Q1. A uniform cylinder has mass M = 4.0 kg, radius R = 0.060 m, and moment of inertia for rotation about an axis through its center given by . The cylinder rolls without slipping up an incline. The translational speed of the cylinder at the bottom of the incline is 4.00 m/s. What maximum height above its initial position does the cylinder reach before starting to slide back down?

Background

Topic: Conservation of Energy in Rotational Motion

This question tests your understanding of how to apply conservation of mechanical energy to a rolling object, accounting for both translational and rotational kinetic energy as it moves up an incline.

Key Terms and Formulas

• Translational Kinetic Energy:

• Rotational Kinetic Energy:

• Moment of Inertia for Cylinder:

• Relationship for Rolling Without Slipping:

• Potential Energy at Height h:

Step-by-Step Guidance

1. Write the conservation of energy equation, equating the total kinetic energy at the bottom to the potential energy at the maximum height:

2. Substitute the moment of inertia for a cylinder and the rolling condition into the equation:

   and

   So,

3. Simplify the rotational kinetic energy term and combine it with the translational kinetic energy:

   Total kinetic energy:

4. Set the total kinetic energy equal to the gravitational potential energy at height and solve for $h$:

   Divide both sides by to isolate .

Try solving on your own before revealing the answer!

Q2. A uniform sphere has mass = 2.0 kg, radius = 0.060 m, and moment of inertia for rotation about an axis through its center given by . If it is rolling without slipping on a horizontal surface, and the translational speed of the sphere is 2.00 m/s, what is its total kinetic energy?

Background

Topic: Rotational and Translational Kinetic Energy

This question tests your ability to calculate the total kinetic energy of a rolling object, which includes both translational and rotational components.

Key Terms and Formulas

• Translational Kinetic Energy:

• Rotational Kinetic Energy:

• Moment of Inertia for Sphere:

• Rolling Without Slipping:

Step-by-Step Guidance

1. Calculate the translational kinetic energy using the given mass and speed:

2. Express the rotational kinetic energy in terms of using and :

3. Simplify the rotational kinetic energy expression:

4. Add the translational and rotational kinetic energies to get the total kinetic energy:

Try solving on your own before revealing the answer!