The radius of gyration of a disc of mass 100 g and radius 5 cm about an axis passing through its centre of gravity and perpendicular to the plane is

Options

A. 0.5 cm
B. 2.5 cm
C. 3.54 cm
D. 6.54 cm

Explanation

To find the radius of gyration of a disc of mass 100 g and radius 5 cm about an axis passing through its center of gravity and perpendicular to the plane, 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: Calculate the moment of inertia of the disc
The moment of inertia (I) of a solid disc about an axis passing through its center and perpendicular to its plane is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
where:
- \( m \) is the mass of the disc,
- \( r \) is the radius of the disc.

Given:
- Mass \( m = 100 \, \text{g} = 0.1 \, \text{kg} \) (convert grams to kilograms),
- Radius \( r = 5 \, \text{cm} = 0.05 \, \text{m} \) (convert centimeters to meters).

Now substituting the values:
\[ I = \frac{1}{2} \times 0.1 \, \text{kg} \times (0.05 \, \text{m})^2 \]
\[ I = \frac{1}{2} \times 0.1 \times 0.0025 \]
\[ I = \frac{1}{2} \times 0.00025 \]
\[ I = 0.000125 \, \text{kg m}^2 \]

### Step 3: Relate moment of inertia to radius of gyration
The radius of gyration (k) is related to the moment of inertia (I) by the formula:
\[ I = m k^2 \]
Rearranging this gives:
\[ k^2 = \frac{I}{m} \]
\[ k = \sqrt{\frac{I}{m}} \]

### Step 4: Substitute the values to find k
Now substituting the values of I and m:
\[ k = \sqrt{\frac{0.000125 \, \text{kg m}^2}{0.1 \, \text{kg}}} \]
\[ k = \sqrt{0.00125 \, \text{m}^2} \]
\[ k = 0.03536 \, \text{m} \]

### Step 5: Convert k back to centimeters
To express the radius of gyration in centimeters:
\[ k = 0.03536 \, \text{m} \times 100 \, \text{cm/m} = 3.536 \, \text{cm} \]

### Final Answer
The radius of gyration of the disc about the specified axis is approximately **3.54 cm**.

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