Radius of gyration of a body about an axis at a distance 6 cm from it COM is 10 cm . Its radius of gyration about a parallel axis passing through its COM is (n) cm . find value of n.

Answer

To find the radius of gyration (n) of a body about a parallel axis passing through its center of mass (COM), we can use the parallel axis theorem. Let's break down the solution step by step:

### Step 1: Understand the Given Data
- The radius of gyration (k) about an axis at a distance of 6 cm from the COM is given as 10 cm.
- We need to find the radius of gyration (k') about a parallel axis that passes through the COM.

### Step 2: Apply the Parallel Axis Theorem
The parallel axis theorem states that:
\[ I_1 = I_2 + m \cdot r^2 \]
Where:
- \( I_1 \) is the moment of inertia about the first axis.
- \( I_2 \) is the moment of inertia about the second axis (through the COM).
- \( m \) is the mass of the body.
- \( r \) is the distance between the two axes.

### Step 3: Express Moment of Inertia in Terms of Radius of Gyration
The moment of inertia can also be expressed in terms of the radius of gyration:
\[ I = m \cdot k^2 \]
Thus, we can write:
- For the first axis:
\[ I_1 = m \cdot (10 \text{ cm})^2 = m \cdot (0.1 \text{ m})^2 = m \cdot 0.01 \text{ m}^2 \]
- For the second axis:
\[ I_2 = m \cdot (k')^2 = m \cdot (k')^2 \]

### Step 4: Substitute into the Parallel Axis Theorem
Substituting the expressions for \( I_1 \) and \( I_2 \) into the parallel axis theorem:
\[ m \cdot 0.01 = m \cdot (k')^2 + m \cdot (0.06 \text{ m})^2 \]

### Step 5: Cancel the Mass (m)
Since mass (m) is common in all terms, we can cancel it out:
\[ 0.01 = (k')^2 + (0.06)^2 \]

### Step 6: Calculate \( (0.06)^2 \)
Calculating \( (0.06)^2 \):
\[ (0.06)^2 = 0.0036 \]

### Step 7: Rearrange the Equation
Now, we can rearrange the equation:
\[ (k')^2 = 0.01 - 0.0036 \]
\[ (k')^2 = 0.0064 \]

### Step 8: Take the Square Root
Taking the square root to find \( k' \):
\[ k' = \sqrt{0.0064} = 0.08 \text{ m} \]

### Step 9: Convert to Centimeters
Convert \( k' \) to centimeters:
\[ k' = 0.08 \text{ m} = 8 \text{ cm} \]

### Conclusion
Thus, the value of \( n \) is:
\[ n = 8 \text{ cm} \]