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This 13 page Class Notes was uploaded by Jackson Wisozk on Monday October 26, 2015. The Class Notes belongs to PHYS 1205 at University of Oklahoma taught by Michael Strauss in Fall. Since its upload, it has received 32 views. For similar materials see /class/229269/phys-1205-university-of-oklahoma in Physics 2 at University of Oklahoma.
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Date Created: 10/26/15
ContextRich Problems Solutions Outline FOCUS the PROBLEM Draw a picture of the situation including ALL the information given in the problem Questions What is the problem asking you to find Approach Outline the approach you will use DESCRIBE the PHYSICS Draw physics diagrams and define ALL quantities uniquely Which of your defined quantities is your Target variables Quantitative Relationships Write equations you will use to solve this problem PLAN the SOLUTION Construct Specific Equations Same Number as Unknowns Check Unis EXECUTE the PLAN Calculate Target Quantityies EVALUATE the ANSWER ls Answer Properly Stated ls Answer Unreasonable ls Answer Complete extra space if needed Centripetal Force 1 Introduction Tu Ml ease the aeeelerahoh aets hot to therease or deerease the magmtude of the veloertv veetor but rather to ehahge rts dlrectum Newton s seeohd Law tells us that m the absehee of any outsrde r ob ee wl pee a SD ght llne efore e observe he drreetdoh ofmohon ehahgrhg we know th e ls a Foree aehhg The drreetdoh of that Foree ls the drreetdoh that the veloertv veetors tums toward 1n the ease of erreular rhotroh we ear see that the We ear show that m order for an object to eohtrhue rhovrhg m a errele that ls at a eohstaht radus at eohstaht speed the Foree must exaetlv rhateh the mass speed and radlus of the errele aeeordhg to the equatroh Fmquot R As shown rh Flgure l lf the F ove outward from the erreular path Ifthe object travels too slowv theh rt wlll fall thward toward the eehter For a glven speed and magutude of Foree whreh ls exacd oree ls too small or the object travelrhg too fast then rt wlll l or th Foree ls too large radlus there ls only one am the object rhovrhg m a errele From the t v hght for ke hg geometry of the srtuatroh we eould prove that thrs magm ude ls mVQR t t we d w l we mrght ealeulate the mohoh of eleetrohs or protohs m a pamele aeeelerator The Forees aehhg rh these r beeause eases are gravrtahohal forces frretroh or electromagnetic forces Whatever therr ong h L l L Fates F and eall tr we s the aeeelerah or they produee C2nmp2m1AceIerananac Speed loa last Dr Fare tow smau gt adv13 ctrou Speed e Pth of 39w slaw Satslmu nr Farce 2 ton large mv R FlgureA ll L L Frthmtrttr l the galnational Foree provldes the deslred Cehthpetal aeeeleratroh The word Centripetal comes from Latin and means quotcenter seekingquot because the Forces and accelerations must point exactly to the center of the circle in order for there to be circular motion Because we see that these Forces and accelerations produce circular motion we know their magnitudes must be We can also relate the centripetal acceleration and Force to the period of rotation T since 272R V T it follows that 2 2 2 F mac zmv zmmzm4 zR R RT T OUR EXPERIMENT In this experiment we will measure the force required to keep a mass moving at constant angular velocity in a circle of constant radius This force will be applied by a spring suspended between the mass and a post at the center of the circle about which the mass rotates The general outline of the apparatus is shown in Figures 2 and 3 A Bob is suspended from a crossa1m such that with no other masses or springs attached to it it hangs straight down over a Pointer When the system is at rest and we attach a spring to the Bob a hanging mass is needed to keep the spring stretched and the Bob directly over the pointer If we remove this mass then the spring will pull the Bob inward towards the axis However if we begin to spin the Bob about the axis its inertia will try to make it go in a straight line and the Spring will have to pull inward to keep it moving in a circle Thus the spring provides the Centripetal Force By Newton s Third Law the Bob applies an equal and opposite Force to the spring and so the Spring will begin to stretch outward As the rotational speed increases the spring must apply more and more Force according to the equation F mvzR The spring therefore stretches until at exactly the right rotational velocity the Bob once again hangs over the Pointer Figure 46 For this experiment we must rotate the apparatus at constant speed while trying to measure the period of rotation This is difficult to achieve and often the experimental results have significant uncertainties for this experiment Try as best you can to measure the frequency of rotation for a constant speed by keeping the bob rotating at a constant radius soutm durum vi ram nbnve miniquot ppsrsrns ariiom sprinp NunHumquot sppsrrtus 7 m or rMALmlI Giff ominrnn rm I Inward Farce I i so ln sob J in s torrid Xmas Z39m it Ever scrairru as sync it and nut to rim is canltlnl Figure 2 The Cmtripetal Force Apparatus 2 Procedure OPERATION OF THE CENTRIPETAL FORCE APPARATU e rotor or axis of the apparatus is rolled by hand betwem your thumb and fore ngas Whm the rotational speed is just right the bob hanging from the crossarm will hang straight the pointa the Cemripetal Force required for unifmm circular motion is Lhm supplied by the spring The purpose of the countweight is to balance the bob so that the apparatus spins smoothly When propaly balanced it applies no force INITIAL SETUP Renove the bob weigh it and record its mass If additional masses are available to add to the bob now is the time to weigh them as well Weigh the mass holda hook and any masses you will use with it as well Rana39nbaquot to record an estimated uncertainty for these values according to how well you think you can read the scales Bob Mass Mass Huldaquot Hook CALIBRATION OF THE SPRING the spring is an quot39ideal spring the force it can apply is equal to F 7k x AR whae k is a as the p 39 or ula i la and AR is the distance the spring stretches or compresses from its relaxed equilibrium lmgth In othaquot words as you pull on or compre s spring the force with which it quotpushes back increases linearly This is known as quotHook39s Law but it is not an actual Law of Physics but maquoter a model for the spring which holds fairly well as long as the spring is not compressed or stretched too much or too little Set up the apparatus as shown in Figure move the mass holda hook Move the crossarm so that the black bob is hanging straight down Line up the va39tical pointaquot directly Lindaquot the tip of the bob as a refamce for your measuranmt Measure the distance from the pointa to the 0mm of the spindle as accurately as you can This is the equilibrium distance the unstretched 181 of the spring Record this lmgth as quotRnquot This distance Rn corresponds to a force meg ofzero being applied to the spring Ru Counterweigm Crossarm I Pointer Figure 3 Cehtnp etal Force Apparatus T Tan a k 1 with Welght othe hahghg mass Thls welght plus the holderl 15 equal to the Tensloh m the smng that goes ova the ulle and so th5 IS the Force belhg applled to the 5 mg For the hon rotatlng apparatus the bob angs SLmlgit down w en e hahglhg mass welght 15 equal to the ens on m e spnhg so by measurlng th welght requlred to keep the mass vailcal at varlous lengths fothe stretched spnhg you will b ableto callmeeLhe spnng e e Spring Constant Adjust the crossaim to move the mass outward so that it is again hanging straight down Lineup the Pointer under the Bob and use it as a reference to measure the distance from the aXis to the Bob Record this new distance R1 The Spring stretches until the force with which it pulls on the weight F1 7k R1 7 R0 7k AR balances the weight M g Therefore record R1 the weights and the value ofF1 Mg R1 Mass F1 2 Repeat this procedure for several different distances R1 Move the pointer to a distance at least 20 greater than the relaxed length of the spring and then find the weight required to balance force of the spring Mark the various positions of the Pointer on the base platform to refer to these distances again later BE CAREFUL NOT TO DAMAGE THE SPRING BY STRETCHING IT TOO FAR R1 Mass F1 2 R1 Mass F1 I R1 Mass F1 2 MEASURING CENTRIPETAL FORCE 1 Hang the Bob straight down Now detach the weights hook and Spring so that the Bob hangs freely down Move the vertical Pointer to some distance R1 for which you have previously measured the force that balances the spring F1 Readjust the crossarm so that the mass again hangs straight down at this distance when the spring is not attached Lock the crossarm into position Adjust the position of the counterweight if necessary so that when you rotate the aXis its motion seems smooth and balanced 2 Rotate the apparatus and let the spring provide the inward centripetal force Reconnect the spring The stretched spring will pull the bob inward Practice rotating the aXis until you can achieve a smooth motion that will cause the bob to pass directly over the pointer as the apparatus rotates By doing this you have determined a speed such that the inward pull of the stretched spring provides exactly enough Centripetal Force to keep the bob moving in a circle at constant radius R R1 and at constant speed 3 Note that the only Forces on the bob are the upward force of the string which balances the downward force of the weight of the bob and the inward force of the stretched spring The forces are therefore quotunbalancedquot There is an inward force on the Bob applied by the spring The bob seems to quotwantquot to go outward simply because its own inertia is trying to keep it moving in a straight line and an inward force is required to force it to move in a circle The necessary force is called the Centripetal Force and the spring is what is providing the Centripetal Force in this case 4 Spin the bob at a roughly constant speed so that the point of the bob passes over the Pointer at the distance you have chosen Use your stopwatch to measure the average period of revolution T To do this accurately measure the time it takes to complete 10 turns if possible then divide the total time by the number of turns Repeat this process four more times until you get nearly repeatable results Have one lab partner checking to make sure that the point of the bob is hanging over the vertical marker through all turns counted during each timing Ifthis can be achieved it will mean that the bob was spinning with constant velocity Trial 1 Time Trial 2 Time Trial 3 Time Trial 4 Time Trial 5 Time Average Time 5 Using the stretched distance R1 for this trial and results from your previous measurements determine the corresponding value of force F1 Mweigmg F1 6 You now know the radius of the circle in which the bob is traveling and from the Period T you have measured the speed with which it rotates For this rst trial calculate the Centripetal Force that is needed to keep your bob moving in a circle at the radius and speed you have determined This should be the Force applied by the Spring Centripetal Force Using propagation of errors calculate the uncertainty in the centripetal force based on the 7 uncertainties in the radius period and mass What uncertainty radius period mass contributes the most to the uncertainty on the centripetal force 8 Do the two values of force agree within your uncertainty Discuss with your lab partners any sources of systematic or random uncertainty 9 Now qualitatively observe what happens when you rotate the bob at a speed much greater than the speed you measured previously In your lab notebook discuss what you observe What is the net force on the bob If you say there is a net outward force discuss this with your lab partners and your TA Why does the bob move the way it does 10 Make two additional measurements of the rotational period for different values of R1 For each trial remember to first adjust pointer to the distance and then the crossarm with the spring detached so that the bob hangs straight down at the chosen distance of revolution If the apparatus starts rocking reset the position of the counterweight and begin again 11 Before you make your measurements predict the new value you will measure for rotational period R1 I Period Prediction R1 I Period Prediction R1 Trial 1 Time Trial 2 Time Trial 3 Time Trial 4 Time Trial 5 Time Average Time R1 Trial 1 Time Trial 2 Time Trial 3 Time Trial 4 Time Trial 5 Time Average Time 2 Plot the centripetal force proportional to the stretch of the spring to 4iIZRTQ where R is the radius of the circular orbit and T is its period The slope the above graph should be the mass m of the bob You should compare that to the mass you determine by weighing it Mass of Bob from Slope of line Mass of Bob based on direct weighing 3 Questions 0 How accurate was your ability to measure the mass What was the largest source of error Be speci c How could the error systematic and random be minimized 0 Do your two methods of determining the mass measure the same type of mass that is gravitational or inertial Discuss 0 As the radius of the circular orbit in your experiment increased how did each of the following change velocity period centripetal force 0 Assume you wish to design a centrifuge that will spin samples at a high rate of speed in order to separate them by density However you want to make sure your centrifuge does not spin so fast that it breaks up Can you design based on this experiment a switch or governor that will shut off the electric motor when the centrifuge spins too rapidly
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