The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is.

Options

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

Answer

To solve the problem of finding the ratio of the radii of gyration of a circular disc to that of a circular ring, we can follow these steps:

### Step 1: Understand the concept of radius of gyration
The radius of gyration (k) is defined as:
\[ k = \sqrt{\frac{I}{m}} \]
where \( I \) is the moment of inertia and \( m \) is the mass of the object.

### Step 2: Identify the moment of inertia for both shapes
1. **For the circular disc**:
The moment of inertia \( I_d \) of a circular disc about its central axis is given by:
\[ I_d = \frac{1}{2} m r^2 \]

2. **For the circular ring**:
The moment of inertia \( I_r \) of a circular ring about its central axis is given by:
\[ I_r = m r^2 \]

### Step 3: Calculate the radius of gyration for both shapes
1. **For the circular disc**:
\[ k_d = \sqrt{\frac{I_d}{m}} = \sqrt{\frac{\frac{1}{2} m r^2}{m}} = \sqrt{\frac{1}{2} r^2} = \frac{r}{\sqrt{2}} \]

2. **For the circular ring**:
\[ k_r = \sqrt{\frac{I_r}{m}} = \sqrt{\frac{m r^2}{m}} = \sqrt{r^2} = r \]

### Step 4: Find the ratio of the radii of gyration
Now, we can find the ratio of the radius of gyration of the disc to that of the ring:
\[ \frac{k_d}{k_r} = \frac{\frac{r}{\sqrt{2}}}{r} = \frac{1}{\sqrt{2}} \]

### Step 5: Express the ratio in a simplified form
Thus, the ratio of the radii of gyration \( k_d : k_r \) can be expressed as:
\[ k_d : k_r = 1 : \sqrt{2} \]

### Conclusion
The final answer is that the ratio of the radii of gyration of a circular disc to that of a circular ring, each of the same mass and radius, is:
\[ \text{Answer: } 1 : \sqrt{2} \]

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