Experiment 09: Angular momentum

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Cameron Tyler
6 years ago
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1 Expeiment 09: Angula momentum

2 Goals Investigate consevation of angula momentum and kinetic enegy in otational collisions. Measue and calculate moments of inetia. Measue and calculate non-consevative wok in an inelastic collision.

3 Appaatus Connect output of phototansisto to channel A of 750. Connect output of tachomete geneato to channel B of 750. Connect powe supply. Red button is pessed: Powe is applied to moto. Red button is eleased: Roto coasts: Read output voltage using DataStudio. Use black sticke o tape on white plastic oto fo geneato calibation.

4 Calibate tachomete-geneato Spin moto up to full speed, let it coast. Measue and plot voltages fo 0.5 s peiod. Sample Rate: 5000 Hz, and Sensitivity: Low. Time 10 peiods to measue ω. Then calculate ω fo 1 V output. Aveage the output voltage ove the same 10 peiods. 4

5 Measue oto I R Plot only the geneato voltage fo est of expeiment. Use a 55 gm weight to acceleate the oto. Settings: Sensitivity: Low Sample ate 500 Hz. Delayed stat: None Auto Stop: 4 seconds Stat DataStudio and let the weight dop.

6 Undestand gaph output to measue I R Geneato voltage while measuing I R. What is happening: 1. Along line A-B?. At point B? 3. Along line B-C? How do you use this gaph to find I R?

7 Measue I R esults Measue and ecod α up and α down. Fo you epot, calculate I R : τ f IR α down I R m( g α ) α up α up down

8 Fast collision Sensitivity Sample Rate Delayed Stat Auto Stop Low 00 Hz 1 sec Falls below 0.5V Find ω 1 (befoe) and ω (afte), estimate δt fo collision. Calculate 1 I m W ω( o + i )

9 Slow collision Find ω 1 and ω, measue δt, fit o measue to find a c. Keep a copy of you esults fo the homewok poblem.

10 Keple Poblem and Planetay Motion 8.01t Nov 15, 004

11 Keple s Laws 1. The obits of planets ae ellipses; and the sun is at one focus. The adius vecto sweeps out equal aeas in equal time 3. The peiod T is popotional to the adius to the 3/ powe T~ 3/

12 Keple Poblem Find the motion of two bodies unde the influence of a gavitational foce using Newtonian mechanics F G mm ( ) 1 1, ˆ

13 Reduction of Two Body Poblem Reduce two body poblem to one body of mass µ moving about a cental point unde the influence of gavity with position vecto coesponding to the vecto fom mass m to mass m 1 F µ ˆ ( ) mm 1 m + m 1 d µ dt 1,

14 Solution of One Body Poblem Solving the poblem means finding the distance fom the oigin ( t) and angle as functions of time θ ( t) Equivalently, finding the distance fom the cente as a function of angle θ ( ) Solution: 0 1 ε cos θ

15 Constants of the Motion d dθ Velocity v ˆ + θˆ dt dt Angula Momentum L µ v µ tangential Enegy 1 Gm1m E µ v 1 d dθ Gm m E µ + dt dt E d L Gm1m dt 1 µ + µ d dθ v + dt dt 1 dθ dt

16 Reduction to One Dimensional Motion Reduce the one body poblem in two dimensions to a one body poblem moving only in the -diection but unde the action of a epulsive foce and a gavitational foce

17 PRS Question Suppose the potential enegy of two paticles (educed mass µ) is given by whee is the elative distance between the paticles. The foce between the paticles is 1. attactive and has magnitude U () 1 L µ L F µ. epulsive and has magnitude F L µ 3. attactive and has magnitude F L µ 3 4. epulsive and has magnitude F L µ 3

18 One Dimensional Desciption Enegy Kinetic Enegy Effective Potential Enegy 1 d 1 L Gm1m E µ + K + U effective dt µ K U 1 d µ dt effective L Gm m 1 µ Repulsive Foce Gavitational foce F F centifugal gavitational d L L d µ µ 3 du gavitational Gm m d 1

19 Enegy Diagam U effective L Gm m 1 µ Case 4: Cicula Obit E Eo Case 3: Elliptic Obit Eo < E < 0 Case : Paabolic Obit E 0 Case 1: Hybebolic Obit E > 0

20 PRS Question The adius and sign of the enegy fo the lowest enegy obit (cicula obit) is given by 1. enegy is positive, 0 L µgm m 1. enegy is positive, 0 L µgm m 1 3. enegy is negative, 0 L µgm m 1 4. Enegy is negative, 0 L µgm m 1

21 PRS Answe: Cicula Obit The lowest enegy state coesponds to a cicula obit whee the adius can be found by finding the minimum of effective potential enegy Enegy of cicula obit E 0 dueffective L Gm m + d µ 0 µ ( U ) L µgm m ( Gm m ) 1 0 effective L

22 PRS Question If the eath slows down due to tidal foces will the moon s angula momentum 1. incease. decease 3. cannot tell fom the infomation given

23 PRS Question If the eath slows down due to tidal foces will the adius of the moon s obit 1. incease. decease 3. cannot tell fom the infomation given

24 Obit Equation Solution: whee the two constants ae adius of cicula obit 0 1 ε cos θ 0 L µgm m 1 eccenticity ε 1+ EL µ Gm m ( ) 1 1

25 Enegy and Angula Momentum Enegy: ( ) 1 E E ε 0 1 whee E 0 is the enegy of the gound state E ( U ) Angula momentum µ 0 ( Gm m ) 1 0 effective L Gmm L ( µ ) 1/ 0 1 whee 0 is the adius of the gound state

26 Semi-Majo axis Popeties of Ellipse: 0 minimum ( θ π) 1 + ε maximum 0 ( θ 0) 1 ε 1 1 Gm m a maximum + minimum + 1 ε 1+ ε 1 ε E ( ) location of the cente of the ellipse Semi-Mino axis 0 x0 maximum a ε εa 1 ε ( ) b a x a 1/ 1/ 0 0 Aea A πab πa 0 3/ 1/

27 Keple s Laws: Equal Aea Aea swept out in time t ( ) A 1 θ θ + t t t da dt 1 dθ dt dθ dt L µ Equal Aea Law: da L dt µ constant

28 Keple s Laws: Peiod Aea A πab πa 0 3/ 1/ Integal of Equal Aea Law obit µ da L T 0 dt Peiod T µ µπ a A L L 3/ 1/ 0 Peiod squaed popotional to cube of the majo axis but depends on both masses T 4 µ 4 π µ a 4 π a π a 0 L Gm1m G m1 m ( + )

29 Two Body Poblem Revisited Elliptic Case: Each mass obits aound cente of mass with m + m m ( ) µ R cm 1 m1+ m m1+ m m µ m

Central Force Motion

Central Force Motion Cental Foce Motion Cental Foce Poblem Find the motion of two bodies inteacting via a cental foce. Examples: Gavitational foce (Keple poblem): m1m F 1, ( ) =! G ˆ Linea estoing foce: F 1, ( ) =! k ˆ Two