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Chapt. 10: Angular Momentum

Angular momentum conservation And applications

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Angular Momentum Conservation

where and

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Ii ωi = If ωf

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Spinning skater: Because the torque exerted by the ice is small, the angular momentum of the skater is approximately constant. When she reduces her moment of inertia by drawing in her arms, her angular speed increases.

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A student sitting on a stool that rests on a turntable with frictionless bearings is holding a rapidly spinning bicycle Wheel (a). The rotation axis of the wheel is initially horizontal, and the magnitude of the spin-angular-momentum vector of the spinning wheel is

What will happen if the student suddenly tips the axle of the wheel (b) so that after the rotation the spin axis of the wheel is vertical and the wheel is spinning counterclockwise (when viewed from above)?

Answer: The turntable, stool, and student will be rotating clockwise with an angular momentum about the vertical axis of the turntable of magnitude

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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 .

•  What is the relation ship between and ?

•  analogous to an inelastic collision: p1 + p2(=0) p3

z z

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Example: Two Disks •  First realize that there are no external torques acting on

the two-disk system of combined mass M. –  Angular momentum will be conserved!

•  Initially, the total angular momentum is due only to the disk on the bottom:

z

2

1

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z

Example: Two Disks

•  Since Li = Lf

z z

Li Lf

An inelastic collision, since E is not

conserved (friction)!

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Example: bullet hitting a 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. Knowing that the initial speed of the bullet is v1, and the final speed is v2.

•  What is the angular speed ωf of the stick immediately after the collision? (Ignore gravity)

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Example: bullet hitting a stick

Final angular momentum:

where

Conservation of angular momentum around pivot axis:

Initial angular momentum:

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Example: throw ball from stool

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Angular momentum

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Angular momentum

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A 25-kg child in a playground runs with an initial speed of 2.5 m/s along a path tangent to the rim of a merry-go-round, whose radius is 2.0 m. The merry-go-round, which is initially at rest, has a moment of inertia of 500 kg · m2 The child then jumps on to it. Find the final angular velocity of the child and the merry-go-round together.

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Summary of Dynamics •  Dynamics (cause – effect)

–  Force – Linear acceleration •  Change in linear momentum, Fnet = dp/dt

–  Torque – angular acceleration •  Change in angular momentum, τ = dL/dt

•  Conservation of momentum –  When net external force = zero

•  Conservation of angular momentum –  When net external torque = zero

•  Conservation of Energy –  Mechanical energy = kinetic + potential –  Non-conservative forces e.g. friction

•  Energy lost to the environment, e.g., heat

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Dynamical Reprise: Linear •  The change of motion of an object is described by Newton

as F=ma where F is the force, m is the mass, and a is the acceleration.

•  For a set of discrete point particles, all forces act on the center of mass

•  The center of mass is a calculable property of the object.

If F=0, then there is no change of motion -- if at rest, the object will remain at rest.

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Dynamical Reprise: Rotation

•  About a fixed rotation axis, you can always write where is the torque, I is the moment of inertia, and is the angular acceleration.

•  For a set of discrete point particles, •  The parallel axis theorem lets you calculate the moment of

inertia about an axis parallel to an axis through the CM if you know ICM :

IPARALLEL = ICM + MD2

L

D M x

CM

ICM IPARALLEL