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 is the sphere’s angular velocity at the bottom of the incline?

Verified step by step guidance

1. Determine the total mechanical energy at the top of the incline. Since the sphere is released from rest, its initial energy is purely gravitational potential energy, given by \( U = mgh \), where \( m \) is the mass of the sphere, \( g \) is the acceleration due to gravity, and \( h \) is the height of the incline. Use trigonometry to find \( h \) as \( h = L \sin(\theta) \), where \( L \) is the length of the incline and \( \theta \) is the angle of the incline.
2. At the bottom of the incline, the sphere's total mechanical energy is converted into both translational kinetic energy and rotational kinetic energy. The translational kinetic energy is given by \( K_{\text{trans}} = \frac{1}{2}mv^2 \), and the rotational kinetic energy is given by \( K_{\text{rot}} = \frac{1}{2}I\omega^2 \), where \( I \) is the moment of inertia of the sphere and \( \omega \) is its angular velocity. For a solid sphere, \( I = \frac{2}{5}mr^2 \).
3. Use the rolling without slipping condition to relate the translational velocity \( v \) to the angular velocity \( \omega \) using \( v = r\omega \), where \( r \) is the radius of the sphere. Substitute \( \omega = \frac{v}{r} \) into the rotational kinetic energy expression.
4. Apply the conservation of mechanical energy principle: \( mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \). Substitute the expressions for \( I \) and \( \omega \) into this equation. Simplify to solve for \( v \), the translational velocity of the sphere at the bottom of the incline.
5. Once \( v \) is determined, use the rolling without slipping condition \( \omega = \frac{v}{r} \) to calculate the angular velocity \( \omega \) of the sphere at the bottom of the incline.

Key Concepts

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the potential energy of the sphere at the top of the incline is converted into kinetic energy as it rolls down. The total mechanical energy is the sum of potential energy (due to height) and kinetic energy (due to motion), which helps in determining the angular velocity at the bottom.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation. For a solid sphere, the moment of inertia is given by the formula I = (2/5)mr², where m is the mass and r is the radius. This concept is crucial for calculating the rotational kinetic energy of the sphere, which contributes to its total kinetic energy as it rolls down the incline.

Rolling Motion

Rolling motion occurs when an object rotates about an axis while simultaneously translating along a surface. For the sphere in this problem, rolling without slipping means that the point of contact with the incline is momentarily at rest. This relationship between linear velocity and angular velocity (v = rω) is essential for finding the angular velocity at the bottom of the incline.