The masses of two uniform discs are in the ratio `1 : 2` and their diameters in the ratio `2 : 1`. The ratio of their moment, of inertia about the axis passing through their respective centres and perpendicular to their planes is

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `1:4`

Explanation

To find the ratio of the moments of inertia of two uniform discs about their respective centers and perpendicular to their planes, we can follow these steps:

### Step 1: Understand the given ratios
We are given:
- The ratio of the masses of the two discs: \( m_1 : m_2 = 1 : 2 \)
- The ratio of the diameters of the two discs: \( d_1 : d_2 = 2 : 1 \)

### Step 2: Relate diameters to radii
Since the radius is half of the diameter, we can find the ratio of the radii:
- \( r_1 : r_2 = \frac{d_1}{2} : \frac{d_2}{2} = 2 : 1 \)

### Step 3: Write the formula for the moment of inertia
The moment of inertia \( I \) of a uniform disc about an axis through its center and perpendicular to its plane is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
where \( m \) is the mass and \( r \) is the radius of the disc.

### Step 4: Calculate the moments of inertia for both discs
Using the formula for both discs:
- For disc 1:
\[ I_1 = \frac{1}{2} m_1 r_1^2 \]
- For disc 2:
\[ I_2 = \frac{1}{2} m_2 r_2^2 \]

### Step 5: Substitute the mass and radius ratios into the moment of inertia formulas
Using the ratios:
- \( m_1 = m \) and \( m_2 = 2m \) (from the mass ratio)
- \( r_1 = 2r \) and \( r_2 = r \) (from the radius ratio)

Substituting these into the equations for \( I_1 \) and \( I_2 \):
\[ I_1 = \frac{1}{2} m (2r)^2 = \frac{1}{2} m (4r^2) = 2mr^2 \]
\[ I_2 = \frac{1}{2} (2m) (r)^2 = \frac{1}{2} (2m) (r^2) = mr^2 \]

### Step 6: Find the ratio of the moments of inertia
Now we can find the ratio \( \frac{I_1}{I_2} \):
\[ \frac{I_1}{I_2} = \frac{2mr^2}{mr^2} = 2 \]

### Final Result
Thus, the ratio of their moments of inertia is:
\[ I_1 : I_2 = 2 : 1 \]