Physics 2211: Lecture 38

1. Physics 2211: Lecture 38 • Rolling Motion • Mass rolling down incline plane • Energy solution • Newton’s 2nd Law solution • Angular Momentum

2. Translational & rotational motion combined • For a solid object which rotates about its center or mass and whose CM is moving: VCM 

3. R R vcm But Rolling Motion • Rolling without slipping

4. v = 0 = 0 K = 0 R M h v = R Rolling Motion • Objects of differentIrolling down an inclined plane: find their speed at the bottom of the plane and their acceleration on the plane.

5. 1 hoop 1/2 disk 2/5 sphere Use v = R and I= cMR2 c = Rolling Motion The rolling speed is always lower than in the case of simple sliding since the kinetic energy is shared between CM motion and rotation.

6. 1 hoop 1/2 disk 2/5 sphere c = R M h v Rolling Motion

7. Rolling Motion(alternate approach) • An object with mass M, radius R, and moment of inertia I rolls without slipping down a plane inclined at an angle  with respect to horizontal. What is its acceleration? • Consider CM motion and rotation about the CM separately when solving this problem (like we did with the lastproblem)... I M R 

8. M y R x   Rolling Motion(alternate approach) • Static friction fcauses rolling. It is an unknown, so we must solve for it. • First consider the free body diagram of the object and use FNET = MaCM . In the x direction • Now consider rotation about the CMand use =I realizing that = Rfanda = R

9. I R a M For a sphere, c =:  Rolling Motion(alternate approach) • We have two equations: • We can combine these to eliminate f:

10. ExampleRotations • Two uniform cylinders are machined out of solid aluminum. One has twice the radius of the other. • If both are placed at the top of the same ramp and released, which is moving faster at the bottom? (1) bigger one (2) smaller one (3) same

11. 1 hoop 1/2disk 2/5 sphere c = h ExampleSolution Answer: (3) same speed does not depend on size, as long as the shape is the same!!

12. y x z y x z Direction of Rotational Motion • In general, the rotation variables (w, a, t, etc.) are vectors (have direction) • If the plane of rotation is in the x-y plane, then the convention is • CCW rotation is in the + zdirection • CW rotation is in the- zdirection

13. y x z y x z Direction of Rotation:The Right Hand Rule • To figure out in which direction the rotation vector points, curl the fingers of your right hand the same way the object turns, and your thumb will point in the direction of the rotation vector! • We normally pick the z-axis to be the rotation axis as shown. • = z • = z • = z • For simplicity we omit the subscripts unless explicitly needed.

14. (since and are perpendicular) We see that is in the z direction. m2 j Using vi = ri, we get m1 i  m3 Analog of ! Angular momentum of a rigid bodyabout a fixed axis: • Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin is the sum of the angular momenta of each particle:

15. z  Angular momentum of a rigid bodyabout a fixed axis: • In general, for an object rotating about a fixed (z) axis we can write LZ = I • The direction of LZ is given by theright hand rule (same as ). • We will omit the Z subscript for simplicity,and write L= I

16. In the absence of external torques Total angular momentum is conserved Conservation of Angular Momentum • where

17. z z f Example: Two Disks • A disk of mass Mand radius R rotates around the z axis with angular velocity i. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity f. i

18. z 2 1 i 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:

19. z 2 1 or f Example: Two Disks • First realize that there are no external torques acting on the two-disk system. • Angular momentum will be conserved! • Finally, the total angular momentum is dueto both disks spinning:

20. z z Li Lf f Example: Two Disks • Since Li = Lf An inelastic collision, since K is not conserved (friction)!

21. Example: Rotating Table • A student sits on a rotating stool with his arms extended and a weight in each hand. The total moment of inertia is Ii, and he is rotating with angular speed i. He then pulls his hands in toward his body so that the moment of inertia reduces to If. What is his final angular speed f? f i If Ii

22. Example: Rotating Table • Again, there are no external torques acting on the student-stool system, so angular momentum will be conserved. • Initially: Li = Iii • Finally: Lf = If f f i If Ii Lf Li

23. ExampleAngular Momentum • A student sits on a freely turning stool and rotates with constant angular velocity w1. She pulls her arms in, and due to angular momentum conservation her angular velocity increases to w2. In doing this her kinetic energy: (1) increases (2) decreases (3) stays the same w2 w1 I2 I1 L L

24. L is conserved and K2 > K1 K increases! I2 < I1 w2 w1 I2 I1 L L ExampleSolution (using L = I)

25. ExampleSolution • Since the student has to force her arms to move toward her body, she must be doing positive work! • The work/kinetic energy theorem states that this will increase the kinetic energy of the system! w2 w1 I2 I1 L L

26. Angular Momentum of aFreely Moving Particle • We have defined the angular momentum of a particle about the origin as • This does not demand that the particle is moving in a circle! • We will show that this particle has a constant angular momentum! y x

27. m Angular Momentum of aFreely Moving Particle • Consider a particle of mass mmoving with speed valong the line y = -d. What is its angular momentum as measured from the origin (0,0)? y x d

28. Angular Momentum of aFreely Moving Particle • We need to figure out • The magnitude of the angular momentum is: • Since and are both in the x-y plane, will be in the z direction (right hand rule): y x d  

29. Angular Momentum of aFreely Moving Particle • So we see that the direction of is along the z axis, and its magnitude is given by LZ = pd = mvd. • L is clearly conserved since d is constant (the distance of closest approach of the particle to the origin) and p is constant (linear momentum conservation). y x d

30. Example: Bullet hitting stick • A uniform stick of mass M and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. The initial speed of the bullet is v1, and the final speed is v2. • What is the angular speed F of the stick after the collision? (Ignore gravity) M F D m D/4 v1 v2 initial final

31. Example: Bullet hitting stick • Conserve angular momentum around the pivot (z) axis! • The total angular momentum before the collision is due only to the bullet (since the stick is not rotating yet). M D m D/4 v1 initial

32. Example: Bullet hitting stick • Conserve angular momentum around the pivot (z) axis! • The total angular momentum after the collision has contributions from both the bullet and the stick. • where I is the moment of inertia of the stick about the pivot. F D/4 v2 final

33. Example: Bullet hitting stick • Set Li = Lf and using M F D m D/4 v1 v2 initial final