Physics 231 Lecture 21

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Logan Brooks
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1 Physics 3 Lectue Main points o today s lectue: Angula momentum: L Newton s law o univesal gavitation: GMm F PE GMm Keple s laws and the elation between the obital peiod and obital adius. T π GM 4 3

2 Rolling down the hill Stating ti om est, an object with moment o mass M, moment o inetia, and adius R olls down an incline o height h. What is the velocity at the bottom? (Hint use consevation o enegy.) h KE total KEtans + KEot Mv + Bigge smalle v Consevation o enegy: What is is do v v you o a a use cylinde? hoop? sphee? i the (/MR (MR (/5MR object slides ) ) ) without KE total Mv Mgh olling? + v a) v gh Rolling motion: v R R 4 b ) v gh v Mgh Mv + 3 R M+ v R 0 c) v gh Mghg gh v 7 v M + d) v gh R + MR

3 Conceptual quiz A solid disk and a ing oll down an incline.the ing is slowe than the disk i a) m ing m disk, whee m is the inetial mass. b) ing disk, whee is the adius. c) m ing m disk and ing disk. d) The ing is always slowe egadless o the elative values o m and. v Mgh gh M + R + MR ing : MR ing : v gh disk : MR 4 disk: v gh 3

(Recall that i thee ae no extenal oces, the linea momentum will be conseved.

4 Angula momentum n analogy to the linea momentum p mv we have the angula momentum L. Fo a single paticle m moving a cicle, the magnitude o L is: L m mvt pt v t thee no extenal toques, the angula momentum will be conseved. (Recall that i thee ae no extenal oces, the linea momentum will be conseved.) Fo example, this means that an isolated object will keep the same value o the angula momentum thoughout a poblem. n linea motion, consevation o p mv o an isolated object meant that the velocity v must emain constant. n angula motion, consevation o L o an isolated object does not mean must emain constant, because can change! Fig 8.4, p.9 Slide 5 thee ae no extenal toques L L

5 Example A woman stands at the cente o platom. The woman and the platom otate with an angula speed o 5.00 ad/s. Fiction is negligible. He ams ae outstetched and she is holding a dumbbell in each hand. n this position, the total moment o inetia o the otating system (platom, woman and dumbbells) is 5.4 kg m. By pulling in he ams, the moment o inetia is educed to 3.8 kg m. Find he new angula speed ad/s kg m 38k 3.8 kg m? ad / s 7.ad / s 3.8

6 quiz Adi dive leaps om the 0 m platom and executes a tiple ti owad somesault dive. While in the tuck, the dive has a moment o inetia o about 3.3 kg m and duing the inal layout, the dive stetches he body and inceases he moment o inetia to 0 kg m. the dive achieves an angula velocity o ev/s while in the tuck, what is the angula velocity duing the layout? 0 5 ad/s a) 6 ev/s kg m b) 66 ev/s 3.8 kg m c) 0.66 ev/s? d 0.7 ev/s ad / s 0.66ad / s 0

Anothe τα Example A cylindical 5.00-kg eel with a adius o 0.

00-kg bucket alls into a well, making a light ope unwind om the eel (a) What is the linea

R M eel a bucket bucket T a) Fo bucket: T m g m a T mbucketg+ mbucketa Fo eel: τ TR eel α T eel α R

7 Anothe τα Example A cylindical 5.00-kg eel with a adius o m and a ictionless axle, stats om est and speeds up uniomly as a 3.00-kg bucket alls into a well, making a light ope unwind om the eel (a) What is the linea acceleation o the alling bucket? (b) What is the angula acceleation o the eel? R M eel a bucket bucket T a) Fo bucket: T m g m a T mbucketg+ mbucketa Fo eel: τ TR eel α T eel α R also: α a / R and eel MeelR MReelR a T T M a R R Reel T mbucketg + mbucketa Meela mbucketg Meela mbucketa mbucket g 3kg 9.8m / s 5.3m / s mbucket + M 5.5kg eel / / / α a R m bucket 5.3m / s 8.9ad / s 0.6m

8 Newton s law o univesal gavitation All objects (even light photons) eel a gavitational oce attacting them to othe objects. This oce is popotional to the two masses and invesely popotional to the squae o the distance between them. F G m m G x 0 Nm² /kg²

9 Example A spaceship is on a jouney to the moon. The masses o the eath and moon ae, espectively, 5.98x0 4 kg and 7.36x0 kg. The distance between the centes o the eath and the moon is 3.85x0 6 m. At what point, as measued om the cente o the eath, does the gavitational oce exeted on the cat by the eath balance the gavitational oce exeted by the moon? This point lies on a line between the centes o the eath and the moon. GM moonm ship F moon GMeathm Feath Moon eath _ ship GMmoonmship GMeathmship eath _ ship Eath M M moon eath eath _ ship eath _ ship M M eath moon eath _ ship moon _ship x0 m eath _ ship eath _ ship M eath Mmoon x0 kg x0 kg ship x0 m 6 eath _ ship 3.47x0 m

Physics 1114: Unit 5 Hand-out Homework (Answers)

Physics 1114: Unit 5 Hand-out Homewok (Answes) Poblem set 1 1. The flywheel on an expeimental bus is otating at 420 RPM (evolutions pe minute). To find (a) the angula velocity in ad/s (adians/second),