An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls, without slipping, to the bottom. What fraction of its kinetic energy is rotational?

Verified step by step guidance

1. Step 1: Begin by identifying the physical principles involved. The sphere rolls without slipping, so both translational and rotational kinetic energy are present. Use the conservation of energy principle to analyze the motion, as the sphere's potential energy at the top is converted into kinetic energy at the bottom.
2. Step 2: Write the expression for the total kinetic energy at the bottom of the incline. It consists of translational kinetic energy \( K_{trans} = \frac{1}{2} m v^2 \) and rotational kinetic energy \( K_{rot} = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
3. Step 3: Determine the moment of inertia \( I \) for a solid sphere. The formula for the moment of inertia of a solid sphere about its center is \( I = \frac{2}{5} m r^2 \), where \( r \) is the radius of the sphere. Convert the diameter to radius: \( r = \frac{8.0 \text{ cm}}{2} = 4.0 \text{ cm} = 0.04 \text{ m} \).
4. Step 4: Relate the angular velocity \( \omega \) to the linear velocity \( v \) using the rolling without slipping condition: \( v = r \omega \). Substitute \( \omega = \frac{v}{r} \) into the rotational kinetic energy expression \( K_{rot} = \frac{1}{2} I \omega^2 \). This gives \( K_{rot} = \frac{1}{2} \left( \frac{2}{5} m r^2 \right) \left( \frac{v^2}{r^2} \right) = \frac{1}{5} m v^2 \).
5. Step 5: Calculate the fraction of kinetic energy that is rotational. The total kinetic energy is \( K_{total} = K_{trans} + K_{rot} = \frac{1}{2} m v^2 + \frac{1}{5} m v^2 \). The fraction of rotational kinetic energy is \( \frac{K_{rot}}{K_{total}} = \frac{\frac{1}{5} m v^2}{\frac{1}{2} m v^2 + \frac{1}{5} m v^2} \). Simplify this expression to find the fraction.

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. For a solid sphere, it is calculated using the formula I = (2/5)mr², where m is the mass and r is the radius. This concept is crucial for understanding how mass distribution affects rotational motion and energy.

Kinetic Energy in Rolling Motion

In rolling motion, an object possesses both translational and rotational kinetic energy. The total kinetic energy (KE) of a rolling sphere is given by KE = (1/2)mv² + (1/2)Iω², where v is the linear velocity and ω is the angular velocity. Understanding this distinction is essential for determining the fraction of energy that is rotational.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the gravitational potential energy lost by the sphere as it rolls down the incline is converted into kinetic energy, both translational and rotational. This concept is fundamental for analyzing the energy distribution in the system.