A ring and a disc have same mass and same radius. Ratio of moments of inertia of the ring about a tangent in its plane to that of the disc about its diameter is

Options

A. `2:1`
B. `4:1`
C. `6:1`
D. `8:1`

Answer

To solve the problem of finding the ratio of moments of inertia of a ring about a tangent in its plane to that of a disc about its diameter, we can follow these steps:

### Step 1: Define the Mass and Radius
Let the mass of both the ring and the disc be \( m \) and the radius be \( r \).

### Step 2: Moment of Inertia of the Ring
The moment of inertia \( I \) of a ring about an axis through its center and perpendicular to its plane is given by:
\[ I_{\text{ring, center}} = m r^2 \]
To find the moment of inertia about a tangent in its plane, we can use the **Parallel Axis Theorem**, which states:
\[ I = I_{\text{cm}} + m d^2 \]
where \( I_{\text{cm}} \) is the moment of inertia about the center of mass, and \( d \) is the distance from the center of mass to the new axis. For a tangent, \( d = r \):
\[ I_{\text{ring, tangent}} = I_{\text{ring, center}} + m r^2 = m r^2 + m r^2 = 2 m r^2 \]

### Step 3: Moment of Inertia of the Disc
The moment of inertia \( I \) of a disc about an axis through its center and perpendicular to its plane is given by:
\[ I_{\text{disc, center}} = \frac{1}{2} m r^2 \]
Using the **Perpendicular Axis Theorem**, for a disc, the moment of inertia about its diameter (which is one of the axes in its plane) is:
\[ I_{\text{disc, diameter}} = \frac{1}{2} I_{\text{disc, center}} = \frac{1}{2} \left(\frac{1}{2} m r^2\right) + \frac{1}{2} \left(\frac{1}{2} m r^2\right) = \frac{1}{4} m r^2 + \frac{1}{4} m r^2 = \frac{1}{2} m r^2 \]

### Step 4: Calculate the Ratio
Now, we can find the ratio of the moments of inertia:
\[ \text{Ratio} = \frac{I_{\text{ring, tangent}}}{I_{\text{disc, diameter}}} = \frac{2 m r^2}{\frac{1}{2} m r^2} = \frac{2}{\frac{1}{2}} = 4 \]

### Final Answer
Thus, the ratio of the moments of inertia of the ring about a tangent in its plane to that of the disc about its diameter is:
\[ \text{Ratio} = 4:1 \]

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