The moment of inertia of a body about a given axis is 1.2 kg `xx " metre"^(2)`. Initially, the body is at rest. In order to produce a rotating kinetic energy of 1500 joules, an angular acceleration of 25 radian/ `sec^(2)` must be applied about that axis for a duration of

Options

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

Explanation

To solve the problem step by step, we need to find the time required to achieve a rotating kinetic energy of 1500 joules given the moment of inertia and angular acceleration.

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

### Step 2: Substitute the given values into the equation.
We know:
- \( KE = 1500 \, \text{joules} \)
- \( I = 1.2 \, \text{kg} \cdot \text{m}^2 \)

Substituting these values into the kinetic energy formula:
\[ 1500 = \frac{1}{2} \times 1.2 \times \omega^2 \]

### Step 3: Solve for \( \omega^2 \).
To isolate \( \omega^2 \), we first multiply both sides by 2:
\[ 3000 = 1.2 \times \omega^2 \]

Now, divide both sides by 1.2:
\[ \omega^2 = \frac{3000}{1.2} \]

Calculating this gives:
\[ \omega^2 = 2500 \]

### Step 4: Find \( \omega \).
Taking the square root of both sides:
\[ \omega = \sqrt{2500} = 50 \, \text{radians/second} \]

### Step 5: Use the angular acceleration to find the time.
We know the angular acceleration (\( \alpha \)) is given as:
\[ \alpha = 25 \, \text{radians/second}^2 \]

Using the formula for angular velocity:
\[ \omega_f = \omega_i + \alpha t \]
where:
- \( \omega_f \) is the final angular velocity,
- \( \omega_i \) is the initial angular velocity (which is 0 since the body is at rest),
- \( t \) is the time.

Substituting the known values:
\[ 50 = 0 + 25t \]

### Step 6: Solve for \( t \).
Rearranging the equation gives:
\[ 50 = 25t \]
Dividing both sides by 25:
\[ t = \frac{50}{25} = 2 \, \text{seconds} \]

### Final Answer:
The duration required to produce a rotating kinetic energy of 1500 joules is **2 seconds**.

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