Question

The moment of inertia and rotational kinetic energy of a rigid body are 4 $kgm^2$ and 50 J respectively. The angular velocity of the body (in rad $s^{-1}$) is:

• A. 10
• B. 20
• C. 25
• D. 5
• E. 15

Hint:

This is the rotational analog of linear kinetic energy $K = \frac{1}{2} mv^2$, where $I$ replaces $m$ and $\omega$ replaces $v$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem relates the rotational kinetic energy of a rigid body to its moment of inertia and angular velocity. We need to use the formula for rotational kinetic energy to find the unknown angular velocity.
Step 2: Key Formula or Approach:
The rotational kinetic energy (\( K_{rot} \)) of a body is given by: \[ K_{rot} = \frac{1}{2} I \omega^2 \] where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. We need to rearrange this formula to solve for \( \omega \).
\[ \omega^2 = \frac{2 K_{rot}}{I} \implies \omega = \sqrt{\frac{2 K_{rot}}{I}} \] Step 3: Detailed Explanation:
We are given: - Moment of inertia, \( I = 4 \text{ kg m}^2 \) - Rotational kinetic energy, \( K_{rot} = 50 \text{ J} \) Substitute these values into the rearranged formula for angular velocity: \[ \omega = \sqrt{\frac{2 \times 50}{4}} \] \[ \omega = \sqrt{\frac{100}{4}} \] \[ \omega = \sqrt{25} \] \[ \omega = 5 \text{ rad/s} \] Step 4: Final Answer:
The angular velocity of the body is 5 rad s\(^{-1}\).