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Physics 211: Lecture 22, Pg 1 Physics 211: Lecture 22 Today’s Agenda l Angular Momentum: è Definitions & Derivations è What does it mean? l Rotation about a fixed axis L = I è Example: Two disks è Student on rotating stool l Angular momentum of a freely moving particle è Bullet hitting stick è Student throwing ball Original: Mats Selen Modified:W.Geist
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Physics 211: Lecture 22, Pg 2 Lecture 22, Act 1 Rotations A girl is riding on the outside edge of a merry-go-round turning with constant . 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? (c) (b)(a) (d)
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Physics 211: Lecture 22, Pg 3 Lecture 22, Act 1 Solution l Just before release, the velocity of the ball is tangent to the circle it is moving in.
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Physics 211: Lecture 22, Pg 4 Lecture 22, Act 1 Solution l After release it keeps going in the same direction since there are no forces acting on it to change this direction.
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Physics 211: Lecture 22, Pg 5 Angular Momentum: Definitions & Derivations l We have shown that for a system of particles Momentum is conserved if l What is the rotational version of this?? F The rotational analogue of force F is torque p l Define the rotational analogue of momentum p to be angular momentum p = mv
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Physics 211: Lecture 22, Pg 6 Definitions & Derivations... L l First consider the rate of change of L: So(so what...?)
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Physics 211: Lecture 22, Pg 7 Definitions & Derivations... l Recall that l Which finally gives us: l Analogue of !! EXT
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Physics 211: Lecture 22, Pg 8 What does it mean? l where and l In the absence of external torques Total angular momentum is conserved
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Physics 211: Lecture 22, Pg 9 i j Angular momentum of a rigid body about a fixed axis: l 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: rr1rr1 rr3rr3 rr2rr2 m2m2 m1m1 m3m3 vv2vv2 vv1vv1 vv3vv3 L We see that L is in the z direction. Using v i = r i, we get I (since r i and v i are perpendicular) Analogue of p = mv!! Rolling chain
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Physics 211: Lecture 22, Pg 10 Angular momentum of a rigid body about a fixed axis: In general, for an object rotating about a fixed (z) axis we can write L Z = I The direction of L Z is given by the right hand rule (same as ). We will omit the Z subscript for simplicity, and write L = I z
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Physics 211: Lecture 22, Pg 11 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. ii z ff z
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Physics 211: Lecture 22, Pg 12 Example: Two Disks l First realize that there are no external torques acting on the two-disk system. è Angular momentum will be conserved! l Initially, the total angular momentum is due only to the disk on the bottom: 00 z 2 1
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Physics 211: Lecture 22, Pg 13 Example: Two Disks l First realize that there are no external torques acting on the two-disk system. è Angular momentum will be conserved! l Finally, the total angular momentum is due to both disks spinning: ff z 2 1
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Physics 211: Lecture 22, Pg 14 Example: Two Disks l Since L i = L f ff z ff z LiLi LfLf An inelastic collision, since E is not conserved (friction)! Wheel rim drop
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Physics 211: Lecture 22, Pg 15 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 I i, 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 I f. What is his final angular speed f ? ii IiIi ff IfIf
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Physics 211: Lecture 22, Pg 16 Example: Rotating Table... l Again, 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 f = I f f ii IiIi ff IfIf LiLi LfLf Student on stool
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Physics 211: Lecture 22, Pg 17 Lecture 22, Act 2 Angular Momentum A student sits on a freely turning stool and rotates with constant angular velocity 1. She pulls her arms in, and due to angular momentum conservation her angular velocity increases to 2. In doing this her kinetic energy: (a) increases (b) decreases (c) stays the same 11 22 I2I2 I1I1 L L
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Physics 211: Lecture 22, Pg 18 Lecture 22, Act 2 Solution l (using L = I ) l L is conserved: I 2 < I 1 K 2 > K 1 K increases! 11 22 I2I2 I1I1 L L
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Physics 211: Lecture 22, Pg 19 Lecture 22, Act 2 Solution l Since the student has to force her arms to move toward her body, she must be doing positive work! l The work/kinetic energy theorem states that this will increase the kinetic energy of the system! 11 I1I1 22 I2I2 L L
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Physics 211: Lecture 22, Pg 20 Angular Momentum of a Freely Moving Particle l We have defined the angular momentum of a particle about the origin as l This does not demand that the particle is moving in a circle! è We will show that this particle has a constant angular momentum! y x v
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Physics 211: Lecture 22, Pg 21 Angular Momentum of a Freely Moving Particle... l 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)? x v m d y
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Physics 211: Lecture 22, Pg 22 Angular Momentum of a Freely Moving Particle... l We need to figure out l The magnitude of the angular momentum is: rpL l Since r and p are both in the x-y plane, L will be in the z direction (right hand rule): y x pv p=mv d r
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Physics 211: Lecture 22, Pg 23 Angular Momentum of a Freely Moving Particle... L l So we see that the direction of L is along the z axis, and its magnitude is given by L Z = pd = mvd. l 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 p d
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Physics 211: Lecture 22, Pg 24 Example: Bullet hitting stick l 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 v 1, and the final speed is v 2. What is the angular speed F of the stick after the collision? (Ignore gravity) v1v1 v2v2 M FF initialfinal m D D/4
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Physics 211: Lecture 22, Pg 25 ACT What is the angular momentum around the pivot of the bullet stick system before the collision? A) zero B)(m+M) v 1 D/4 C) mv 1 D/4 D) none of the above v1v1 v2v2 M FF initialfinal m D D/4
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Physics 211: Lecture 22, Pg 26 Example: Bullet hitting stick... l Conserve angular momentum around the pivot (z) axis! l The total angular momentum before the collision is due only to the bullet (since the stick is not rotating yet). v1v1 D M initial D/4m
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Physics 211: Lecture 22, Pg 27 ACT What is the angular momentum around the pivot of the bullet - stick system after the collision? A) zero B)(m+M) v 2 D/4 C) mv 2 D/4 - I ω F D) mv 2 D/4 + Iω F v1v1 v2v2 M FF initialfinal m D D/4
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Physics 211: Lecture 22, Pg 28 Example: Bullet hitting stick... l Conserve angular momentum around the pivot (z) axis! l 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. v2v2 FF final D/4
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Physics 211: Lecture 22, Pg 29 Example: Bullet hitting stick... l Set L i = L f using v1v1 v2v2 M FF initialfinal m D D/4
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Physics 211: Lecture 22, Pg 30 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? top view: initial final d v M I I FF
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Physics 211: Lecture 22, Pg 31 Example: Throwing ball from stool... l Conserve angular momentum (since there are no external torques acting on the student-stool system): è L i = 0 L f = 0 = I F - Mvd top view: initial final d v M I I FF
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Physics 211: Lecture 22, Pg 32 Lecture 22, Act 3 Angular Momentum l 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
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Physics 211: Lecture 22, Pg 33 Lecture 22, Act 3 Solution l The angular momentum is due to the girl, the merry-go-round and the ball. L NET = L MGR + L GIRL + L BALL Initial: v R m Final: v m same
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Physics 211: Lecture 22, Pg 34 Lecture 22, Act 3 Solution Since L BALL is the same before & after, must stay the same to keep “the rest” of L NET unchanged.
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Physics 211: Lecture 22, Pg 35 Lecture 22, Act 3 Conceptual answer l 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. l Just like dropping a weight from a level coasting car does not affect the speed of the car.
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Physics 211: Lecture 22, Pg 36 Lecture 22, Act 3 Angular Momentum l 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 throws the ball perpendicular to the tangential direction, the angular velocity of the merry-go-round will: (a) increase (b) decrease (c) stay the same
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Physics 211: Lecture 22, Pg 37 Recap of today’s lecture l Angular Momentum: è Definitions & Derivations è What does it mean? l Rotation about a fixed axis L = I è Example: Two disks è Student on rotating stool l Angular momentum of a freely moving particle è Bullet hitting stick è Student throwing ball
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