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 Vernier calipers, we can follow these steps:

### Step 1: Understand the Vernier Scale
The smallest division on the main scale (MSD) is 1 mm. Since 10 Vernier divisions (VD) coincide with 9 main scale divisions, we can calculate the value of one Vernier division.

### Step 2: Calculate the Value of One Vernier Division
Since 10 VD = 9 MSD, we can express this as:
\[ \text{Value of one VD} = \frac{9 \text{ MSD}}{10} = \frac{9 \times 1 \text{ mm}}{10} = 0.9 \text{ mm} \]

### Step 3: Determine the Reading from the Vernier Scale
The zero mark of the Vernier scale lies between 2.0 cm and 2.1 cm. This means the main scale reading (MSR) is:
\[ \text{MSR} = 2.0 \text{ cm} = 20 \text{ mm} \]

The 5th division of the Vernier scale coincides with a main scale division. Since each Vernier division is 0.9 mm, the additional length contributed by the Vernier scale is:
\[ \text{Vernier Scale Reading (VSR)} = 5 \times 0.9 \text{ mm} = 4.5 \text{ mm} \]

### Step 4: Calculate the Total Diameter
Now, we can find the total diameter of the sphere by adding the main scale reading and the Vernier scale reading:
\[ \text{Diameter} = \text{MSR} + \text{VSR} = 20 \text{ mm} + 4.5 \text{ mm} = 24.5 \text{ mm} \]

### Step 5: Convert to Centimeters
Finally, we convert the diameter from millimeters to centimeters:
\[ \text{Diameter} = \frac{24.5 \text{ mm}}{10} = 2.45 \text{ cm} \]

Thus, the diameter of the sphere is **2.45 cm**.

---