A tennis racket can be idealized as a uniform ring of mass M and radius R, attached to a uniform rod also of mass M and length L. The rod and the ring are coplanar, and the line of the rod passes through the centre of the ring. The moment of inertia of the object (racket) about an axis through the centre of the ring and perpendicular to its plane is

Options

A. `(1)/(12)M(6R^(2)+L^(2))`
B. `(1)/(12)M(18R^(2)+L^(2))`
C. `(1)/(3)M(6R^(2)+L^(2)+3LR)`
D. None of these

Explanation

To find the moment of inertia of the tennis racket, which consists of a uniform ring and a uniform rod, we will follow these steps:

### Step 1: Identify the components
The tennis racket consists of:
- A uniform ring with mass \( M \) and radius \( R \).
- A uniform rod with mass \( M \) and length \( L \).

### Step 2: Calculate the moment of inertia of the ring
The moment of inertia \( I_{\text{ring}} \) of a uniform ring about an axis perpendicular to its plane and passing through its center is given by the formula:
\[ I_{\text{ring}} = M R^2 \]

### Step 3: Calculate the moment of inertia of the rod about its own center
The moment of inertia \( I_{\text{rod, cm}} \) of a uniform rod about an axis through its center and perpendicular to its length is given by:
\[ I_{\text{rod, cm}} = \frac{1}{12} M L^2 \]

### Step 4: Use the Parallel Axis Theorem for the rod
To find the moment of inertia of the rod about the axis through the center of the ring, we will use the Parallel Axis Theorem. The distance \( d \) from the center of the rod to the center of the ring is:
\[ d = \frac{L}{2} + R \]
According to the Parallel Axis Theorem:
\[ I_{\text{rod}} = I_{\text{rod, cm}} + M d^2 \]
Substituting the values:
\[ I_{\text{rod}} = \frac{1}{12} M L^2 + M \left(\frac{L}{2} + R\right)^2 \]

### Step 5: Expand the equation for the rod's moment of inertia
Expanding \( \left(\frac{L}{2} + R\right)^2 \):
\[ \left(\frac{L}{2} + R\right)^2 = \left(\frac{L^2}{4} + LR + R^2\right) \]
Thus,
\[ I_{\text{rod}} = \frac{1}{12} M L^2 + M \left(\frac{L^2}{4} + LR + R^2\right) \]

### Step 6: Combine the moments of inertia
Now, we can combine the moments of inertia of the ring and the rod:
\[ I_{\text{total}} = I_{\text{ring}} + I_{\text{rod}} \]
Substituting the values:
\[ I_{\text{total}} = M R^2 + \left(\frac{1}{12} M L^2 + M \left(\frac{L^2}{4} + LR + R^2\right)\right) \]

### Step 7: Simplify the total moment of inertia
Combining all terms:
\[ I_{\text{total}} = M R^2 + \frac{1}{12} M L^2 + M \frac{L^2}{4} + M LR + M R^2 \]
\[ I_{\text{total}} = 2 M R^2 + \left(\frac{1}{12} M L^2 + \frac{3}{12} M L^2\right) + M LR \]
\[ I_{\text{total}} = 2 M R^2 + \frac{4}{12} M L^2 + M LR \]
\[ I_{\text{total}} = 2 M R^2 + \frac{1}{3} M L^2 + M LR \]

### Final Result
Thus, the moment of inertia of the tennis racket about the specified axis is:
\[ I_{\text{total}} = M \left(2 R^2 + \frac{1}{3} L^2 + LR\right) \]

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