Question

A body is rotating about its own axis. Its rotational kinetic energy is \( x \) and its angular momentum is \( y \). Hence its moment of inertia about its own axis is

• A. \( \frac{x}{y^2} \)
• B. \( \frac{y^2}{x} \)
• C. \( \frac{x}{y} \)
• D. \( \frac{y^2}{x^2} \)

Hint:

The moment of inertia can be determined by rearranging the formula for rotational kinetic energy and using the relationship between kinetic energy and angular momentum.

Answer 1

The Correct Option is B

Step 1: Understanding the relationship between kinetic energy and angular momentum.
The rotational kinetic energy \( K \) of a rotating object is given by the formula:
\[ K = \frac{1}{2} I \omega^2, \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. The angular momentum \( L \) is related to the moment of inertia and angular velocity by:
\[ L = I \omega. \]
Thus, we can express \( \omega \) in terms of \( L \) and \( I \):
\[ \omega = \frac{L}{I}. \]

Step 2: Substituting into the equation for kinetic energy.
Substitute the expression for \( \omega \) into the formula for kinetic energy:
\[ K = \frac{1}{2} I \left( \frac{L}{I} \right)^2 = \frac{1}{2} \frac{L^2}{I}. \]

Step 3: Relating the given quantities to kinetic energy.
We are given that the rotational kinetic energy is \( x \) and the angular momentum is \( y \). Thus, we have:
\[ x = \frac{1}{2} \frac{y^2}{I}. \]
Solving for \( I \): \[ I = \frac{y^2}{2x}. \]
Final Answer:
Thus, the moment of inertia is:
\[ \boxed{\frac{y^2}{2x}}. \]