Calculate the rotational kinetic energy of a body of mass 2 kg rotating on a circular path of diameter 4 m at the rate of 50 rotations in 20 seconds.

Answer

To calculate the rotational kinetic energy of the body, we can follow these steps:

### Step 1: Identify the given data
- Mass of the body (m) = 2 kg
- Diameter of the circular path = 4 m
- Radius (r) = Diameter / 2 = 4 m / 2 = 2 m
- Rotations in 20 seconds = 50

### Step 2: Calculate the moment of inertia (I)
The moment of inertia for a point mass rotating about an axis is given by the formula:
\[ I = m \cdot r^2 \]
Substituting the values:
\[ I = 2 \, \text{kg} \cdot (2 \, \text{m})^2 = 2 \cdot 4 = 8 \, \text{kg m}^2 \]

### Step 3: Calculate the frequency (f)
The frequency (f) is the number of rotations per second. Given that the body makes 50 rotations in 20 seconds:
\[ f = \frac{50 \, \text{rotations}}{20 \, \text{seconds}} = 2.5 \, \text{rotations/second} \]

### Step 4: Convert frequency to angular velocity (ω)
Angular velocity (ω) in radians per second can be calculated using the formula:
\[ \omega = 2 \pi f \]
Substituting the frequency:
\[ \omega = 2 \pi \cdot 2.5 = 5 \pi \, \text{radians/second} \]

### Step 5: Calculate the rotational kinetic energy (KE)
The rotational kinetic energy is given by the formula:
\[ KE = \frac{1}{2} I \omega^2 \]
Substituting the values of I and ω:
\[ KE = \frac{1}{2} \cdot 8 \cdot (5 \pi)^2 \]
\[ KE = 4 \cdot 25 \pi^2 \]
\[ KE = 100 \pi^2 \]

### Step 6: Calculate the numerical value
Using the approximation \( \pi \approx 3.14 \):
\[ KE \approx 100 \cdot (3.14)^2 \]
\[ KE \approx 100 \cdot 9.8596 \]
\[ KE \approx 985.96 \, \text{J} \]
Rounding to the nearest whole number:
\[ KE \approx 986 \, \text{J} \]

### Final Answer
The rotational kinetic energy of the body is approximately **986 Joules**.

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