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

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)

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. 

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. 

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

Physics 211: Lecture 22, Pg 6 Definitions & Derivations... L l First consider the rate of change of L: So(so what...?)

Physics 211: Lecture 22, Pg 7 Definitions & Derivations... l Recall that l Which finally gives us: l Analogue of !!   EXT

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

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

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

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. ii z ff z

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: 00 z 2 1

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: ff z 2 1

Physics 211: Lecture 22, Pg 14 Example: Two Disks l Since L i = L f ff z ff z LiLi LfLf An inelastic collision, since E is not conserved (friction)! Wheel rim drop

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 ? ii IiIi ff IfIf

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 ii IiIi ff IfIf LiLi LfLf Student on stool

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 11 22 I2I2 I1I1 L L

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! 11 22 I2I2 I1I1 L L

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! 11 I1I1 22 I2I2 L L

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

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

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 

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

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 FF initialfinal m D D/4

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 FF initialfinal m D D/4

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

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 FF initialfinal m D D/4

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 FF final D/4

Physics 211: Lecture 22, Pg 29 Example: Bullet hitting stick... l Set L i = L f using v1v1 v2v2 M FF initialfinal m D D/4

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 FF

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 FF

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 

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

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. 

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. 

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 

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