A ring of mass 10 kg and diameter 0.4 meter is rotating about its geometrical axis at 1200 rotations per minute. Its moment of inertia and angular momentum will be respectively:-

Options

A. 0.4kg `m^(2)` and 50.28 Joule sec
B. 0.4 kg `m^(2)` and 0.4 Joule sec
C. 50.28 kg `m^(2)` and 0.4 Joule sec
D. 0.4 kg `m^(2)` and zero

Answer

To solve the problem, we need to calculate the moment of inertia and angular momentum of a ring. Here’s a step-by-step solution:

### Step 1: Calculate the Radius of the Ring
The diameter of the ring is given as 0.4 meters. To find the radius (r), we use the formula:
\[ r = \frac{\text{diameter}}{2} \]
\[ r = \frac{0.4 \, \text{m}}{2} = 0.2 \, \text{m} \]

**Hint:** Remember that the radius is half of the diameter.

### Step 2: Calculate the Moment of Inertia (I)
The moment of inertia (I) for a ring rotating about its geometrical axis is given by the formula:
\[ I = m \cdot r^2 \]
Where:
- \( m \) = mass of the ring = 10 kg
- \( r \) = radius = 0.2 m

Substituting the values:
\[ I = 10 \, \text{kg} \cdot (0.2 \, \text{m})^2 \]
\[ I = 10 \, \text{kg} \cdot 0.04 \, \text{m}^2 \]
\[ I = 0.4 \, \text{kg} \cdot \text{m}^2 \]

**Hint:** The moment of inertia depends on both the mass and the square of the radius.

### Step 3: Convert Rotations per Minute to Revolutions per Second
The ring is rotating at 1200 rotations per minute (rpm). To convert this to revolutions per second (f), we use:
\[ f = \frac{\text{rpm}}{60} \]
\[ f = \frac{1200}{60} = 20 \, \text{revolutions per second} \]

**Hint:** To convert from minutes to seconds, divide by 60.

### Step 4: Calculate Angular Velocity (ω)
Angular velocity (ω) in radians per second can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of f:
\[ \omega = 2\pi \cdot 20 \]
\[ \omega = 40\pi \, \text{radians per second} \]

**Hint:** Remember that one revolution corresponds to \( 2\pi \) radians.

### Step 5: Calculate Angular Momentum (L)
Angular momentum (L) is given by the formula:
\[ L = I \cdot \omega \]
Substituting the values we have:
\[ L = 0.4 \, \text{kg} \cdot \text{m}^2 \cdot (40\pi) \]
\[ L = 0.4 \cdot 40\pi \]
\[ L = 16\pi \, \text{kg} \cdot \text{m}^2/\text{s} \]

Calculating the numerical value:
\[ L \approx 16 \cdot 3.14 \approx 50.24 \, \text{kg} \cdot \text{m}^2/\text{s} \]

**Hint:** Angular momentum is the product of moment of inertia and angular velocity.

### Final Answers:
- Moment of Inertia (I) = 0.4 kg·m²
- Angular Momentum (L) ≈ 50.24 kg·m²/s