Question

A rigid body rotates about a fixed axis under a constant moment (M). If the Mass Moment of Inertia (I) of the body is doubled while the moment remains the same, the angular acceleration ($\alpha$) will

• A. double
• B. remain the same
• C. be halved
• D. quadruple

Hint:

Moment of inertia is the rotational analog of mass.
Just as doubling the mass halves the linear acceleration for a constant force ($a = F/m$), doubling the moment of inertia halves the angular acceleration for a constant torque ($\alpha = M/I$).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks how the angular acceleration ($\alpha$) of a rotating rigid body changes if its mass moment of inertia ($I$) is doubled while the driving moment ($M$) is held constant.

Step 2: Key Formula or Approach:
We apply Newton's second law for rotational motion:
\[ M = I \alpha \]
where:
$M$ is the applied torque or moment.
$I$ is the mass moment of inertia.
$\alpha$ is the angular acceleration.

Step 3: Detailed Explanation:

• Rearranging the rotational equation of motion to solve for angular acceleration gives:
\[ \alpha = \frac{M}{I} \]

• Since the moment $M$ remains constant, the angular acceleration $\alpha$ is inversely proportional to the mass moment of inertia $I$:
\[ \alpha \propto \frac{1}{I} \]

• Let the initial moment of inertia be $I_1$ and the initial angular acceleration be $\alpha_1 = \frac{M}{I_1}$.

• The new moment of inertia is doubled: $I_2 = 2I_1$.

• The new angular acceleration is:
\[ \alpha_2 = \frac{M}{I_2} = \frac{M}{2I_1} = \frac{1}{2}\alpha_1 \]

• Thus, doubling the moment of inertia reduces the angular acceleration to half of its initial value.

Step 4: Final Answer:
The angular acceleration ($\alpha$) will be halved.