The smallest division on the main scale of a vernier callipers is `1mm`, and 10 vernier divisions coincide with 9 mainn scalel divisions. While measuring the diameter of a spehre the zero mark of the vernier scale lies between 2.0 and 2.1 cm and the fifth division of hte vernier main scale coincide with a main scale division. Then diameter of the sphere is

Options

A. `2.05cm`
B. `3.05cm`
C. `2.50cm`
D. none of these

Answer

To find the diameter of the sphere using the given information about the Vernier calipers, we will follow these steps:

### Step 1: Understand the given data
- The smallest division on the main scale (MSD) = 1 mm = 0.1 cm
- 10 Vernier divisions (VD) coincide with 9 main scale divisions (MSD).
- The zero mark of the Vernier scale lies between 2.0 cm and 2.1 cm.
- The 5th Vernier division coincides with a main scale division.

### Step 2: Calculate the least count (LC) of the Vernier calipers
The least count is calculated using the formula:
\[ \text{Least Count (LC)} = \text{Value of 1 MSD} - \text{Value of 1 VD} \]

First, we find the value of 1 Vernier division:
\[ \text{Value of 1 VD} = \frac{9 \text{ MSD}}{10} = \frac{9 \times 1 \text{ mm}}{10} = 0.9 \text{ mm} = 0.09 \text{ cm} \]

Now, substituting this into the least count formula:
\[ \text{LC} = 1 \text{ mm} - 0.9 \text{ mm} = 0.1 \text{ mm} = 0.01 \text{ cm} \]

### Step 3: Identify the main scale reading (MSR)
From the problem, we know that the zero mark of the Vernier scale lies between 2.0 cm and 2.1 cm. Therefore, the main scale reading (MSR) is:
\[ \text{MSR} = 2.0 \text{ cm} \]

### Step 4: Determine the Vernier scale reading (VSR)
The 5th division of the Vernier scale coincides with a main scale division. The value of the Vernier scale reading can be calculated as:
\[ \text{VSR} = 5 \times \text{LC} = 5 \times 0.01 \text{ cm} = 0.05 \text{ cm} \]

### Step 5: Calculate the total reading (Diameter of the sphere)
The diameter of the sphere is given by the sum of the main scale reading and the Vernier scale reading:
\[ \text{Diameter} = \text{MSR} + \text{VSR} = 2.0 \text{ cm} + 0.05 \text{ cm} = 2.05 \text{ cm} \]

### Final Answer
The diameter of the sphere is **2.05 cm**.

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