A flywheel is making `(3000)/(pi)` revolutions per minute about its axis. If the moment of inertia of the flywheel about that axis is 400 `kgm^(2)`, its rotational kinetic energy is :

Options

A. `2xx10^(6)J`
B. `3xx10^(3)J`
C. `5xx10^(3)J`
D. `12xx10^(3)J`

Answer

To find the rotational kinetic energy of the flywheel, we can follow these steps:

### Step 1: Convert Revolutions Per Minute to Radians Per Second
The angular velocity (ω) in radians per second can be calculated using the formula:
\[ \omega = \frac{2\pi n}{60} \]
where \( n \) is the number of revolutions per minute.

Given \( n = \frac{3000}{\pi} \) revolutions per minute, we substitute this value into the formula:
\[ \omega = \frac{2\pi \left(\frac{3000}{\pi}\right)}{60} \]

### Step 2: Simplify the Expression
Now, simplify the expression:
\[ \omega = \frac{2 \times 3000}{60} = \frac{6000}{60} = 100 \text{ radians/second} \]

### Step 3: Use the Formula for Rotational Kinetic Energy
The rotational kinetic energy (K.E.) is given by the formula:
\[ K.E. = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia.

Given \( I = 400 \, \text{kgm}^2 \) and \( \omega = 100 \, \text{radians/second} \), we can substitute these values into the formula:
\[ K.E. = \frac{1}{2} \times 400 \times (100)^2 \]

### Step 4: Calculate the Kinetic Energy
Now, calculate the kinetic energy:
\[ K.E. = \frac{1}{2} \times 400 \times 10000 = 200 \times 10000 = 2000000 \, \text{Joules} \]

### Step 5: Convert to Kilojoules
To convert joules to kilojoules, we divide by 1000:
\[ K.E. = \frac{2000000}{1000} = 2000 \, \text{kJ} \]

### Final Answer
The rotational kinetic energy of the flywheel is:
\[ K.E. = 2 \times 10^6 \, \text{Joules} \quad \text{or} \quad 2000 \, \text{kJ} \]