To solve the problem step by step, we will analyze the given data and calculate the required quantities.
### Given:
- Mass of the ring (m) = 100 kg
- Diameter of the ring = 2 m, hence the radius (r) = 1 m (since radius = diameter / 2)
- Rotational speed = \( \frac{300}{\pi} \) rpm
### Step 1: Calculate the Moment of Inertia (I)
The moment of inertia (I) for a ring about an axis passing through its center and perpendicular to its plane is given by the formula:
\[ I = m r^2 \]
Substituting the values:
\[ I = 100 \, \text{kg} \times (1 \, \text{m})^2 = 100 \, \text{kg m}^2 \]
### Step 2: Convert Angular Speed to Radians per Second
The angular speed in radians per second (\( \omega \)) can be calculated from the given rpm using the conversion factor:
\[ \omega = \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{minute}} \]
First, convert rpm to seconds:
\[ \omega = \frac{300}{\pi} \, \text{rpm} \times \frac{2\pi}{60} \]
Calculating this:
\[ \omega = \frac{300 \times 2\pi}{60\pi} = \frac{600}{60} = 10 \, \text{radians/second} \]
### Step 3: Calculate Kinetic Energy (KE)
The kinetic energy (KE) of a rotating body is given by:
\[ KE = \frac{1}{2} I \omega^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \times 100 \, \text{kg m}^2 \times (10 \, \text{radians/second})^2 \]
\[ KE = \frac{1}{2} \times 100 \times 100 = 5000 \, \text{J} = 5 \, \text{kJ} \]
### Step 4: Calculate Angular Deceleration (α) due to Retarding Torque
Given a retarding torque (\( \tau \)) of 200 Nm, we can find the angular deceleration (\( \alpha \)) using:
\[ \alpha = \frac{\tau}{I} \]
Substituting the values:
\[ \alpha = \frac{200 \, \text{Nm}}{100 \, \text{kg m}^2} = 2 \, \text{radians/second}^2 \]
### Step 5: Calculate Time to Come to Rest
Using the first equation of motion for angular motion:
\[ \omega_f = \omega_i - \alpha t \]
Where:
- \( \omega_f = 0 \) (final angular speed when it comes to rest)
- \( \omega_i = 10 \, \text{radians/second} \)
- \( \alpha = 2 \, \text{radians/second}^2 \)
Rearranging the equation to find time (t):
\[ 0 = 10 - 2t \]
\[ 2t = 10 \]
\[ t = 5 \, \text{seconds} \]
### Conclusion
All calculated values are consistent with the statements provided in the question:
- Moment of inertia = 100 kg m²
- Kinetic energy = 5 kJ
- Time to come to rest = 5 seconds
Thus, the correct option is D, which states that all of these are given.
