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Physics 111: Lecture 22 Today’s Agenda. Angular Momentum: Definitions & Derivations What does it mean? Rotation about a fixed axis L = I Example: Two disks Student on rotating stool Angular momentum of a freely moving particle Bullet hitting stick Student throwing ball.
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Physics 111: Lecture 22Today’s Agenda • Angular Momentum: • Definitions & Derivations • What does it mean? • Rotation about a fixed axis • L = I • Example: Two disks • Student on rotating stool • Angular momentum of a freely moving particle • Bullet hitting stick • Student throwing ball
Lecture 22, Act 1Rotations • A girl is riding on the outside edge of a merry-go-round turning with constant w. She holds a ball at rest in her hand and releases it. Viewed from above, which of the paths shown below will the ball follow after she lets it go? (a) (b) (c) (d) w
Lecture 22, Act 1Solution • Just before release, the velocity of the ball is tangent to the circle it is moving in. w
w Lecture 22, Act 1Solution • After release it keeps going in the same direction since there are no forces acting on it to change this direction.
The rotational analogue of force F is torque • Define the rotational analogue of momentum p to be angular momentum p = mv Angular Momentum:Definitions & Derivations • We have shown that for a system of particles Momentum is conserved if • What is the rotational version of this??
So (so what...?) Definitions & Derivations... • First consider the rate of change of L:
Definitions & Derivations... • Recall that EXT • Which finally gives us: • Analogue of !!
In the absence of external torques What does it mean? • where and Total angular momentum is conserved
Angular momentum of a rigid bodyabout a fixed axis: Rolling chain • 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: (since riand vi are perpendicular) v1 We see that L is in the z direction. m2 j Using vi = ri, we get r2 m1 r1 i v2 r3 m3 v3 I Analogue of p = mv!!
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 z
z z f Example: Two Disks • A disk of mass M and 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
z 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: 2 1 0
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: z 2 1 f
z z f Example: Two Disks Wheel rim drop • Since Li = Lf An inelastic collision, since E is not conserved (friction)! Li Lf f
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
Student on stool Example: Rotating Table... Drop mass from stool • Again, there are no external torques acting on the student-stool system, so angular momentum will be conserved. • Initially: Li = Iii • Finally: Lf = If f f i If Ii Lf Li
Lecture 22, Act 2Angular 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: (a) increases (b) decreases (c) stays the same w2 w1 I2 I1 L L
L is conserved: K2 > K1 K increases! I2 < I1 w2 w1 I2 I1 L L Lecture 22, Act 2Solution (using L = I)
Lecture 22, Act 2Solution • 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
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 v
m Angular Momentum of aFreely Moving Particle... • Consider a particle of mass m moving with speed v along the line y = -d. What is its angular momentum as measured from the origin (0,0)? y x d v
Angular Momentum of aFreely Moving Particle... • We need to figure out • The magnitude of the angular momentum is: • Since r and p are both in the x-y plane, L will be in the z direction (right hand rule): y x r d p=mv
Angular Momentum of aFreely Moving Particle... • So we see that the direction of L 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 (momentum conservation). y x d p
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
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
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
Example: Bullet hitting stick... • Set Li = Lf using M F D m D/4 v1 v2 initial final
Example: Throwing ball from stool • A student sits on a stool which is free to rotate. The moment of inertia of the student plus the stool is I. She throws a heavy ball of mass M with speed v such that its velocity vector passes a distance d from the axis of rotation. • What is the angular speed F of the student-stool system after she throws the ball? M v F d I I top view: initial final
Example: Throwing ball from stool... • Conserve angular momentum (since there are no external torques acting on the student-stool system): • Li = 0 • Lf = 0 = IF - Mvd M v F d I I top view: initial final
Lecture 22, Act 3Angular Momentum • A student is riding on the outside edge of a merry-go-round rotating about a frictionless pivot. She holds a heavy ball at rest in her hand. If she releases the ball, the angular velocity of the merry-go-round will: (a) increase (b) decrease (c) stay the same w2 w1
Initial: Final: same v v m m R w w Lecture 22, Act 3Solution • The angular momentum is due to the girl, the merry-go-round and the ball. LNET = LMGR +LGIRL + LBALL
Lecture 22, Act 3Solution • Since LBALL is the same before & after, w must stay the same to keep “the rest” of LNET unchanged. w w
Lecture 22, Act 3Conceptual answer • Since dropping the ball does not cause any forces to act on the merry-go-round, there is no way that this can change the angular velocity. • Just like dropping a weight from a level coasting car does not affect the speed of the car. w2 w
Recap of today’s lecture • Angular Momentum: (Text: 10-2, 10-4) • Definitions & Derivations • What does it mean? • Rotation about a fixed axis (Text: 10-2, 10-4) • L = I • Example: Two disks • Student on rotating stool • Angular momentum of a freely moving particle (Text: 10-2, 10-4) • Bullet hitting stick • Student throwing ball • Look at textbook problems Chapter 10: # 21, 29, 35, 41, 47, 49