The moment of inertia of a disc of mass `M` and radius `R` about a tangent to its rim in its plane is

Options

A. `2/3 MR^(2)`
B. `3/2 MR^(2)`
C. `4/5 MR^(2)`
D. `5/4 MR^(2)`

Answer

To find the moment of inertia of a disc of mass \( M \) and radius \( R \) about a tangent to its rim in its plane, we can follow these steps:

### Step 1: Moment of Inertia about the Center
The moment of inertia \( I \) of a disc about an axis through its center and perpendicular to its plane is given by the formula:
\[ I = \frac{1}{2} M R^2 \]

### Step 2: Identify the Relevant Theorems
To find the moment of inertia about the tangent line, we can use the **Parallel Axis Theorem**. The 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.

### Step 3: Calculate the Distance
The distance \( d \) between the center of the disc and the tangent line at the rim is equal to the radius \( R \) of the disc.

### Step 4: Apply the Parallel Axis Theorem
Using the Parallel Axis Theorem:
\[ I_{tangent} = I_{center} + M d^2 \]
Substituting the values we have:
\[ I_{tangent} = \frac{1}{2} M R^2 + M R^2 \]

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ I_{tangent} = \frac{1}{2} M R^2 + 1 M R^2 = \frac{1}{2} M R^2 + \frac{2}{2} M R^2 = \frac{3}{2} M R^2 \]

### Conclusion
Thus, the moment of inertia of the disc about a tangent to its rim in its plane is:
\[ I_{tangent} = \frac{3}{2} M R^2 \]

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