COLLEGE PHYSICS A
COLLEGE PHYSICS A PHY 2053C
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This 29 page Class Notes was uploaded by Garett Kovacek on Thursday September 17, 2015. The Class Notes belongs to PHY 2053C at Florida State University taught by Staff in Fall. Since its upload, it has received 476 views. For similar materials see /class/205538/phy-2053c-florida-state-university in Physics 2 at Florida State University.
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Date Created: 09/17/15
PHY 2053C College Physics A Moti y 5 es 3 3 on Forces Enery Heat W v w 39sLetm u men s msh Linear Momentum commons v Cenmrrofrmass women 9 Rotational Momentum memos 2 A Remember momenta add uq yr de39 a Collision of tw We caH the ve ocmes before common VMVZ afterconswon w M m m 2 quotM WWW w 39 momentum Ques n Elastic Collisio 1dimensional m1 71 l l39lnz 1 momentum pm Virnz V2 conservation v1 I 2 energy conservation 4 v rel velocity before Vi V2 V1 39 V2 39 rel velocity a er Elastic colliSIo 1dimensional V V relative velocity before39 relative velocity after This result is specific to e astic collisions where energ and momentum are conserved It is inde en ent of the masses Simultaneous Cons Energy and Mo 1 Conswons conservation 75 mmzvzlm Vtwmzvzeg 2 own V1 V 2 one eqLaDO 2 Elastwc comsron conservation of energv x m 3v 7mv393mgx9 wwn ouk v1v392tw ecunons In tne touowrng We Wm use both Dropemes to calculate the two unknowns A common n wnrcn energy rs conserved rs caHed n an elastic collislo warpe Newton39s Cradle Why does urIy pa bomce mm Newt aH baHs nave the same mass m 1 Momentum conservatwon m V X m 2 ergvconervatwon 1 therefore mv1ltxwrjv1 gt Inelastic colllsions Two ra road cars couroe and couple togetner g m m Momentum is always conserved mvr2mv gt v IZ12m5 Energv s not Ill KE V 2 2 J KE 2 mg l tne Missing KE goes nto heat or vibrations lmv an KE Important Points from Last Lecture Q In aH coursrons momentum rs conserved 1n elastic coursrons momentum and energy are conserved I r 9 dam 3 9 m gtlt m 3 We do not have to change much 5 x ed obj ts move transatone mono7 as mass was concentrated 7 the quotEs7151 afmass p07t 1n thrs orcture the center of massquot moves We a free projectile rndeoendent ot the rotationa component Center of Mass Loca on The center of mass rs a oosrtron the mas werghtedaverage ot the oosrtrons ot obJeCE that make do a svstem or odv Jed 20b mltrr7plt2 quot1quot X X X 30mm PM m t zr The same rs true for the yrcoordmate ofcm m Haydn yCM x c m1 m2 m11o mm250 mm360 m M sd lreash mass equals man my 0 m 50 quot1 3 m m3 60m 0m 325 m the tota n r er to thd the center ot ravlty suspend the bodv from two o mm V The effect of many small weight I weight acthg on of mass LG WlH alwava be below both Dlvot DolnE together l5 the center of Translatlonal m ion the extended object s mass the centerrofrmass w n kw ig 39g 3 gt2 Cenler39ufmass xw WWW mm m M mln5 u asses can t they Rotate w Extended Bodies ampquot quot Dole Extended obJectS can do one lhlng Whlch Dolnt m CE move as if all of as concentrated ll39l gm ylmm Hum lquoth mn hile they translate Why Rotational Kinematics What is the difference between the circular motion from chapter 5 and rotational m tion7 39 tquot objects moving in circuiar Here We are taiking about extended rigid bodies Gupta 5 om rim n i ha m M Rotat39onal Kinematics Rotationai motion is described bv angles of orientation eanguiarcoord ate measure n radans r this means that the distance traveied is given as I rs NOTE that this tormuia is not true tor angies measured in the customan degrees The transiation radians e degrees is 39 21rradE 36039 lt9 360 21 1 rad 57339 Angular Velocity in describing angui r motion the angie in radians serves as e coor nate dispiacernent in anaiogv We detine average angular velocity as Linear and Angular K ematics mary Displacemen X m angle Shad Veo ty vms angular vel wrads acceleration a ms angular acc u rad52 How do the angular kinematics trans ate to linearquotknematc5 7 i 5 ngm ammmpmm what peed 5 yaw rangevma 5 Vlznrm yew rangema acaaEa o tar yaw rada mentnpetay amEra an Angular Kinematic Equations Non wmw me angu ar coordinate a ve ocity w and acce erauon u come me kinematic equations vv at const a t must a 2x v I at2 V2at2 I 57E 1 7L 5 7E Note that we use a as a coordinate it goes beyond d1equcirdeor2 mum h awhee goes back to m 0H ma p0 non after one m mm me coordinate conunues and measures mm mm as nuUpEs 02quot Question 2 yame Roll39ng Motion 7 be mangJa C What5 me reap a r v39e urnfora veooww and rollingw e 1 Tne oornt of Contact between Whe m and ground P rs rnornentanw at rest Tne awe rs rnowng at ve octtv V Now go rnto wneet s trarne reterence Tne oornt P rs rn atvetocrty v Now we see of ovtng Vmrw V rs tne same rn ootn trarnes of re Gwen rot freq rrevsgtm2n reds quotEM r r 4 1m 1r ran1 Stay tune Wednesday chapters cont39d Rotattona Dvnarmcs Moment of Inertra Angu ar Momentum Rotattona Energv Friday mPA7 Recihh39on Monday Chapter 9 statrc Equmbnurn 43 Experiment VI Static Equilibrium of Rigid Bodies Torques Goals 0 Study the relationship between force lever arm and torque study how torques add 0 Study the conditions for static equilibrium of a rigid body 0 Study the concept of center of gravity Introduction and Background T 0rques The torque I created by a force F with respect to a certain axis of rotation is L39 F X R L 61 where R L is the socalled lever arm which is the perpendicular shortest distance between the axis of rotation and the line of force Torques are vectors Strictly speaking the direction of a torque should be perpendicular to either into or out of the plane de ned by the force and its lever arm However for simplicity we will de ne the direction of a torque as either clockwise CW or counterclockwise CCW depending which way the torque would make the object rotate about the axis of rotation chosen Either CW or CCW can be chosen as the direction as long as consistency is maintained We will choose CW as the direction here The resultant of multiple torques can then be obtained as the algebraic sum of the individual torques Static Equilibrium In order for a rigid body to stay in equilibrium the net force and the net torque on the object must both be zero We usually only deal with motion of objects in two dimensions in which case the conditions for static equilibrium are 2Fx0 ZFy 0 62 L39 0 63 The second condition should hold for any choice of the axis of rotation Center of Gravity For a rigid body with nite size the force of gravity acts on all parts of the body But for the purpose of studying the translational motion of the body as a whole or the static equilibrium of the body we can assume that the entire weight of the body acts at a single point This point is the center of gravity When we draw the force diagram for a rigid body we can put a single force of gravity at the center of gravity Experimental Setup Equipment Meter Clqmp Meier stick stick fulcrum clamps V weightsmanbalance mmmummmmHmmmmmmmmmxmmmx Setup The experimental weighl 39 E Fulcrum gt 5 web setup is shown SChematlca y m Flgure Figure 61 7 Arrangement for balancing torques 44 61 The apparatus consists of a meter stick suspended from a fulcrum Weights may be hung from the meter stick at various positions along the stick by means of special clamps in order to apply torques Experimental Procedure and Data Analysis A T 0rques N E 5 Adjust the location of the fulcrum so that the meter stick with no weights hanging on it is in static equilibrium balance in a horizontal position The position of the fulcrum is roughly the center of gravity of the meter stick Note that this position is not necessarily in the middle of the stick 50 cm mark With a clamp hang a 200g mass on the stick 20 cm from the fulcrum Hang another 200g mass with a clamp on the opposite side of the stick Find a position so that the meter stick is again in equilibrium in a horizontal position Record the position of the fulcrum the masses added to each side of the stick and their positions to the nearest millimeter mark Keep the rst 200g xed and repeat the procedure changing the second mass successively to values of 500 g 400 g 300 g 250 g 150 g and 100 g In each case adjust the position of the second mass to achieve equilibrium and record the data Remember that you need to include the mass of the clamps Weigh them with a balance and include them in the masses you record Calculate the weight W from the mass Calculate the lever arm L from the positions of the mass and the fulcrum Start the Excel template named Torque and input the values of W and L into it Since you have kept the added torque on one side of the stick xed from the second condition of static equilibrium you should realize that the torque produced by any weight on the other side must be equal in magnitude and opposite in direction Check whether your results are consistent with this conclusion by analyzing WL versus W with the template including a Linear Regression Fit and a graph for WL versus W o What value of the slope do you obtain from your computer analysis What is the expected value What is the uncertainty in the slope Is your slope the same as the expected value within the uncertainty 0 How does the intercept compare with the fixed torque on the other side of the stick Now perform a Linear Regression Fit and plot a graph for W versus lL o What should the slope of this plot give you What is its value 0 What is the expected value for the intercept What did you obtain 0 From the results can you draw any conclusion as to how the force required to balance a fixed torque depend on its lever arm 45 B Resultant of Multiple T 0rques 1 Maintain the fulcrum at the same position and balance the meter stick now with three weights Put a single weight on one side and the other two on the other side Record the masses and the positions Calculate the magnitude of various torques around the fulcrum and nd their vector sum 0 Does your vector sum have the value you expect C Center of Gravity 1 So far the fulcrum has been placed only at the center of gravity Now move the fulcrum to a point between the 10 cm and 25 cm marks Balance the meter stick by applying a single mass between 100 to 500 g to the shorter side Record the mass its position and the fulcrum position Weigh the stick without clamp and record its mass Although the meter stick has two sections on either side of the fulcrum according to the concept of center of gravity we can still regard the entire weight acting at a single point center of gravity and it is the torque generated by this entire weight acting at center of gravity that is balanced by the torque from the hanging weight on the other side Use the second condition for equilibrium to calculate the distance from the center of gravity to the fulcrum and thus the position of the center of gravity and compare this point with the previously determined position of center of gravity 2 Draw a freebody force diagram for the meter stick for the last experimental situation and answer the following o What must be the force exerted by the fulcrum on the meter stick Show your calculation 0 Remember the second condition for static equilibrium should hold for any choice of axis of rotation Now take the 0 cm mark as the axis of rotation and calculate the net torque about it Is your result as expected within experimental error Conclusions Brie y discuss whether you have accomplished the goals listed at the beginning PHY 2053C College Physics A Motion Forces Energy Heat Waves Conserved quantities Work Force and Displacement Kinetic and Potential Energy Conservation of Energy Conservation of Momentum Inertial Reference Frames So far we had 5 ecified Frames of Reference with coordinate axes fixed relative to the earth39s surface According to Newton any object including moving ones which is not subject to external net forces may serve as the Point of Reference We call those Inertial Frames of Reference v Ur 5395 b o 8398 in W m on reference frame bl Ground rel rrence frame We can not choose points of reference which are accelerated and have them be inertial gt Apparent Weightlessness 39 How does NASA simulate weghtessness KC 135 air lane vomit comet 7 ies parabolic freefall path neutral bnuyancy water tank 7 balance bouyancy with weights If we use accelerated frames of reference objects appear to change their weight eg a person in an elevator with constant velocity will feel their normal weight wm g inertial reference 39mne a person standing in an elevator accelerating upward or downward will feel an increased wn7 95 or decreased wn7 95 weight naninertial reference 39mne If the elevator cable breaks both the elevator and the person will experience free fall the person will be apparently weightless naninertial reference frame wn7 39ga 0 WorkEnergy Principle Work can become several forms of energy they are the same kind of stuff 7 this chapter we WY talk about Kinetic energy the energy of motion Gravitational potential energy Elastic potential energy potential energy associated with position tater Chapter 14 we WIl see that Heat is also energy Energy is stored work which can be retrieved or changed from one form to another units same as work Joule The net work done m an object is equal to its change in kinetic energy Kinetic Energy Assume a constant net force applied to a car over a distance d What is the work done to change Me car39s speed l l V u I 1 2 2 39 Fma with ad from V V 28dl 2 2 V V 2 WnelFneldrnadrn 2211 1 1 Wnel nwi 5nw lltE2 KE1 We de ne as the translational kinetic energyquot of an object Note that this is the net workm object change in energy Retrieving Kinetic Energy The hammer39s kinetic energy KEl1 iS USEd up as work on the nail it slows to a stop rapidly as the nail is struck As it strikes the nail the nail exerts a force on the hammer slowing it down with a constant force F V1 0 Whthm 2d d Wh mvf KEh from Action Reaction WnFnd thKEh The work done on the nail is equal to KEH Therefore all KE is used up as work on the nail Potential Energy Gravitational Lift a brick of mass m from y1 to y2 What is Me work done on the brick 7 Wextfixgglmghcos0umgh yz But Wext can be retrieved take the hand away and the brick will retrieve the work done as kinetic energy d h v22 h g 1 1 gt K E m v23m2ghmgh Y1 Work had been stored by raising the brick39s position This form of energy is Gravitational Potential Energy Potential Energy Elastic Springs l HOOKe S Law If a person stretches or compresses a spring with a force FP the spring exerts an opposing restoring force Equilibrium position F5 FP DfSPrinQ which is proportional to the elongation or Force of compression AX spring kA m 4XX i liiillliiiimll y restoring means the spring tries to go back to position to x0 The constant k is called spring constant units Nm Potential Energy Elastic Springs How much work is done by compressing Me spring 7 The force FPkAx increases while we are pushing But we can use the average force to calculate the work done 1 FP7kAxf IrAx kAx 3 2 Jilliliiiliiiliiiilll by compressing it Work had been stored in the compressed spring 07 39 Aquot this is called the Elastic Potential Energy Potential Energy Properties g Potential Energy is an energy associated with the position of an object gravitational elm Notes i The gravitational PE depends on the height above a certain reference level You may choose any point as the y 0 point But be consistent ii Therefore the value of PE is not unique even for a given position But APE is the physically meaningful quantity and APE does NOT depend on the choice of the reference level Conservation of Energy We Vjust looked at an example where potential energy of the rock is converted to all pr kinetic energy as it falls Initially attop PE Epmgh XV 7 V Finally at floor KE EK 12mV2 For those and all points inbetween part of the energy is kinetic and part is potential but I the total energy is constant alHKE Question 4 Water Slides Question 5 Water SIidesZ Momentum Momentum is defined as mass times velocity Momentum is a vector Units kg ms unit does not have special name Newton wrote his second law as When mass is constant this is equivalent to gt A5 m A17 ZF At At m a review amp pre View Conserved quantities energy amp momentum The Total Energy is a conserved quantity You can not change energy from the inside of alsystem EKEPE mv2mghc0nstant You can change energy from the outside by performing work on an object AE Wch059 The Total Momentum is also a condserved quantity You can e not change momentum from the InSI i 2 my constant You can change the momentum of an object from the outside by applying a force gt A p F At Example Energy conservation I LooptheLoop A small block of magm10 kg SI es Wl ou rlctlon along the looptheloop track shown The block starts from the point P at I 39 39 39 t a distance h58m above bottom of the loop of radius R20 m What is the kinetic energy of me mass at pointA 0n the loop m SoutOh Point P EPE mgh 10 kg 98 msz58 m 5684J PointA 5684JEPE KE mg2 RKE a KE A5684 J mg2 R 5684 J 1 kg 98 m5240m 5684J 3923 1764J How fast is the mass moving at pointA 7 KEAmi gt VA1188mS Conservative and I Nonconservative Forces The conservatio applies only if no f H Friction is a non quot meaning that Fricti energy into heat cannot be directly mechanical energy The work done by forces such as fric mechanical energy Conservative and E Nonconservative Forces Friction is a nonconservativequot force The work done by all nonconservative forces change the total mechanical energy I EXAMPLE Let a block of m1 kg slide down a path with 2m height difference and no friction Then it encounters a 4m long level surface which excerts 10N of friction force on it What is its energy at Me end of Mat paw 3 E1 4kg2m98msz 784J potential E E2 E1 784J my kinetic m E Ef au m mMAJ quotquot quot2 a l 4m friction Momentu m l Momentum is defined as mass times velocity Momentum is a vector Units kg ms unit does not have special name Newton wrote his second law as I When mass is constant this is equivalent to Ap m A17 At At quotm 2F Change of Momentum quotimpulsequot This definition of momentum is in accord with the momentum used in everyday life lore momentum an oty39ect has the harder I stop or to get it going 5T1 At gtimpulsequot Ie change of momentum equals orcequot times quottime appliedquot This is also lled impulse g Conservation of Momentum omentum is a conserved quantity just like energy if there is no external force acting That sounds like old news what is it good for Even for many objects total momentum does not change at any time if they are an isolated system 1 ij N 2 You cannot change the momentum of a svstem from the inside Question 1 g Stay tuned Friday CAPA6 Recitation Monday Chapter 7 Linear Momentum Ke eQ39A nPI Im 39 an PHY2053C
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