The main scale of Vernier calipers reads in millimeter. Its Vernier is divided into 10 divisions, which coincide with 9 divisions of the main scale. When there is nothing between its jaws, the fifth division of its Vernier scale coincides with a division on the main scale. Also, zero of the Vernier scale lies on the right side of the zero of the main scale. When a coin in tightly gripped by its jaws to measure the diameter, the zero of the Vernier scale is observed to be slightly left to the 3.4 cm and the third Vernier scale division coincides with a main scale division. The diameter of the coin is

Options

A. 3.27 cm
B. 3.38 cm
C. 3.33 cm
D. 3.23 cm

Answer

To find the diameter of the coin using the Vernier calipers, we need to follow these steps:

### Step 1: Understand the Least Count of the Vernier Calipers
The least count (LC) of the Vernier calipers is determined by the formula:
\[ \text{Least Count} = \text{Value of 1 Main Scale Division (MSD)} - \text{Value of 1 Vernier Scale Division (VSD)} \]
Given that the Vernier scale has 10 divisions that coincide with 9 divisions of the main scale, we can calculate the least count:
\[ \text{Value of 1 MSD} = 1 \text{ mm} = 0.1 \text{ cm} \]
\[ \text{Value of 1 VSD} = \frac{9 \text{ MSD}}{10} = 0.09 \text{ cm} \]
Thus,
\[ \text{Least Count} = 0.1 \text{ cm} - 0.09 \text{ cm} = 0.01 \text{ cm} \]

### Step 2: Determine the Zero Error
When there is nothing between the jaws, the fifth division of the Vernier scale coincides with a division on the main scale, indicating a zero error. Since the zero of the Vernier scale is to the right of the zero of the main scale, the zero error is positive:
\[ \text{Zero Error} = +5 \times \text{Least Count} = +5 \times 0.01 \text{ cm} = +0.05 \text{ cm} \]

### Step 3: Read the Main Scale Reading (MSR)
When measuring the diameter of the coin, the zero of the Vernier scale is slightly left of 3.4 cm. This means the main scale reading is:
\[ \text{Main Scale Reading} = 3.4 \text{ cm} \]

### Step 4: Read the Vernier Scale Reading (VSR)
The third division of the Vernier scale coincides with a division on the main scale. Therefore, the Vernier scale reading is:
\[ \text{Vernier Scale Reading} = 3 \times \text{Least Count} = 3 \times 0.01 \text{ cm} = 0.03 \text{ cm} \]

### Step 5: Calculate the True Reading
The true reading can be calculated using the formula:
\[ \text{True Reading} = \text{Main Scale Reading} + \text{Vernier Scale Reading} - \text{Zero Error} \]
Substituting the values:
\[ \text{True Reading} = 3.4 \text{ cm} + 0.03 \text{ cm} - 0.05 \text{ cm} \]
\[ \text{True Reading} = 3.43 \text{ cm} - 0.05 \text{ cm} = 3.38 \text{ cm} \]

### Final Answer
The diameter of the coin is:
\[ \text{Diameter of the coin} = 3.38 \text{ cm} \]

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