If a disc of mass m and radius r is reshaped into a ring of radius`2r`, the mass remining the same, the radius of gyration about centrodal axis perpendicular to plane goes up by a factor of

Options

A. `sqrt(2)`
B. 2
C. `2 sqrt(2)`
D. 4

Explanation

To solve the problem, we need to determine the radius of gyration of both the disc and the ring and then find the factor by which the radius of gyration increases when the disc is reshaped into a ring.

### Step 1: Calculate the Radius of Gyration for the Disc

The formula for the radius of gyration \( k_d \) of a disc about an axis perpendicular to its plane through its center is given by:

\[ k_d = \sqrt{\frac{I_d}{m}} \]

where \( I_d \) is the moment of inertia of the disc and \( m \) is its mass.

The moment of inertia \( I_d \) of a disc is given by:

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

Substituting this into the equation for \( k_d \):

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

### Step 2: Calculate the Radius of Gyration for the Ring

Now, we need to find the radius of gyration \( k_r \) of the ring. The ring has a radius of \( 2r \) and the mass \( m \) remains the same.

The moment of inertia \( I_r \) of a ring about an axis perpendicular to its plane through its center is given by:

\[ I_r = m R^2 \]

For our ring, \( R = 2r \):

\[ I_r = m (2r)^2 = 4mr^2 \]

Now, substituting this into the equation for \( k_r \):

\[ k_r = \sqrt{\frac{I_r}{m}} = \sqrt{\frac{4mr^2}{m}} = \sqrt{4r^2} = 2r \]

### Step 3: Find the Factor by which the Radius of Gyration Increases

Now, we need to find the factor by which the radius of gyration increases when the disc is reshaped into a ring:

\[ \text{Factor} = \frac{k_r}{k_d} = \frac{2r}{\frac{r}{\sqrt{2}}} \]

Simplifying this:

\[ \text{Factor} = 2r \cdot \frac{\sqrt{2}}{r} = 2\sqrt{2} \]

### Conclusion

The radius of gyration about the centroidal axis perpendicular to the plane goes up by a factor of \( 2\sqrt{2} \).

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