A wheel 2 kg having practically all the mass concentrated along the circumference of a circle of radius 20 cm is rotating on its axis with angular velocity of 100 rad/s, then the rotational kinetic energy of wheel is

Options

A. 4 J
B. 70 J
C. 400 J
D. 800 J

Answer

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

### Step 1: Identify the parameters
We have the following parameters:
- Mass of the wheel (m) = 2 kg
- Radius of the wheel (r) = 20 cm = 0.2 m (convert cm to m)
- Angular velocity (ω) = 100 rad/s

### Step 2: Determine the moment of inertia (I)
For a ring (which is the case here since the mass is concentrated along the circumference), the moment of inertia is given by the formula:
\[ I = m r^2 \]
Substituting the values:
\[ I = 2 \, \text{kg} \times (0.2 \, \text{m})^2 \]
\[ I = 2 \times 0.04 \]
\[ I = 0.08 \, \text{kg m}^2 \]

### Step 3: Use the formula for rotational kinetic energy (K.E.)
The formula for rotational kinetic energy is:
\[ K.E. = \frac{1}{2} I \omega^2 \]
Substituting the values we found:
\[ K.E. = \frac{1}{2} \times 0.08 \, \text{kg m}^2 \times (100 \, \text{rad/s})^2 \]
\[ K.E. = \frac{1}{2} \times 0.08 \times 10000 \]
\[ K.E. = 0.04 \times 10000 \]
\[ K.E. = 400 \, \text{J} \]

### Step 4: Conclusion
The rotational kinetic energy of the wheel is 400 Joules.

---