler39s Laws Circular motion about a fixed AXIS Angular Displacement Use polar coordinates r6 Distance r doesn39t change Three possible units for 6 Revolutions Degrees 1 rev 360 deg Radians 1 rev 291 rad TL O 5 Reference line Reference 0 line Angular Displacement cont 0 Distance Traveled by a point distance r from axis R f S 27171 N N counts revolutions 612603 1 6 9 is in radians a 9 Fad 231 N revolutions Reference line b o 2003 Thomson BrooksCole Example 71 An automobile wheel has a radius of 42 cm If a car drives 10 km through what angle has the wheel rotated a In revolutions a N 3789 b In radians b 0 238x104 radians c In degrees c 0 136x106 degrees Angular Speed in rads Can also be given in Revolu l39ionss Degreess Linear Tangential Speed 0139 r X 2003 Thoms on BrooksCole v rA9 t At At w in rads Example 72 A race car engine can turn at a maximum rate of 12000 rpm revolutions per minute a What is the angular velocity in radians per second b If helicopter blades were attached to the crankshaft while it turns with this angular velocity what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound DATA The speed of sound is 343 ms a 1256 rads b 27 cm Angular Acceleration Denoted by a oo in rads 0L r39clds2 Every point on rigid object has same on and CL Rotational Linear Correspondence A6 6 Ax 000 e 120 00f 9 VJ 06 e a tet Rotational Linear Correspondence cont39d Constant a Rotational Motion Linear Motion a 0 v v A6 0 fgtt 0 ft 2 1 2 1 2 A6w0t at Axv0t at 2 2 1 2 2 A6wft Eat Axvft at 2 2 2 2 a v f w0aA6 fv0an 2 2 2 o UDSUOQ Example 73 A pottery wheel is accelerated uniformly from rest to a rotation speed of 10 rpm in 30 seconds a What was the angular acceleration in rads2 b How many revolutions did the wheel undergo during that time a 00349 rads2 b 250 revolutions Linear movement of a rotating point Distance h x rAH V P 39 speed As Different points V m M x have different Acceleration 0 linear speeds a ra Angles must be in radians Special Case Rolling Motion Wheel radius r39 rolls without slipping Angular motion of wheel gives linear motion of car Distance x rAH Speed 1 r00 Acceleration ClVOC Chapter 4 Classical Mechanics Forces and Mass does quot0 apply for very Tiny objecTs lt aTomic sizes objecTs moving near The speed of lighT Newton39s First Law Forces If The neT force 2F exerTed on an objecT is Usually a push or pull zero The objecT conTinues in iTs original sTaTe of Vector moTion ThaT is if 2F 0 an objecT aT resT remains aT resT and an objecT moving wiTh some velociTy conTinues wiTh The same velociTy ConTrasT wiTh ArisToTle EiTher conTacT or field force Contact and Field Forces Fundamental Field Forces Types Trong nuclear force ElecTromagneTic force Weak nuclear force GraviTy Euguieai UOSLUOILL io uogsmp e Gugqsuqnd aloa sxoma zoo

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