The main scale of a vernier calipers reads `10 mm` in `10` divisions. Ten divisions of vernier scale coincide with nine divisions of the main scale. When the two jaws of the calipers touch each other, the fifth division of the vernier coincides with `9` main scale divisions and zero of the vernier is to the right of zero of main scale, when a cylinder is tighty placed between the two jaws, the zero of the vernier scale lies slighty to the left of `3.2 cm` and the fourth vernier division coincides with a main scale division. Find diameter of the cylinder.

Explanation

To find the diameter of the cylinder using the given information about the vernier calipers, we can follow these steps:

### Step 1: Determine the Main Scale Division (MSD)
The main scale of the vernier calipers reads 10 mm in 10 divisions. Therefore, we can calculate the value of one main scale division (MSD):
\[ \text{MSD} = \frac{10 \text{ mm}}{10} = 1 \text{ mm} \]

### Step 2: Determine the Vernier Scale Division (VSD)
It is given that 10 divisions of the vernier scale coincide with 9 divisions of the main scale. Thus, we can find the value of one vernier scale division (VSD):
\[ 10 \text{ VSD} = 9 \text{ MSD} \implies \text{VSD} = \frac{9 \text{ MSD}}{10} = \frac{9 \text{ mm}}{10} = 0.9 \text{ mm} \]

### Step 3: Calculate the Least Count (LC)
The least count of the vernier calipers can be calculated using the formula:
\[ \text{LC} = \text{MSD} - \text{VSD} \]
Substituting the values we found:
\[ \text{LC} = 1 \text{ mm} - 0.9 \text{ mm} = 0.1 \text{ mm} = 0.01 \text{ cm} \]

### Step 4: Determine the Zero Error
When the jaws of the calipers touch each other, the fifth division of the vernier coincides with 9 main scale divisions, and the zero of the vernier is to the right of zero of the main scale. This indicates a positive zero error. The zero error can be calculated as:
\[ \text{Zero Error} = 5 \times \text{LC} = 5 \times 0.01 \text{ cm} = 0.05 \text{ cm} \]

### Step 5: Read the Measurement with the Cylinder
When the cylinder is placed between the jaws, it is stated that the zero of the vernier scale lies slightly to the left of 3.2 cm, and the fourth vernier division coincides with a main scale division. Therefore, we can interpret the reading:
- The main scale reading (MSR) is approximately 3.1 cm (since it is slightly left of 3.2 cm).
- The vernier reading (VR) is given by:
\[ \text{VR} = 4 \times \text{LC} = 4 \times 0.01 \text{ cm} = 0.04 \text{ cm} \]

### Step 6: Calculate the Observed Value
Now, we can calculate the observed value (OV) of the diameter of the cylinder:
\[ \text{OV} = \text{MSR} + \text{VR} = 3.1 \text{ cm} + 0.04 \text{ cm} = 3.14 \text{ cm} \]

### Step 7: Calculate the True Value
Finally, we need to account for the positive zero error to find the true value (TV) of the diameter:
\[ \text{TV} = \text{OV} - \text{Zero Error} = 3.14 \text{ cm} - 0.05 \text{ cm} = 3.09 \text{ cm} \]

### Conclusion
The diameter of the cylinder is:
\[ \text{Diameter} = 3.09 \text{ cm} \]