Problem 52 - College Physics

Question

A solid ball of mass \(m\) rolls along a horizontal surface with a translational speed of \(v\). What percent of its total kinetic energy is translational?

Step-by-Step Solution

Answer

71.43%

Step 1: Understand the Kinetic Energy Components

The total kinetic energy of a rolling ball consists of its translational kinetic energy and rotational kinetic energy. Translational kinetic energy relates to the motion of the center of mass of the ball, while rotational kinetic energy relates to the motion around its center.

Step 2: Determine Translational Kinetic Energy

The translational kinetic energy (K_t) of a ball is given by the formula: \[ K_t = \frac{1}{2} m v^2 \]where \(m\) is the mass of the ball, and \(v\) is the translational speed.

Step 3: Determine Rotational Kinetic Energy

The rotational kinetic energy (K_r) of a solid ball is given by the formula: \[ K_r = \frac{1}{2} I \omega^2 \]Here, \(I\) is the moment of inertia of the ball, and \(\omega\) is the angular velocity. The moment of inertia for a solid ball is \(I = \frac{2}{5} m R^2\).

Step 4: Relate Angular Velocity to Translational Speed

For a rolling object without slipping, \(\omega\) is related to the translational speed \(v\) by the equation:\[ \omega = \frac{v}{R} \]Substitute this into the expression for rotational kinetic energy.

Step 5: Calculate Rotational Kinetic Energy Using \(\omega\)

Substitute \(\omega = \frac{v}{R}\) into the rotational kinetic energy formula:\[ K_r = \frac{1}{2} \left( \frac{2}{5} m R^2 \right) \left( \frac{v}{R} \right)^2 = \frac{1}{5} m v^2 \]

Step 6: Determine Total Kinetic Energy

The total kinetic energy (K_t + K_r) of the ball is the sum of translational and rotational components:\[ K_{total} = K_t + K_r = \frac{1}{2} m v^2 + \frac{1}{5} m v^2 \]

Step 7: Calculate Percentage of Translational Kinetic Energy

Find the percentage of the total kinetic energy that is translational:\[ \text{Percentage} = \frac{K_t}{K_{total}} \times 100 = \frac{\frac{1}{2} mv^2}{\frac{1}{2} mv^2 + \frac{1}{5} mv^2} \times 100 \] Simplify the expression:\[ \text{Percentage} = \frac{\frac{1}{2}}{\frac{1}{2} + \frac{1}{5}} \times 100 = \frac{\frac{1}{2}}{\frac{7}{10}} \times 100 = \frac{5}{7} \times 100 \approx 71.43 \% \]

Key Concepts

Translational Kinetic Energy

In physics, kinetic energy associated with the linear motion of a body's center of mass is known as translational kinetic energy. This form of energy results from the movement along a path, without accounting for any rotation the body might have. For any object that moves in a straight line, the translational kinetic energy can be calculated using the formula: \[ K_t = \frac{1}{2} m v^2 \]Where:

• \( m \) is the mass of the object.
• \( v \) represents the velocity at which the object moves.

This equation tells us that the energy depends on the mass and the square of the velocity, meaning both the heaviness and speed of the object influence its energy level.
Translational kinetic energy is a critical concept in physics, helping us understand how and why objects move as they do. It's essential when analyzing scenarios like a rolling ball, where motion isn't limited to straight-line travels.

Rotational Kinetic Energy

When objects rotate around an axis, not only do they have translational kinetic energy due to their center of mass moving, but they also possess rotational kinetic energy. This second type of kinetic energy considers the rotational movement around an axis, making it vital in understanding objects that roll or spin.
For the rotational kinetic energy of a solid ball, the formula is:\[ K_r = \frac{1}{2} I \omega^2 \]Here:

• \( I \) is the moment of inertia, which is a measure of an object's resistance to changes in its rotation. For a solid spherical ball, \( I = \frac{2}{5} m R^2 \).
• \( \omega \) is the angular velocity, or how fast the object spins around its axis, related to the translational speed by \( \omega = \frac{v}{R} \).

This formula shows that rotational kinetic energy also depends on the distribution of the object's mass (moment of inertia) and how quickly it spins (angular velocity).
Understanding both translational and rotational kinetic energy allows us to calculate the total kinetic energy of rolling objects, crucial for solving complex physics problems.

Moment of Inertia

The moment of inertia is a pivotal concept when studying rotational motion, akin to mass in translational motion. It measures how much an object resists changing its rotation and depends significantly on the object's mass distribution about the axis of rotation.
For a solid ball (or sphere), the moment of inertia is calculated as:\[ I = \frac{2}{5} m R^2 \]Where:

• \( m \) denotes the mass of the ball.
• \( R \) stands for the radius of the ball.

This indicates that objects with more mass concentrated further from the axis have higher moments of inertia, making them harder to spin up or slow down.
The moment of inertia is critical when determining rotational kinetic energy for spinning or rolling objects. By analyzing how the mass is distributed and calculating its inertia, we can better predict the rotational behavior of objects, which is essential for balancing energy equations in physics problems, such as a rolling ball's total kinetic energy calculation.