Two uniform, thin identical rods each of mass M and length `l` are joined together to form a cross. What will be the moment of inertia of the cross about an axis passing through the point at which the two rods are joined and perpendicular to the plane of the cross ?

Options

A. `(Ml^(2))/(12)`
B. `(Ml^(2))/(6)`
C. `(Ml^(2))/(4)`
D. `(ML^(2))/(3)`

Explanation

To find the moment of inertia of the cross formed by two uniform, thin identical rods about an axis passing through their joint point and perpendicular to the plane of the cross, we can follow these steps:

### Step 1: Identify the components
We have two identical rods, each with mass \( M \) and length \( l \). They are positioned to form a cross, meaning they intersect at their midpoints.

### Step 2: Moment of Inertia for a Single Rod
The moment of inertia \( I \) of a uniform rod about an axis passing through its center of mass and perpendicular to its length is given by the formula:
\[ I_{CM} = \frac{1}{12} M l^2 \]
where \( M \) is the mass of the rod and \( l \) is its length.

### Step 3: Calculate the Moment of Inertia for Each Rod
Since both rods are identical, the moment of inertia for each rod about its center of mass is:
\[ I_{CM1} = \frac{1}{12} M l^2 \quad \text{(for Rod 1)} \]
\[ I_{CM2} = \frac{1}{12} M l^2 \quad \text{(for Rod 2)} \]

### Step 4: Total Moment of Inertia about the Joint Point
The total moment of inertia about the point where the two rods are joined (point O) can be calculated by adding the moment of inertia of both rods. Since the axis of rotation is through the joint point and perpendicular to the plane of the cross, we can directly add the two moments of inertia:
\[ I_O = I_{CM1} + I_{CM2} = \frac{1}{12} M l^2 + \frac{1}{12} M l^2 = \frac{2}{12} M l^2 = \frac{1}{6} M l^2 \]

### Conclusion
Thus, the moment of inertia of the cross about the axis passing through the point at which the two rods are joined and perpendicular to the plane of the cross is:
\[ I_O = \frac{1}{6} M l^2 \]

### Final Answer
The correct answer is \( \frac{M l^2}{6} \).

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