Problem 21 - University Physics with Modern Physics

Question

What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? (a) a uniform solid cylinder; (b) a uniform sphere; ( c) a thin walled, hollow sphere; (d) a hollow cylinder with outer radius \(R\) and inner radius \(R / 2\) .

Step-by-Step Solution

Answer

(a) \(\frac{1}{3}\), (b) \(\frac{2}{7}\), (c) \(\frac{2}{5}\), (d) \(\frac{3}{7}\).

Step 1: Identify kinetic energy components

The total kinetic energy for an object rolling without slipping is the sum of translational kinetic energy and rotational kinetic energy. It can be expressed as \( KE_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \), where \( m \) is the mass, \( v \) is the linear velocity, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity.

Step 2: Relate linear and angular velocities

Since the object is rolling without slipping, the linear velocity \( v \) is related to the angular velocity \( \omega \) by the equation \( v = \omega R \), where \( R \) is the radius of the object. Substitute \( \omega = \frac{v}{R} \) into the rotational kinetic energy term.

Step 3: Define fraction of rotational kinetic energy

The fraction of rotational kinetic energy to the total kinetic energy is given by \( \text{Fraction}_{rotational} = \frac{\frac{1}{2}I\omega^2}{\frac{1}{2}mv^2 + \frac{1}{2}I\omega^2} \). Substitute \( \omega = \frac{v}{R} \) for each object.

Step 4: Calculate for solid cylinder

For a solid cylinder, the moment of inertia \( I = \frac{1}{2} mR^2 \). Substitute this value in:\[ \text{Fraction}_{solid, cylinder} = \frac{\frac{1}{2} \cdot \frac{1}{2} mR^2 \cdot \left( \frac{v}{R} \right)^2}{\frac{1}{2}mv^2 + \frac{1}{2} \cdot \frac{1}{2} mR^2 \cdot \left( \frac{v}{R} \right)^2} = \frac{1/4}{3/4} = \frac{1}{3} \]

Step 5: Calculate for uniform sphere

For a uniform sphere, the moment of inertia \( I = \frac{2}{5}mR^2 \). Substitute this in:\[ \text{Fraction}_{uniform, sphere} = \frac{\frac{1}{2} \cdot \frac{2}{5} mR^2 \cdot \left( \frac{v}{R} \right)^2}{\frac{1}{2}mv^2 + \frac{1}{2} \cdot \frac{2}{5} mR^2 \cdot \left( \frac{v}{R} \right)^2} = \frac{1/5}{7/10} = \frac{2}{7} \]

Step 6: Calculate for hollow sphere

For a thin, hollow sphere, the moment of inertia \( I = \frac{2}{3}mR^2 \). Substitute in:\[ \text{Fraction}_{hollow, sphere} = \frac{\frac{1}{2} \cdot \frac{2}{3} mR^2 \cdot \left( \frac{v}{R} \right)^2}{\frac{1}{2}mv^2 + \frac{1}{2} \cdot \frac{2}{3} mR^2 \cdot \left( \frac{v}{R} \right)^2} = \frac{1/3}{5/6} = \frac{2}{5} \]

Step 7: Calculate for hollow cylinder

The hollow cylinder, with an outer radius \( R \) and inner radius \( \frac{R}{2} \), has an inertia of \( I = \frac{3}{4}mR^2 \). Substitute this in:\[ \text{Fraction}_{hollow, cylinder} = \frac{\frac{1}{2} \cdot \frac{3}{4} mR^2 \cdot \left( \frac{v}{R} \right)^2}{\frac{1}{2}mv^2 + \frac{1}{2} \cdot \frac{3}{4} mR^2 \cdot \left( \frac{v}{R} \right)^2} = \frac{3/8}{7/8} = \frac{3}{7} \]

Key Concepts

Moment of Inertia

The moment of inertia is a crucial concept in rotational dynamics. Think of it as the rotational equivalent of mass in linear motion. It measures how much torque is required for a desired angular acceleration about a rotational axis. The moment of inertia depends not only on the mass but also on how that mass is distributed relative to the axis of rotation.

For different objects, the moment of inertia varies based on their shapes and mass distributions:

• **Solid Cylinder:** The moment of inertia is calculated using the formula \( I = \frac{1}{2}mR^2 \). This reflects the mass being concentrated more towards the axis.
• **Uniform Sphere:** Here, \( I = \frac{2}{5}mR^2 \), indicating a more even distribution of mass.
• **Thin-Walled, Hollow Sphere:** With \( I = \frac{2}{3}mR^2 \), the mass is concentrated more towards the sphere's surface.
• **Hollow Cylinder:** Having an outer and inner radius, its moment of inertia is \( I = \frac{3}{4}mR^2 \), reflecting the distributed mass between the shell and the hollow part.

Understanding the moment of inertia is vital in determining how different objects accelerate when subjected to rotational forces.

Rolling Without Slipping

Rolling without slipping is a condition where an object rolls on a surface without sliding. It's crucial for understanding the energy distribution in rolling objects, as it establishes a specific relationship between linear and angular velocities.

In this case, the point of contact between the object and the surface is momentarily at rest. This condition is mathematically expressed as:

• \( v = \omega R \)

where \( v \) is the linear velocity, \( \omega \) is the angular velocity, and \( R \) is the radius of the object.

This means the linear speed of the object is dictated by its rate of rotation and its size. When an object rolls without slipping:

• The translational kinetic energy is \( \frac{1}{2}mv^2 \).
• The rotational kinetic energy is \( \frac{1}{2}I\omega^2 \).

This relationship lets us calculate the energy associated with both the movement and the spinning of the object, ensuring calculations accurately reflect real-world motions.

Fraction of Kinetic Energy

The concept of the fraction of kinetic energy aims at understanding how much of the total kinetic energy of a rolling object is due to its rotation. It's significant for evaluating energy distribution and is useful for design and analysis in mechanical systems.

To find this fraction, you typically start with the expression:

• Total Kinetic Energy, \( KE_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \)
• Fraction of rotational kinetic energy, \( \text{Fraction}_{rotational} = \frac{\frac{1}{2}I\omega^2}{\frac{1}{2}mv^2 + \frac{1}{2}I\omega^2} \)

By substituting \( \omega = \frac{v}{R} \), you can calculate this fraction for different shapes:

• For a **solid cylinder**, it is \( \frac{1}{3} \).
• A **uniform sphere** has a fraction of \( \frac{2}{7} \).
• The **hollow sphere** results in \( \frac{2}{5} \).
• Lastly, a **hollow cylinder** gives \( \frac{3}{7} \).

These ratios tell us how much of their motion energy is spinning compared to moving along a path. Recognizing this helps in fields like biomechanics, vehicle dynamics, and robotics, where efficient energy utilization is critical.