A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minute, its kinetic energy would be :

Question

Accepted Answer

To determine the kinetic energy of a mass moving in a circular path, we need to consider both its rotational and translational motion. The kinetic energy (KE) of an object moving in a circular path can be calculated using the formula for rotational kinetic energy:

$$ KE = \frac{1}{2} m v^2 $$

where $ m $ is the mass of the object and $ v $ is the linear velocity. The linear velocity $ v $ can be related to the angular velocity $ \omega $ by the equation:

$$ v = \omega r $$

where $ r $ is the radius of the circular path. The angular velocity $ \omega $ in radians per second can be calculated from the revolutions per minute (rpm) using the conversion:

$$ \omega = \frac{2\pi \times \text{rpm}}{60} $$

Given:
- Mass $ m = 5 \, \text{kg} $
- Radius $ r = 1 \, \text{m} $
- Revolutions per minute = 300

First, calculate the angular velocity $ \omega $:

$$ \omega = \frac{2\pi \times 300}{60} = 10\pi \, \text{rad/s} $$

Now, calculate the linear velocity $ v $:

$$ v = \omega r = 10\pi \times 1 = 10\pi \, \text{m/s} $$

Substitute $ v $ into the kinetic energy formula:

$$ KE = \frac{1}{2} \times 5 \times (10\pi)^2 $$

$$ KE = \frac{1}{2} \times 5 \times 100\pi^2 $$

$$ KE = \frac{1}{2} \times 500\pi^2 $$

$$ KE = 250\pi^2 $$

Thus, the kinetic energy of the mass is $ 250\pi^2 $, which corresponds to Option 3.

**Explanation of why other options are incorrect:**

1. **Option 1: $100\pi^2$**
   - This value is incorrect because it underestimates the kinetic energy. The calculation shows that the kinetic energy is $ 250\pi^2 $, not $ 100\pi^2 $.

2. **Option 2: 0**
   - This is incorrect because a moving mass cannot have zero kinetic energy unless it is stationary. Since the mass is moving with 300 revolutions per minute, it has a non-zero kinetic energy.

3. **Option 4: $6\pi^2$**
   - This value is incorrect because it significantly underestimates the kinetic energy. The correct calculation yields $ 250\pi^2 $, not $ 6\pi^2 $.

Therefore, the correct answer is Option 3: $ 250\pi^2 $.