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Dynamic Applications of Torque

When dealing with systems that have both rotating parts and translating parts, we must be careful to account for all forces and torques correctly and to keep the signs straight.

For a particle of mass m moving in a circle of radius r:

For more general motion,

Example: Angular Momentum

(a) What is the angular momentum of a 0.13 kg Frisbee, considered to be a uniform disk of radius 7.5 cm, spinning with w = 11.5 rad/s?

(b) What is the angular momentum of a 95 kg person running with a speed of 5.1 m/s around a circular track of radius 25 m?

An object is moving in a straight line with momentum p. It has non-zero angular momentum:

(a) always; (b) sometimes; (c) never.

The value of the angular momentum L depends on what we take to be the center of rotation.

An object is moving in a straight line with momentum p. It has non-zero angular momentum:

(a) always; (b) sometimes; (c) never.

L>0

L>0

The angular momentum L is take to be positive if the angular position is increasing with time, i.e., if the motion associated with L is a counterclockwise rotation.

Example: Jumping On

Running with a speed of 4.10 m/s, a 21.2 kg child heads toward the rim of a merry-go-round of radius 2.00 m, as shown.

What is the child’s angular momentum L with respect to the center of the merry-go-round?

Example: The Spin Angular Momentum of the Earth

What is the angular momentum of the Earth as it rotates on its axis? (Assume a uniform sphere.)

Example: The Orbital Angular Momentum of the Earth

What is the angular momentum of the Earth as it orbits the Sun?

Looking at the rate at which angular momentum changes,

Therefore, if t= 0, then L is constant with time.

If the net external torque on a system is zero, the angular momentum is conserved.

Example: A Windmill

In a light wind, a windmill experiences a constant torque of 255 N m.

If the windmill is initially at rest, what is its angular momentum after 2.00 s?

Notice that you do not need to know the moment of inertia of the windmill to do this calculation.

5.33 kg m2

1.60 kg m2

Conservation of Angular Momentum

If the net external torque on a system is zero, the angular momentum is conserved.

The most interesting consequences occur in systems that are able to change shape:

Example: Spinning the Wheel

You are sitting on a stool on a frictionless turntable holding a bicycle wheel. Initially, neither the wheel nor the turntable is spinning. You hold the axel vertical with one hand and spin the wheel counterclockwise with the other hand.

You observe that the stool and turntable begin to rotate clockwise. Then you stop the wheel with your free hand.

What happens to the turntable rotation?

Example: A Stellar Performance

A star of radius Ri = 2.3 x 108 m rotates initially with an angular speed of wi = 2.4 x 10-6 rad/s.

If the star collapses to a neutron star of radius Rf = 20.0 km, what will be its final angular speed wf ?

If the moment of inertia increases, the angular speed decreases, so the angular momentum does not change.

Angular momentum is conserved in rotational collisions:

Example: Two Interacting Disks

A 20 cm diameter 2.0 kg solid disk is rotating at 200 rpm. A 20 cm diameter 1.0 kg circular loop is dropped straight down on the rotating disk. Friction causes the loop to accelerate until it is “riding” on the disk. What is the final angular velocity of the combined system?

Example: A Rotating Disk

Disk 1 is rotating freely and has angularvelocity wi and moment of inertia I1about its symmetry axis, as shown.It drops onto disk 2 of moment ofinertia I2, initially at rest. Becauseof kinetic friction, the two diskseventually attain a common angularvelocity wf.

(a) What is wf?

(b) What is the ratio of final to initial kinetic energy?

Example: Mean Gene in the Mud

You and three of your friends have been bullied byGene, so you make a plan. A nearby park has a merry-go-round with a 3.0 m diameter turntable that has a130 kg m2 moment of inertia. Initially all five of you stand near the rim while the turntable rotates at 20rev/min. When you give the signal, all four of youmove to within 0.3 m of the center, leaving Gene atthe rim. Gene is quick and strong, so it would require an acceleration of 4.0 g to throw him off into the mud. Assume everybody has a mass of 60 kg

Will the plan work?

Yes!

Example:Ride the Merry-go-Round

A 25 kg child at a playground runs with aninitial speed of 2.5 m/s along a path tangent to the rim of a merry-go-round with a radius of 2.0 m and jumps on. Themerry-go-round, which is initially at rest, has a moment of inertia of 500 kg m2.

Find the angular velocity of the child and merry-go-round.

Example: Wrapping the Post

A puck on a frictionless plane is given an initial speed v0. The puck is attached to a massless string that wraps around a vertical post.

Is angular momentum conserved?

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