The radius of gyration of body is `18cm` when it is rotating about an axis passing through centre of mass of body. If radius of gyration of same body is `30cm` about a parallel axis to first axis then, perpendicular distance between two parallel axes is

Options

A. `12 cm `
B. `16 cm`
C. `24 cm`
D. `36 cm`

Explanation

To solve the problem, we will use the concept of the radius of gyration and the parallel axis theorem. The radius of gyration \( k \) is related to the moment of inertia \( I \) and the mass \( m \) of the body by the formula:

\[ k = \sqrt{\frac{I}{m}} \]

### Step 1: Write down the given data
- Radius of gyration about the center of mass (CM) \( k_{CM} = 18 \, \text{cm} \)
- Radius of gyration about the parallel axis \( k_{P} = 30 \, \text{cm} \)

### Step 2: Calculate the moment of inertia about the center of mass
Using the formula for the radius of gyration:

\[ k_{CM} = \sqrt{\frac{I_{CM}}{m}} \implies I_{CM} = m \cdot k_{CM}^2 \]

Substituting the value of \( k_{CM} \):

\[ I_{CM} = m \cdot (18 \, \text{cm})^2 = m \cdot 324 \, \text{cm}^2 \]

### Step 3: Calculate the moment of inertia about the parallel axis
Using the radius of gyration about the parallel axis:

\[ k_{P} = \sqrt{\frac{I_{P}}{m}} \implies I_{P} = m \cdot k_{P}^2 \]

Substituting the value of \( k_{P} \):

\[ I_{P} = m \cdot (30 \, \text{cm})^2 = m \cdot 900 \, \text{cm}^2 \]

### Step 4: Apply the parallel axis theorem
The parallel axis theorem states:

\[ I_{P} = I_{CM} + m d^2 \]

where \( d \) is the perpendicular distance between the two axes.

### Step 5: Substitute the values into the equation
Substituting the expressions for \( I_{P} \) and \( I_{CM} \):

\[ m \cdot 900 = m \cdot 324 + m d^2 \]

### Step 6: Simplify the equation
Dividing through by \( m \) (assuming \( m \neq 0 \)):

\[ 900 = 324 + d^2 \]

### Step 7: Solve for \( d^2 \)
Rearranging gives:

\[ d^2 = 900 - 324 = 576 \]

### Step 8: Calculate \( d \)
Taking the square root:

\[ d = \sqrt{576} = 24 \, \text{cm} \]

### Final Answer
The perpendicular distance between the two parallel axes is \( 24 \, \text{cm} \).

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