Two spheres of same mass and radius are in contact with each other. If the moment of inertia of a sphere about its diameter is I, then the moment of inertia of both the spheres about the tangent at their common point would be

Options

A. 3I
B. 7I
C. 4I
D. 5I

Explanation

To find the moment of inertia of two spheres about the tangent at their common point, we can follow these steps:

### Step 1: Understand the moment of inertia of a sphere
The moment of inertia \( I \) of a solid sphere about its diameter is given as \( I \).

### Step 2: Use the parallel axis theorem
To find the moment of inertia about a tangent to the sphere at the point of contact, we can use the parallel axis theorem. The parallel axis theorem states that:
\[ I' = I + Md^2 \]
where:
- \( I' \) is the moment of inertia about the new axis (the tangent),
- \( I \) is the moment of inertia about the axis through the center of mass (diameter),
- \( M \) is the mass of the sphere,
- \( d \) is the distance between the two axes.

### Step 3: Calculate the distance \( d \)
For a sphere of radius \( r \), the distance \( d \) from the center of the sphere to the tangent at the point of contact is equal to the radius \( r \).

### Step 4: Apply the parallel axis theorem for one sphere
For one sphere, the moment of inertia about the tangent is:
\[ I' = I + Mr^2 \]

### Step 5: Calculate for both spheres
Since there are two spheres, and they are identical, we can calculate the moment of inertia for both spheres about the tangent. The total moment of inertia \( I_{total} \) will be:
\[ I_{total} = I'_{sphere1} + I'_{sphere2} \]
Substituting the expression from Step 4:
\[ I_{total} = (I + Mr^2) + (I + Mr^2) = 2I + 2Mr^2 \]

### Step 6: Final expression
Thus, the moment of inertia of both spheres about the tangent at their common point is:
\[ I_{total} = 2I + 2Mr^2 \]