A thin rod of length `L` and mass `M` is bent at its midpoint into two halves so that the angle between them is `90^@`. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is.

Options

A. `(ML^(2))/(24)`
B. `(ML^(2))/12`
C. `(ML^(2))/6`
D. `(sqrt(2) ML^(2))/(24)`

Answer

To find the moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod, we can follow these steps:

### Step 1: Understand the Configuration
The rod of length \( L \) and mass \( M \) is bent at its midpoint, forming two halves that create a \( 90^\circ \) angle. Each half of the rod will have a length of \( \frac{L}{2} \) and a mass of \( \frac{M}{2} \).

### Step 2: Moment of Inertia of Each Half
The moment of inertia \( I \) of a thin rod about an axis through one end and perpendicular to its length is given by the formula:
\[ I = \frac{1}{3} m l^2 \]
For each half of the rod:
- Mass \( m = \frac{M}{2} \)
- Length \( l = \frac{L}{2} \)

Substituting these values into the formula gives:
\[ I_{\text{half}} = \frac{1}{3} \left(\frac{M}{2}\right) \left(\frac{L}{2}\right)^2 = \frac{1}{3} \left(\frac{M}{2}\right) \left(\frac{L^2}{4}\right) = \frac{M L^2}{24} \]

### Step 3: Total Moment of Inertia
Since there are two halves of the rod, and they are both contributing to the total moment of inertia about the bending point, we need to add the moments of inertia of both halves:
\[ I_{\text{total}} = I_{\text{half}} + I_{\text{half}} = \frac{M L^2}{24} + \frac{M L^2}{24} = \frac{2M L^2}{24} = \frac{M L^2}{12} \]

### Final Answer
Thus, the moment of inertia of the bent rod about the axis passing through the bending point and perpendicular to the plane defined by the two halves is:
\[ I = \frac{M L^2}{12} \]