The moment of inertia of the body about an axis is 1.2 kg `m^(2)`. Initially the body is at rest. In order to produce a rotational kinetic energy of 1500J, an angualr acceleration of 25 `rad/s^(2)` must be applied about the axis for the duration of

Options

A. 4s
B. 2s
C. 8s
D. 10s

Explanation

To solve the problem step by step, we will follow the process of calculating the final angular velocity needed to achieve the given rotational kinetic energy, and then use the equations of motion to find the time duration for which the angular acceleration must be applied.

### Step 1: Use the formula for rotational kinetic energy
The formula for rotational kinetic energy (KE) is given by:

\[ KE = \frac{1}{2} I \omega^2 \]

Where:
- \( KE \) is the rotational kinetic energy,
- \( I \) is the moment of inertia,
- \( \omega \) is the final angular velocity.

Given:
- \( KE = 1500 \, J \)
- \( I = 1.2 \, kg \cdot m^2 \)

### Step 2: Rearrange the formula to solve for \( \omega^2 \)
Substituting the known values into the rotational kinetic energy formula:

\[ 1500 = \frac{1}{2} \times 1.2 \times \omega^2 \]

Multiply both sides by 2 to eliminate the fraction:

\[ 3000 = 1.2 \times \omega^2 \]

Now, divide both sides by 1.2:

\[ \omega^2 = \frac{3000}{1.2} \]

Calculating the right side:

\[ \omega^2 = 2500 \]

### Step 3: Calculate \( \omega \)
Taking the square root of both sides to find \( \omega \):

\[ \omega = \sqrt{2500} = 50 \, rad/s \]

### Step 4: Use the equation of motion for rotational motion
Now, we will use the first equation of rotational motion:

\[ \omega = \omega_0 + \alpha t \]

Where:
- \( \omega_0 \) is the initial angular velocity (which is 0 since the body is initially at rest),
- \( \alpha \) is the angular acceleration,
- \( t \) is the time duration.

Given:
- \( \omega = 50 \, rad/s \)
- \( \omega_0 = 0 \, rad/s \)
- \( \alpha = 25 \, rad/s^2 \)

### Step 5: Substitute the values into the equation
Substituting the known values into the equation:

\[ 50 = 0 + 25t \]

### Step 6: Solve for \( t \)
Now, solving for \( t \):

\[ 50 = 25t \]

Dividing both sides by 25:

\[ t = \frac{50}{25} = 2 \, seconds \]

### Final Answer
The time duration for which the angular acceleration must be applied is:

\[ t = 2 \, seconds \]

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