The length of a cylinder is measured with the help of a vernier callipers whose smallest division on the main scale is 0.5 mm and nine divisions of the main scale are equal to ten divisions of the vernier scale. It is observed that `78^("th")` division of the main scale coincides with sixth division of the vernier callipers. Find the length of the cylinder.

Options

A. 3.13 cm
B. 3.33 cm
C. 3.63 cm
D. 3.93 cm

Explanation

To solve the problem of measuring the length of a cylinder using a vernier caliper, we will follow these steps:

### Step 1: Identify the Main Scale Reading (MSR)
The problem states that the 78th division of the main scale coincides with the 6th division of the vernier scale. The main scale reading (MSR) is therefore:
\[ \text{MSR} = 78 \text{ divisions} \]

### Step 2: Calculate the Value of Each Main Scale Division
The smallest division on the main scale is given as 0.5 mm. Therefore, the MSR in mm is:
\[ \text{MSR in mm} = 78 \times 0.5 \text{ mm} = 39 \text{ mm} \]

### Step 3: Determine the Least Count of the Vernier Caliper
We know that 9 divisions of the main scale are equal to 10 divisions of the vernier scale. Thus, we can calculate the least count (LC) as follows:
- The length of 9 divisions of the main scale:
\[ 9 \times 0.5 \text{ mm} = 4.5 \text{ mm} \]
- The length of 10 divisions of the vernier scale is also 4.5 mm, so:
\[ \text{Length of one division on the vernier scale} = \frac{4.5 \text{ mm}}{10} = 0.45 \text{ mm} \]

Now, the least count (LC) is given by:
\[ \text{LC} = \text{Value of one main scale division} - \text{Value of one vernier scale division} \]
\[ \text{LC} = 0.5 \text{ mm} - 0.45 \text{ mm} = 0.05 \text{ mm} \]

### Step 4: Determine the Vernier Scale Reading (VSR)
The vernier scale reading (VSR) is given as the 6th division of the vernier scale:
\[ \text{VSR} = 6 \text{ divisions} \]
To convert this to mm:
\[ \text{VSR in mm} = 6 \times 0.05 \text{ mm} = 0.3 \text{ mm} \]

### Step 5: Calculate the Total Length of the Cylinder
Now, we can find the total length of the cylinder by adding the main scale reading and the vernier scale reading:
\[ \text{Total Length} = \text{MSR in mm} + \text{VSR in mm} \]
\[ \text{Total Length} = 39 \text{ mm} + 0.3 \text{ mm} = 39.3 \text{ mm} \]

### Step 6: Convert the Length to Centimeters
To convert the total length from mm to cm:
\[ \text{Total Length in cm} = \frac{39.3 \text{ mm}}{10} = 3.93 \text{ cm} \]

### Final Answer
Thus, the length of the cylinder is:
\[ \text{Length of the cylinder} = 3.93 \text{ cm} \]