Question

A thin spherical shell of radius \( 0.5 \, \text{m} \) and mass \( 2 \, \text{kg} \) is rotating about its axis of symmetry with an angular velocity of \( 10 \, \text{rad/s} \). What is its moment of inertia?

• A. \( 1 \, \text{kg} \cdot \text{m}^2 \)
• B. \(0.5 \, \text{kg} \cdot \text{m}^2 \)
• C. \( 2.0 \, \text{kg} \cdot \text{m}^2 \)
• D. \( 4.0 \, \text{kg} \cdot \text{m}^2 \)

Hint:

Remember: The moment of inertia for a thin spherical shell depends on its mass and radius, and it differs from that of a solid sphere.

Answer 1

The Correct Option is A

Calculate Moment of Inertia: Thin Spherical Shell

Input parameters:

• Spherical shell radius: \( r = 0.5 \, \text{m} \)
• Spherical shell mass: \( m = 2 \, \text{kg} \)

Procedure:

Step 1: Identify the moment of inertia formula

The formula for the moment of inertia of a thin spherical shell about its axis of symmetry is: \[ I = \frac{2}{3} m r^2 \]

Step 2: Apply the given values

\[ I = \frac{2}{3} \times 2 \, \text{kg} \times (0.5 \, \text{m})^2 = \frac{2}{3} \times 2 \times 0.25 \, \text{kg} \cdot \text{m}^2 = \frac{2}{3} \times 0.5 \, \text{kg} \cdot \text{m}^2 = 1.0 \, \text{kg} \cdot \text{m}^2 \]

Result:

The calculated moment of inertia for the spherical shell is \( 1.0 \, \text{kg} \cdot \text{m}^2 \).