The radius of gyration of a disc of radius 25 cm about a centroidal axis perpendicular to disc is a) 18 cm b) 12.5 cm c) 36 cm d) 50 cm

Options

A. 18 cm
B. 12.5 cm
C. 36 cm
D. 50 cm

Explanation

To find the radius of gyration (k) of a disc about a centroidal axis perpendicular to the disc, we can follow these steps:

### Step 1: Identify the given data
- The radius of the disc (r) is given as 25 cm.

### Step 2: Recall the formula for the radius of gyration
The radius of gyration (k) is given by the formula:
\[ k = \sqrt{\frac{I}{m}} \]
where \( I \) is the moment of inertia and \( m \) is the mass of the disc.

### Step 3: Determine the moment of inertia for the disc
The moment of inertia \( I \) of a disc about a centroidal axis perpendicular to the disc is given by:
\[ I = \frac{1}{2} m r^2 \]
where \( r \) is the radius of the disc.

### Step 4: Substitute the moment of inertia into the radius of gyration formula
Substituting \( I \) into the formula for \( k \):
\[ k = \sqrt{\frac{\frac{1}{2} m r^2}{m}} \]

### Step 5: Simplify the expression
The mass \( m \) cancels out:
\[ k = \sqrt{\frac{1}{2} r^2} \]
This simplifies to:
\[ k = \frac{r}{\sqrt{2}} \]

### Step 6: Substitute the value of r
Now, substituting the value of \( r = 25 \) cm:
\[ k = \frac{25}{\sqrt{2}} \]

### Step 7: Calculate the value of k
Calculating \( \sqrt{2} \):
\[ \sqrt{2} \approx 1.414 \]
Now, substituting this value:
\[ k \approx \frac{25}{1.414} \approx 17.68 \, \text{cm} \]

### Step 8: Final answer
The radius of gyration of the disc about the centroidal axis perpendicular to the disc is approximately 17.68 cm. Thus, the correct option is:
**Option A: 18 cm (approximately)**

---