Q1. Two spheres, solid cylinders, or other objects of the same mass but different radii are released from rest at the same height on an inclined plane. They are released simultaneously. Assume the toy oil cans roll without slipping to the bottom of the incline. Which of the following statements is true?

Background

Topic: Rotational Dynamics and Conservation of Energy

This question tests your understanding of rotational inertia (moment of inertia), rolling motion, and how energy conservation applies to objects rolling down an incline without slipping.

Key Terms and Formulas:

• Moment of Inertia (I): A measure of an object's resistance to changes in its rotation. Depends on mass distribution.

• Conservation of Energy: For rolling without slipping, both translational and rotational kinetic energy must be considered.

• Kinetic Energy for Rolling Object:

• Relationship for Rolling Without Slipping:

Step-by-Step Guidance

1. Recall that as the objects roll down the incline, gravitational potential energy is converted into both translational and rotational kinetic energy.

2. Write the energy conservation equation for each object at the top and bottom of the incline:

3. Substitute into the equation to express everything in terms of .

4. Remember that the moment of inertia depends on the shape and radius of the object (e.g., for a solid sphere , for a solid cylinder ).

5. Compare how the distribution of mass (moment of inertia) affects the final speed at the bottom of the incline for objects of the same mass but different radii.

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Q2. A uniform disk with mass 40 kg and radius R = 0.500 m is mounted on an axle through its center. It starts at rest and then starts to rotate with constant angular acceleration . What is its rotational kinetic energy after it has turned through ?

Background

Topic: Rotational Kinetic Energy and Angular Kinematics

This question tests your ability to use rotational kinematics and energy concepts to find the rotational kinetic energy after a disk has rotated through a certain angle with constant angular acceleration.

Key Terms and Formulas:

• Rotational Kinetic Energy:

• Moment of Inertia for a Disk:

• Angular Kinematics (starting from rest):

Step-by-Step Guidance

1. Calculate the moment of inertia for the disk using its mass and radius.

2. Use the angular kinematics equation to find the final angular velocity after the disk has rotated through , starting from rest ().

3. Plug the value of into the rotational kinetic energy formula .

4. Set up the calculation for , but stop before plugging in the final numbers.

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Q3. A solid sphere is rolling without slipping on a horizontal surface. What fraction of its kinetic energy is translational?

Background

Topic: Rolling Motion and Energy Partition

This question tests your understanding of how the total kinetic energy of a rolling object is divided between translational and rotational forms.

Key Terms and Formulas:

• Total Kinetic Energy for Rolling:

• Moment of Inertia for a Solid Sphere:

• Rolling Without Slipping:

Step-by-Step Guidance

1. Write the total kinetic energy as the sum of translational and rotational kinetic energy.

2. Substitute the moment of inertia for a solid sphere and the rolling condition into the equation.

3. Express the rotational kinetic energy in terms of and combine with the translational part.

4. Set up the ratio of translational kinetic energy to total kinetic energy, but do not simplify to a final fraction yet.

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