The moment of inertia of a body about a given axis is `2.4 kg–m^(2)`. To produce a rotational kinetic energy of 750 J, an angular acceleration of 5 rad/s2 must be applied about that axis for

Options

A. 6 sec
B. 5 sec
C. 4 sec
D. 3 sec

Answer

To solve the problem, we need to find the time for which an angular acceleration must be applied to produce a given rotational kinetic energy. We will use the formula for rotational kinetic energy and the relationship between angular acceleration, angular velocity, and time.

### Step 1: Understand the relationship between rotational kinetic energy and angular velocity.
The rotational kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} I \omega^2 \]
where:
- \( KE \) is the rotational kinetic energy,
- \( I \) is the moment of inertia,
- \( \omega \) is the angular velocity.

### Step 2: Rearrange the formula to find angular velocity.
We can rearrange the formula to solve for angular velocity (\( \omega \)):
\[ \omega = \sqrt{\frac{2 KE}{I}} \]

### Step 3: Substitute the known values into the equation.
Given:
- \( KE = 750 \, J \)
- \( I = 2.4 \, kg \cdot m^2 \)

Substituting these values into the equation:
\[ \omega = \sqrt{\frac{2 \times 750}{2.4}} = \sqrt{\frac{1500}{2.4}} = \sqrt{625} = 25 \, rad/s \]

### Step 4: Use the relationship between angular acceleration and time.
The angular velocity (\( \omega \)) is related to angular acceleration (\( \alpha \)) and time (\( t \)) by the equation:
\[ \omega = \alpha t \]
Rearranging this gives us:
\[ t = \frac{\omega}{\alpha} \]

### Step 5: Substitute the known values for angular acceleration.
Given:
- \( \alpha = 5 \, rad/s^2 \)

Substituting the values into the equation:
\[ t = \frac{25}{5} = 5 \, s \]

### Final Answer:
The time for which the angular acceleration must be applied is \( 5 \, seconds \).

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