The moment of inertia of a uniform semicircular wire of mass m and radius r, about an axis passing through its centre of mass and perpendicular to its plane is `mr^(2)(-(k)/(pi^(2)))`. Find the value of k.

Options

A. `(mr^(2))/(2)`
B. `mr^(2)`
C. `mr^(2)(1-(4)/(pi^(2)))`
D. `mr^(2)(1+(4)/(pi^(2)))`

Answer

To find the value of \( k \) in the moment of inertia of a uniform semicircular wire of mass \( m \) and radius \( r \) about an axis passing through its center of mass and perpendicular to its plane, we can follow these steps:

### Step 1: Understand the Geometry
We have a semicircular wire with mass \( m \) and radius \( r \). The center of mass (COM) of the semicircular wire lies along the vertical axis at a distance \( d \) from the center of the semicircle.

### Step 2: Find the Distance to the Center of Mass
The distance \( d \) of the center of mass from the center of the semicircle can be calculated using the formula:
\[ d = \frac{2r}{\pi} \]

### Step 3: Apply the Parallel Axis Theorem
To find the moment of inertia \( I_O \) about the axis through the center of mass, we can use the parallel axis theorem:
\[ I_O = I_{CM} + m d^2 \]
where \( I_{CM} \) is the moment of inertia about the center of mass, and \( m d^2 \) accounts for the shift in the axis.

### Step 4: Calculate the Moment of Inertia About the Center of Mass
The moment of inertia \( I_{CM} \) of a semicircular wire about its center of mass is given by:
\[ I_{CM} = \frac{1}{2} m r^2 \]

### Step 5: Substitute Values into the Parallel Axis Theorem
Now substituting the values into the parallel axis theorem:
\[ I_O = \frac{1}{2} m r^2 + m \left(\frac{2r}{\pi}\right)^2 \]
Calculating \( m \left(\frac{2r}{\pi}\right)^2 \):
\[ m \left(\frac{2r}{\pi}\right)^2 = m \cdot \frac{4r^2}{\pi^2} \]

### Step 6: Combine the Terms
Now we combine the terms:
\[ I_O = \frac{1}{2} m r^2 + \frac{4m r^2}{\pi^2} \]
To combine these, we need a common denominator:
\[ I_O = \frac{\pi^2}{2\pi^2} m r^2 + \frac{4m r^2}{\pi^2} = \frac{\pi^2 + 8}{2\pi^2} m r^2 \]

### Step 7: Express in the Required Form
We need to express this in the form \( m r^2 \left(1 - \frac{k}{\pi^2}\right) \):
\[ I_O = m r^2 \left( \frac{1}{2} + \frac{4}{\pi^2} \right) = m r^2 \left( 1 - \frac{4}{\pi^2} \right) \]

### Step 8: Identify \( k \)
From the expression, we can see that:
\[ 1 - \frac{k}{\pi^2} = \frac{1}{2} + \frac{4}{\pi^2} \]
Thus, we can equate:
\[ k = 4 \]

### Final Answer
The value of \( k \) is \( 4 \).

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