Rolling without slipping Angular Momentum Conservation of Angular Momentum. Physics 201: Lecture 19, Pg 1

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1 Physics 131: Lecture Today s Agenda Rolling without slipping Angular Momentum Conservation o Angular Momentum Physics 01: Lecture 19, Pg 1

2 Rolling Without Slipping Rolling is a combination o rotation and translation. For an object that rolls without slipping, the translation o the center o mass is related to the angular velocity by Physics 01: Lecture 19, Pg

3 Rolling Without Slipping The igure below shows how the velocity vectors at the top, center, and bottom o a rolling wheel are ound. v top = v cm. v bottom = 0. The point on the bottom o a rolling object is instantaneously at rest. Physics 01: Lecture 19, Pg 3

4 Rolling without slipping The linear velocity is the same as the tangential velocity on the edge o the rolling object So we can use the viewer equation v CM = r Physics 01: Lecture 19, Pg 4

5 Rolling Kinetic Energy When a disk rolls we assume it does not slip and the outside o the ball moves with a velocity such that: v = r K TOT = ½mv + ½ =v/r = 1/ mr 1 1 K TOT MV I K TOT 1 1 V MV R 1 V 1 1 MR MV MV 4 K 3 4 MV Physics 01: Lecture 19, Pg 5

6 Give it a try: A hollow cylinder o mass m rolls down an incline o height m. We want to ind the velocity o its center o mass when it reaches the bottom o the incline. Since it s rolling without slipping how would we represent it s total KE? (a) (3/6)mv (b) (3/5)mv (c) (7/6)mv (d) mv m (e) (3/4)mv Physics 01: Lecture 19, Pg 6

7 Give it a try: When a hollow cylinder rolls without slipping with a velocity o v, what kinetic energy does it have? K = + TOT ½mv ½ =v/r = mr 1 1 MV I K TOT 1 1 V 1 1 K TOT MV MR MV MV R K MV Physics 01: Lecture 19, Pg 7

8 Angular Momentum As you can imagine there is a rotational momentum It is called Angular Momentum It is signiied with an L For rotation about a ixed axis the angular momentum is deined as The angular momentum o a body rotating about a ixed axis is the product o the bodies moment o inertia about that axis and angular speed. L = Units kg(m /s) Analog to p = mv Physics 01: Lecture 19, Pg 8

9 Give it a try A solid disk o mass 4 kg and radius 0.15 m is rotating constantly once every two seconds. What is the magnitude o it s angular momentum? (a) 0.8 kg m /s (b) 0.14 kg m /s r () (c) 0k 0. kg m /s (d) 0. kg m /s (e) 054kgm 0.54 /s L Iω L 0.14kg m s I 1 MR 0045k kg m s rad s Physics 01: Lecture 19, Pg 9

10 Conservation o angular momentum Start with Newton s second law or rotations τ t τ net Iα τ I net τ net dω dt d ( Iω ) dt τ ext τ net τ int τ ext dl dt dl dt So i no external torques act, angular momentum will be conserved dl dt Physics 01: Lecture, Pg 10

11 Conservation o Angular Momentum This material is just a dierent way to write Newton s second law. dl dt L 0 tt L 0 L L 0 dl Note: When ext = 0 0 ext i dt L L i This means Angular momentum does not change. Angular Momentum is conserved Physics 01: Lecture, Pg 11

12 Conservation o Angular Momentum We will change this to read when external torques sum to zero. ext dl dt When the external torques on an system sum to zero, the total angular momentum o the system will be conserved. The angular momentum o an isolated system is conserved Physics 01: Lecture, Pg 1

13 Conservation o Linear Momentum When the external torques on an system sum to zero, the total angular momentum o the system will be conserved. L L i I I object 1 object object 1 object I + I = I i + I i i Physics 01: Lecture, Pg 13

14 Demonstrations: Angular momentum What happens to your angular momentum as you pull in your arms? Does it increase, decrease or stay the same? What happens to your angular velocity as you pull in your arms? i I i I Physics 01: Lecture, Pg 14

15 Example: Rotating Table... There are no external torques acting on the student-stool system, so angular momentum will be conserved. Initially: L i = I i i Finally: L = I I i 1 i I i I i I Physics 01: Lecture, Pg 15

16 Give it a try: A disk o mass M and radius R rotates with angular velocity i. A second identical disk, initially not rotating, is dropped on top o the irst. There is riction between the disks, and eventually they rotate together with angular velocity. Which below is true? (a) = ½ i (b) = i (c) = ¼ i z z i Physics 01: Lecture, Pg 16

17 Example: Two Disks First realize that there are no external torques acting on the two-disk system. Angular momentum will be conserved! Initially, the total angular momentum is due only to the disk on the bottom: z L i I MR i i 1 Physics 01: Lecture, Pg 17

18 Example: Two Disks Finally, the total angular momentum is due to both disks spinning: L I I MR 1 1 z 1 Physics 01: Lecture, Pg 18

19 Example: Two Disks Since L = L i MR MR 1 Since L L i i z z 1 i L i L Physics 01: Lecture, Pg 19

20 Rotating Neutron Star A star is dying. Its core collapses such that its radius goes rom 6,000 km to about 40 km. Assume the core does not lose mass. I it were rotating at an angular speed o 0.1 rad/s, what is its speed now? How many times per second does this revolve? Physics 01: Lecture, Pg 0

21 Give it a try: A star is dying. Its core collapses such that its radius goes rom 6, 000 km to about 40 km. I it was rotating at an angular speed o 0.1 rad/s, what is its speed now? Assume no outside torques act and the core does not lose mass. (a) 700 rad/s (b) 4600 rad/s (c) 436 rad/s (d) 18 rad/s (e) 0.1 rad/s Physics 01: Lecture, Pg 1

22 Rotating Neutron Star Rotating Neutron Star g What is the angular momentum or the star beore and ater? What is the angular momentum or the star beore and ater? L i L i i i ω mr L 5 i i ω mr ω mr R i ω ω ω mr L 5 i ω R ω Physics 01: Lecture, Pg s rad ω 700

23 Rotating Neutron Star How many times per second does this revolve? 700 rad s 700 rad s 430Hz Physics 01: Lecture, Pg 3