Find the moment of inertia of a sphere about a tangent to the sphere, while the mass of the sphere is M and the radius of the sphere is R.

Options

A. `2/5 MR^(2)`
B. `6/5 MR^(2)`
C. `4/5 MR^(2)`
D. `7/5 MR^(2)`

Explanation

To find the moment of inertia of a sphere about a tangent to the sphere, 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 own axis is given by the formula:
\[ I = \frac{2}{5} M R^2 \]
where \(M\) is the mass of the sphere and \(R\) is its radius.

### Step 2: Apply the Parallel Axis Theorem
To find the moment of inertia about a tangent to the sphere, we can use the Parallel Axis Theorem. This theorem states that if you know the moment of inertia about an axis through the center of mass, you can find the moment of inertia about any parallel axis by adding the product of the mass and the square of the distance between the two axes.

The formula for the Parallel Axis Theorem is:
\[ I_t = I_{cm} + M d^2 \]
where:
- \(I_t\) is the moment of inertia about the tangent,
- \(I_{cm}\) is the moment of inertia about the center of mass,
- \(M\) is the mass of the sphere,
- \(d\) is the distance from the center of mass to the new axis (which is equal to the radius \(R\) in this case).

### Step 3: Substitute Values into the Equation
We already know:
- \(I_{cm} = \frac{2}{5} M R^2\)
- \(d = R\)

Now we can substitute these values into the Parallel Axis Theorem equation:
\[ I_t = \frac{2}{5} M R^2 + M R^2 \]

### Step 4: Simplify the Equation
Now, we can simplify the equation:
\[ I_t = \frac{2}{5} M R^2 + \frac{5}{5} M R^2 = \frac{2}{5} M R^2 + \frac{5}{5} M R^2 = \frac{7}{5} M R^2 \]

### Final Answer
Thus, the moment of inertia of the sphere about a tangent to the sphere is:
\[ I_t = \frac{7}{5} M R^2 \]