Problem 51 - College Physics

Question

\(\bullet\) 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) \( \frac{1}{4} \), (b) \( \frac{1}{5} \), (c) \( \frac{1}{3} \), (d) \( \frac{3}{8} \).

Step 1: Understanding kinetic energy for rolling objects

For a rolling object, the total kinetic energy is the sum of translational and rotational kinetic energies. The translational kinetic energy is given by \( KE_{trans} = \frac{1}{2}mv^2 \), and the rotational kinetic energy is given by \( KE_{rot} = \frac{1}{2}I\omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. For rolling without slipping, \( v = R\omega \), where \( R \) is the radius of the object.

Step 2: Calculate rotational kinetic energy fraction for a solid cylinder

For a uniform solid cylinder, the moment of inertia is \( I = \frac{1}{2}mR^2 \). Substituting \( I \) into the rotational kinetic energy formula, we have \( KE_{rot} = \frac{1}{2} \cdot \frac{1}{2}mR^2 \cdot \left( \frac{v}{R} \right)^2 = \frac{1}{4}mv^2 \). The total kinetic energy is \( KE_{total} = \frac{1}{2}mv^2 + \frac{1}{4}mv^2 = \frac{3}{4}mv^2 \). Thus, the fraction is \( \frac{1}{4} \).

Step 3: Calculate rotational kinetic energy fraction for a uniform sphere

For a uniform sphere, the moment of inertia is \( I = \frac{2}{5}mR^2 \). Plugging this in, \( KE_{rot} = \frac{1}{2} \cdot \frac{2}{5}mR^2 \cdot \left( \frac{v}{R} \right)^2 = \frac{1}{5}mv^2 \). Total kinetic energy is \( KE_{total} = \frac{1}{2}mv^2 + \frac{1}{5}mv^2 = \frac{7}{10}mv^2 \). The rotational fraction is \( \frac{1}{5} \).

Step 4: Calculate rotational kinetic energy fraction for a thin-walled, hollow sphere

For a thin-walled, hollow sphere, the moment of inertia is \( I = \frac{2}{3}mR^2 \). Therefore, \( KE_{rot} = \frac{1}{2} \cdot \frac{2}{3}mR^2 \cdot \left( \frac{v}{R} \right)^2 = \frac{1}{3}mv^2 \). Total kinetic energy is \( \frac{1}{2}mv^2 + \frac{1}{3}mv^2 = \frac{5}{6}mv^2 \). The rotational fraction is \( \frac{1}{3} \).

Step 5: Calculate rotational kinetic energy fraction for a hollow cylinder

For a hollow cylinder with outer radius \( R \) and inner radius \( R/2 \), the moment of inertia is \( I = \frac{3}{4}mR^2 \). Hence, \( KE_{rot} = \frac{1}{2} \cdot \frac{3}{4}mR^2 \cdot \left( \frac{v}{R} \right)^2 = \frac{3}{8}mv^2 \). The total kinetic energy is \( KE_{total} = \frac{1}{2}mv^2 + \frac{3}{8}mv^2 = \frac{7}{8}mv^2 \). Thus, the fraction is \( \frac{3}{8} \).

Key Concepts

Rotational Kinetic Energy

When an object rolls on a surface, it exhibits rotational kinetic energy alongside translational kinetic energy. This energy is tied to the rotation of the object around its axis. To put it simply, rotational kinetic energy is the energy due to the object's spinning. It's calculated using the equation:\[KE_{rot} = \frac{1}{2} I \omega^2\]Where:

• \( KE_{rot} \) is the rotational kinetic energy.
• \( I \) is the moment of inertia, which depends on the mass distribution of the object.
• \( \omega \) is the angular velocity, a measure of how fast the object is rotating.

Understanding the distribution between rotational and translational kinetic energy is key to solving problems like the one given in the exercise. Depending on the shape of the object, a different fraction of its total kinetic energy will be accounted for by its rotation.

Moment of Inertia

Moment of inertia, often symbolized as \( I \), tells us how resistant an object is to changing its rotational state. Think of it as the rotational equivalent of mass in linear motion. For different shapes, whether they are solid or hollow, the moment of inertia varies because the mass is distributed differently. Here are some common forms:

• For a solid cylinder, it's \( I = \frac{1}{2}mR^2 \).
• For a solid sphere, \( I = \frac{2}{5}mR^2 \).
• For a hollow, thin-walled sphere, \( I = \frac{2}{3}mR^2 \).
• A hollow cylinder with outer radius \( R \) and inner radius \( R/2 \) has \( I = \frac{3}{4}mR^2 \).

The moment of inertia is crucial when calculating rotational kinetic energy, as it directly influences how much energy is stored or needed for rotation at a given speed.

Rolling Without Slipping

Rolling without slipping is a term that describes the motion of an object, like a ball or wheel, that rolls on a surface without sliding or skidding. In this condition, there's a connection between the translational and rotational motion of the object. This leads to the following important relationship:\[v = R \omega\]Where:

• \( v \) is the linear velocity of the object's center of mass.
• \( R \) is the radius of the object.
• \( \omega \) is its angular velocity.

The no-slip condition ensures that for every unit of distance that the object rolls forward, it rotates by a corresponding angular distance. This link is vital for solving problems related to rolling objects, as it allows converting between rotational and translational quantities, enabling the calculation of total kinetic energy.