The M.I. of a disc about its diameter is 2 units. Its M.I. about axis through a point on its rim and in the plane of the disc is

Options

A. 4 units.
B. 6 units.
C. 8 units.
D. 10 units

Answer

To find the moment of inertia (M.I.) of a disc about an axis through a point on its rim and in the plane of the disc, we can follow these steps:

### Step 1: Understand the Given Information
We know that the moment of inertia of the disc about its diameter (Id) is given as 2 units. The formula for the moment of inertia of a disc about its diameter is:

\[ I_d = \frac{1}{4} m r^2 \]

### Step 2: Calculate the Mass Moment of Inertia
Since we have \(I_d = 2\) units, we can express this in terms of mass (m) and radius (r):

\[ \frac{1}{4} m r^2 = 2 \]

From this, we can find \(m r^2\):

\[ m r^2 = 8 \quad \text{(1)} \]

### Step 3: Use the Perpendicular Axis Theorem
According to the perpendicular axis theorem, for a planar body, the moment of inertia about an axis perpendicular to the plane of the body (Iz) is the sum of the moments of inertia about two perpendicular axes in the plane of the body (Ix and Iy):

\[ I_z = I_x + I_y \]

For a disc, both \(I_x\) and \(I_y\) are equal to \(I_d\):

\[ I_z = 2 I_d \]

### Step 4: Apply the Parallel Axis Theorem
Now, we need to find the moment of inertia about an axis through a point on its rim (Ir). We can use the parallel axis theorem, which states:

\[ I_r = I_{cm} + m d^2 \]

Where:
- \(I_{cm}\) is the moment of inertia about the center of mass (which is \(I_d\)),
- \(d\) is the distance from the center of mass to the new axis (which is equal to the radius \(r\) of the disc).

### Step 5: Substitute Values
From the previous steps, we know:

- \(I_{cm} = I_d = \frac{1}{4} m r^2 = 2\) units
- \(d = r\)

Thus, we can write:

\[ I_r = I_d + m r^2 \]

Substituting \(I_d = 2\) and using equation (1) where \(m r^2 = 8\):

\[ I_r = 2 + 8 \]

### Step 6: Calculate the Final Moment of Inertia
Now, we can calculate:

\[ I_r = 10 \text{ units} \]

### Final Answer
The moment of inertia of the disc about an axis through a point on its rim and in the plane of the disc is **10 units**.

---