Least count of a vernier callipers is `0.01 cm` When the two jaws of the instrument touch each other the `5th` division of the vernier scale coincide with a main scale division and the zero of the vernier scale lies to the left of the zero of the main scale. Furthermore while measuring the diameter of a sphere, the zero markof the vernier scale lies between `2.4 cm` and `2.5 cm` and the `6th` vernier division coincides with a main scale division. Calculate the diameter of the sphere.

Explanation

To calculate the diameter of the sphere using the vernier calipers, we will follow these steps:

### Step 1: Understand the given information
- Least count of the vernier calipers = 0.01 cm
- When the jaws touch, the 5th division of the vernier scale coincides with a main scale division, and the zero of the vernier scale lies to the left of the zero of the main scale.

### Step 2: Calculate the systematic error
Since the 5th division of the vernier scale coincides with a main scale division when the jaws touch, and the zero of the vernier scale is to the left of the main scale zero, we have a negative error.

The error can be calculated as:
\[ \text{Error} = -(\text{Vernier division that coincides}) \times \text{Least Count} \]
\[ \text{Error} = -5 \times 0.01 \, \text{cm} = -0.05 \, \text{cm} \]

### Step 3: Measure the diameter of the sphere
- The zero mark of the vernier scale lies between 2.4 cm and 2.5 cm. Therefore, the main scale reading (MSR) is:
\[ \text{MSR} = 2.4 \, \text{cm} \]

- The 6th division of the vernier scale coincides with a main scale division. Thus, the vernier scale reading (VSR) is:
\[ \text{VSR} = 6 \]

### Step 4: Calculate the observed value of the diameter
The formula for the observed value of the diameter (D) is:
\[ D = \text{MSR} + (\text{VSR} \times \text{Least Count}) + \text{Error} \]

Substituting the values:
\[ D = 2.4 \, \text{cm} + (6 \times 0.01 \, \text{cm}) - 0.05 \, \text{cm} \]
\[ D = 2.4 \, \text{cm} + 0.06 \, \text{cm} - 0.05 \, \text{cm} \]
\[ D = 2.4 \, \text{cm} + 0.01 \, \text{cm} \]
\[ D = 2.41 \, \text{cm} \]

### Step 5: Final calculation
Thus, the diameter of the sphere is:
\[ \text{Diameter} = 2.41 \, \text{cm} \]