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 of 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, such that the moment of inertia remains the same.

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

### Step 3: Moment of Inertia for the Circular Disc
For a circular disc of the same mass \( m \) and radius \( r \), the moment of inertia \( I \) 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 4: Relate Moment of Inertia to Radius of Gyration
The moment of inertia can also be expressed in terms of the radius of gyration \( k \):
\[ I = m k^2 \]
From this, we can find the radius of gyration for both the ring and the disc.

### Step 5: Calculate Radius of Gyration for the Ring
Using the moment of inertia for the ring:
\[ m k_r^2 = m r^2 \implies k_r^2 = r^2 \implies k_r = r \]

### Step 6: Calculate Radius of Gyration for the Disc
Using the moment of inertia for the disc:
\[ m k_d^2 = \frac{1}{2} m r^2 \implies k_d^2 = \frac{1}{2} r^2 \implies k_d = \frac{r}{\sqrt{2}} \]

### Step 7: Find the Ratio of the Radii of Gyration
Now we can find the ratio of the radii of gyration \( k_r \) and \( k_d \):
\[ \text{Ratio} = \frac{k_r}{k_d} = \frac{r}{\frac{r}{\sqrt{2}}} = \sqrt{2} \]

### Step 8: Final Answer
Thus, the ratio of the radii of gyration of a circular ring and a circular disc of the same mass and radius about an axis passing through their centers and perpendicular to their planes is:
\[ \sqrt{2} : 1 \]