Physics 2107 Moments of Inertia Experiment 1

Transcription

1 Physics 107 Moments o Inetia Expeiment 1 Read the ollowing backgound/setup and ensue you ae amilia with the theoy equied o the expeiment. Please also ill in the missing equations 5, 7 and 9. Backgound/Setup The moment o inetia, I, o a body is a measue o how had it is to get it otating about some axis. The moment I is to otation as mass m is to tanslation. The lage the alue o I, the moe wok must be done in ode to get the object spinning. This is analogous to the lage the mass, the moe wok must be done in ode to get it moing in a staight line. Alloy im wheels on a bicycle hae a lowe moment o inetia than steel im wheels, theeby making them easie to set spinning and, as a esult, making it easie to acceleate the bicycle. The moment o inetia o a body is always deined with espect to a paticula axis o otation. This is equently the symmety axis o the body, but it can in act be any axis een one that is outside the body. The moment o inetia o a body about a paticula axis is deined as: I m, (1) i i Figue 1 A otating disk is composed o many paticles, two o which ae shown. i whee the sum is oe all the body pats (o index i), m i is the mass o pat I and i is the distance om pat i to the axis o otation. This sum is easy to peom i the object consists o discete point masses (Figue 1). I the body is a continuous object o abitay shape, peoming the sum equies using integal calculus. In this pactical we will detemine the moment o inetia o a numbe o objects and compae the alues obtained using two dieent methods. Fo a disk with an axis though its cente o symmety (Figue ) the moment o inetia is gien by: 1 I m. () mass, m Axis o otation Note that the thickness o the disk has no inluence on the alue o I, which depends only on the adius, and the total mass, m. Figue Disk with axis though its cente. 1.1

2 PY107 Moments o Inetia Expeiment 1 In this expeiment you will detemine I o two dieent systems: (i) a disk and axle olling down an incline and (b) a ball-beaing oscillating on a concae spheical suace. Fo both systems you will detemine I in two ways. Fist, you will measue the mass and adii o the systems unde inestigation and compute I om gien omulae. Then you will compute I by expeimental inestigation and using the pinciple o conseation o enegy. You will be expected to compae the alues that you obtain. Pat 1 Moment o Inetia o a Disk and Axle In this expeiment you must measue I o a disk mounted on an axle. The axle can be thought o as a ey thick disk and you use the same expession to compute I disk I. The total I o the disk axle is the sum I disk I axle. and axle R adius m axle Method 1: Calculate the moment o inetia o the disk and axle about the axis o symmety using the equation: 1 1 I I wheel I axle mdisk R maxle. (3) Measue the masses and adii o the disk and the axle and then compute I using the expession aboe. Note that we ae assuming that the disk is complete i.e. we ae ignoing the missing section though which the axle passes by assuming that the dieence is negligible since << R. m R disk m axle m disk Ensue you include the maximum eo in each o you measuements, e.g. m axle ( 300 ± 0.05) g. All data that you ecod should show the measued alue and its associated maximum eo (and, o couse, units). Result using method 1 o axle disk I 1.

3 PY107 Moments o Inetia Expeiment 1 Method : In the second method you will compute I by timing the wheel as it olls down inclined ails and using the pinciple o conseation o enegy. Conside a wheel consisting o disk and axle, olling down an inclined set o ails ate stating om est at the top. The wheel will moe down the plane with constant acceleation and its total enegy will consist o the sum o the tanslational kinetic enegy, the otational kinetic enegy and the gaitational potential enegy. 1 1 Enegy KE KE PE M Iω Mgh. (4) tans ot Hee, M is the total mass o disk axle, is its tanslational speed, ω is its angula elocity and h is the height o the cente o mass. I the wheel stats om est at position A and olls downwads to position B then the loss in potential enegy must equal the gain in kinetic enegy. A h l B Zeo PE height α Figue 3: Expeimental aangement At point A, the enegy is entiely potential since the wheel is at est: At point B, the enegy is entiely kinetic: Enegy initial PE (5) Enegy inal 1 1 KEtans KEot M Iω, (6) whee is the inal tanslational speed and ω is the inal angula speed. By assuming that iction is ey small, we can assume that the total enegy is constant as the wheel olls down the ails and so the initial enegy is equal to the inal enegy. 1.3

4 PY107 Moments o Inetia Expeiment 1 (7) Fo an axle o wheel that olls without slipping, the angula elocity and the tanslational speed ae elated by: ω. (8) Note that hee, is the adius o the axle, NOT the adius o the lage disk. Using equations (7) and (8) one can ind I in tems o M,, g, and h. I (9) Since the body stats om est and moes with a constant acceleation we can detemine in tems o the distance taelled l and the time taken t. Newton s equations o motion gie that: i at a t 1 l it at l 1 i t i t 0 t l t (10) Substituting this into the expession o I in (9) we get that: gh t I M 1 (11) l Looking at igue 3 and ensuing that the slope o the plane is kept constant at an angleα with the hoizontal we can ewite expession (11) as: g sinα t I M 1 (1) l since sin α h l. Keeping the slope o the ails constant, measue the time t it takes the wheel to moe though dieent distances l along the ails om est using a stopwatch. The same peson should use the stopwatch and elease the wheel and make a ew tial uns to detemine the best pocedue. Fo each distance l take a numbe o measuements o t in ode to detemine the aeage time and estimate the uncetainty in t. Plot a gaph o t esus l and ind the slope. Substitute this alue into (1) and calculate the moment o inetia o the disk plus axle. Ree to Appendix A o the eo analysis. Result using method o axle disk I 1.4

5 PY107 Moments o Inetia Expeiment 1 How does the answe you obtained o method 1 compae with that obtained o method? Mention possible souces o eo that would account o this discepancy. 1.5

6 PY107 Moments o Inetia Expeiment 1 Pat : Moment o Inetia o a Ball-Beaing In this expeiment you must measue I o a ball-beaing. Method 1: Measue caeully the mass m and adius o the ball-beaing and detemine its moment o inetia using the expession: I m (13) 5 Ensue you answe contains estimates on the eo in I. Result using method 1 o ball-beaing I Method : In this method you will detemine I o the ball-beaing using the pinciple o conseation o enegy and eeing to Appendix B. Conside a uniom sphee o mass m and adius olling back and oth without slipping on a concae spheical suace o adius o cuatue R, it will execute small amplitude oscillations in a etical plane. By showing these oscillations ae simple hamonic in natue it is possible to detemine an expession o the peiod and, hence, the moment o inetia o the sphee. As with the peious method, we will ignoe any ictional eects and assume that enegy must be conseed. I we conside the schematic o the poblem shown in igue 4, we can assume that when the oscillations each maximum amplitude (position A) the enegy o the sphee is entiely potential. When it eaches the equilibium position (position B) the enegy will be entiely kinetic. O R y x A B 1.6

7 PY107 Moments o Inetia Expeiment 1 A R O C y x D The cente-o-mass o the sphee moes in a etical cicle o adius R. Applying the geometical theoem AC x CB CD, we see that: ( ( R ) y) y x ( R ) y y x ( R ) y x, y << R. (14) Applying conseation o enegy we hae that: B Figue 4: Expeimental paametes whee ω. 1 1 E m Iω mgy constant, (15) Dieentiating equation (15) with espect to time we obtain: d m dω Iω dy mg d m d m I I d d mg mg d x ( R ) x ( R ) 0 (16) I d I d x g x 1 1 m m R (17) g m Equation (17) has the om x γ x, whee γ. The motion is R m I theeoe simple hamonic in natue. We can deine the peiod o the motion as T π ω π γ. 1.7

8 PY107 Moments o Inetia Expeiment 1 T I ( R ) m π. (18) mg Hence the moment o inetia is gien by: gt I m 1. (19) 4π ( R ) Time the peiod o, say, 10 oscillations o the sphee on the cued suace and, thence, detemine the aeage peiod o the oscillations, T. Make sue you answe is in the SI unit o time, i.e. the second. Measue the adius o the ball-beaing and detemine the adius o cuatue o the suace R using the spheomete as descibed in Appendix B. Note: please do this on the concae spheical suace (athe than the conex one on page 1.10) othewise you may get the wong answe. Ensue you include the eos on each alue and make an estimate on the eo in I detemined using this method. Result using method o ball-beaing I Compae the alues obtained o the moment o inetia o a sphee using both methods compae and comment on whee additional souces o eo may aise. 1.8

9 PY107 Moments o Inetia Expeiment 1 Appendix A Example o Eo Analysis o the Disk Axle Expeiment I M M g sinα t l [ XS 1], 1 whee X ln I ln g sinα t and S is the slope. l ( M ) ln( XS 1) ln M ln ln ( XS 1) ΔI I ΔM M ΔM M Δ Δ ( XS ) Δ XS 1 SΔX XΔS XS 1 Hee g sinα g X, ΔX cosαδα NB: The eo Δα must be expessed in adians. 1.9

10 PY107 Moments o Inetia Expeiment 1 Appendix B Detemination o the adius o cuatue o a conex suace using a spheomete Appaatus Spheomete, lat suace, cued suace Method Step 1 The spheomete is ist inspected to detemine how a (as measued on the etical scale), the scew moes when otated though one complete eolution o the cicula scale. In geneal this will be 0.5 o 1 mm. The alue o one diision on the cicula scale is then known. Step The spheomete is next placed on a slab o glass and the cente leg is adjusted until its point just touches he suace o the glass. This is best detemined by obseing the image o the leg in the glass suace when iewed at an angle. In this position the zeo on the two scales should align. I not, the zeo eo must be detemined by taking the aeage o seeal settings. Step 3 The cente leg is now scewed upwads and the spheomete is placed on the cued suace so that the thee legs ae in contact with the suace. The cente leg is again scewed downwads until it just touches the suace (best detemined when iewed optically as aboe) and the eadings o the two scales ae taken. This pocedue is epeated seeal times and the aeage o the eadings is taken. Step 4 Finally the spheomete is pessed onto a piece o pape and the aeage distance, l, between the cente point and oute legs is detemined. Theoy On placing the spheomete on the cued suace, the thee oute legs stand on the cicumeence o a cicle o adius l, o diamete AB. Let leg A, the spheomete scew and the cente o cuatue O deine a plane pependicula to this cicle. Let the height though which the cente leg is aised be h. I R is the adius o cuatue o the suace, om the popeties o intesecting chods, we hae that: AC CB DC CE In othe wods: l h( R h) l h On eaanging we get that: R. h. A l R D h C O B 1.10

11 PY107 Moments o Inetia Expeiment 1 Results E #1 # #3 #4 Aeage Value Zeo eo eadings Aeage zeo eo Readings on cued suace Aeage Readings o l Aeage h R cm 1.11