The ratio of radii of gyration of a circular ring and a circular disc, of the same mass and radius about an axis passing through their centres and perpendicular to their planes are

Options

A. `sqrt(2):1`
B. `1:sqrt(2)`
C. `3:2`
D. `2:1`

Answer

To find the ratio of the radii of gyration of a circular ring and a circular disc, both having the same mass and radius about an axis passing through their centers and perpendicular to their planes, we can follow these steps:

### Step 1: Understand the concept of radius of gyration
The radius of gyration (k) is defined as the distance from the axis of rotation at which the total mass of the body can be assumed to be concentrated without changing its moment of inertia (I). It is given by the formula:
\[ k = \sqrt{\frac{I}{m}} \]
where \( I \) is the moment of inertia and \( m \) is the mass of the object.

### Step 2: Calculate the moment of inertia of the circular ring
For a circular ring of mass \( m \) and radius \( r \), the moment of inertia about an axis passing through its center and perpendicular to its plane is given by:
\[ I_{\text{ring}} = m r^2 \]

### Step 3: Calculate the radius of gyration of the circular ring
Using the formula for the radius of gyration:
\[ k_{\text{ring}} = \sqrt{\frac{I_{\text{ring}}}{m}} = \sqrt{\frac{m r^2}{m}} = \sqrt{r^2} = r \]

### Step 4: Calculate the moment of inertia of the circular disc
For a circular disc of mass \( m \) and radius \( r \), the moment of inertia about an axis passing through its center and perpendicular to its plane is given by:
\[ I_{\text{disc}} = \frac{1}{2} m r^2 \]

### Step 5: Calculate the radius of gyration of the circular disc
Using the formula for the radius of gyration:
\[ k_{\text{disc}} = \sqrt{\frac{I_{\text{disc}}}{m}} = \sqrt{\frac{\frac{1}{2} m r^2}{m}} = \sqrt{\frac{1}{2} r^2} = \frac{r}{\sqrt{2}} \]

### Step 6: Find the ratio of the radii of gyration
Now, we need to find the ratio of the radius of gyration of the circular ring to that of the circular disc:
\[ \text{Ratio} = \frac{k_{\text{ring}}}{k_{\text{disc}}} = \frac{r}{\frac{r}{\sqrt{2}}} \]

### Step 7: Simplify the ratio
Simplifying the ratio gives:
\[ \text{Ratio} = \frac{r \cdot \sqrt{2}}{r} = \sqrt{2} \]

### Step 8: Finalize the ratio
Thus, the ratio of the radii of gyration of the circular ring to the circular disc is:
\[ \text{Ratio} = \sqrt{2} : 1 \]

### Conclusion
The final answer is that the ratio of the radii of gyration of a circular ring and a circular disc about an axis passing through their centers and perpendicular to their planes is \( \sqrt{2} : 1 \).