Chapter 8
Rotational Motion

 

Dynamics of Rotation

Example 8-20

The figure shows a uniform disk, with mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration of the falling block, the angular acceleration of the disk, and the tension in the cord. The cord does not slip, and there is no friction at the axle.

Solution

 

Rotational Kinetic Energy

The rotational kinetic energy of a rigid object rotating with an angular speed about a fixed axis and having a moment of inertia I is

 

The rotational work done by a constant torque in turning an object through an angle is

 

Total Mechanical Energy

 

Angular Momentum

The angular momentum L of a body rotating about a fixed axis is the product of the body's moment of inertia I and its angular velocity with respect fo that axis:

 

Example 8-21

A thin-walled hollow cylinder (mass = , radius = ) and a solid cylinder (mass = , radius = ) start from rest at the top of an incline. Both cylinders start at the same vertical height . All heights are measured relative to an arbitrarily chosen zero level that passes through the center of mass of a cylinder when it is at the bottom of the incline. Ignoring energy losses due to retarding forces, determine which cylinder has the greatest translational speed upon reaching the bottom.

Solution

 

Example 8-22

For a classroom demonstration, a sits on a piano stool holding a sizable mass in each hand. Initially, the student holds his arms outstretched and spins about the axis of the stool with an angular speed of 3.74 rad/s. The moment of inertia in this case is . While still spinning, the student pulls his arms in to his chest, reducing the moment of inertia to . What is the student's angular speed now?

Solution:

1. Apply the conservation of angular momentum.

2. Solve for the final angular speed:

3. Substitute the numerical values:

 

Example 8-23

A star of radius rotates with an angular speed . If this star collapses to a radius of 20.0 km, find its final angular speed. (Treat the star as if it were a uniform sphere, and assume that no mass is lost as the star collapses.)

Solution:

1. Apply conservation of angular momentum:

2. Write expressions for the initial and final moments of inertia:

3. Solve for the final angular speed:

4. Substitute numerical values:

 

Example 8-24

A 34.0 kg child runs with a speed of 2.80 m/s tangential to the rim of a stationary merry-go-round. The merry-go-round has a moment of inertia of and a radius of 2.31 m. When the child jumps onto the merry-go-round, the entire system begins to rotate. What is the angular speed of the system?

Solution:

1. Write the initial angular momentum of the child:

2. Write the final angular momentum of the system:

3. Set them equal to each other and solve for the angular speed:

4. Substitute numerical values:

The End