Question

The diameter of a spherical bob is measured using a vernier callipers. 9 divisions of the main scale, in the vernier callipers, are equal to 10 divisions of vernier scale. One main scale division is 1 mm. The main scale reading is 10 mm and \(8^{th}\) division of vernier scale was found to coincide exactly with one of the main scale division. If the given vernier callipers has positive zero error of 0.04 cm, then the radius of the bob is ________ \(\times 10^{-2}\) cm.

Hint:

Remember: {Subtract} positive zero errors and {Add} negative zero errors to the observed reading to find the actual measurement.

Answer 1

Correct Answer: 52

Step 1: Understanding the Concept:
The reading of a vernier calliper is given by the sum of the Main Scale Reading (MSR) and the Vernier Scale Reading (VSR $\times$ Least Count). Any zero error must be subtracted from the observed reading to get the true value.
Step 2: Key Formula or Approach:
1. Least Count (LC) = \( 1 \text{ MSD} - 1 \text{ VSD} \).
2. Observed Reading = MSR + (VSR \(\times\) LC).
3. True Reading = Observed Reading \(-\) (Zero Error).
Step 3: Detailed Explanation:
Given: \( 10 \text{ VSD} = 9 \text{ MSD} \implies 1 \text{ VSD} = 0.9 \text{ MSD} \).
\( 1 \text{ MSD} = 1 \text{ mm} = 0.1 \text{ cm} \).
Least Count (LC) = \( 1 \text{ MSD} - 0.9 \text{ MSD} = 0.1 \text{ MSD} = 0.1 \text{ mm} = 0.01 \text{ cm} \).
Observed Diameter:
MSR = 10 mm = 1.0 cm.
VSR = 8.
Observed value = \( 1.0 + (8 \times 0.01) = 1.08 \text{ cm} \).
True Diameter:
True value = \( 1.08 - 0.04 \text{ (positive error)} = 1.04 \text{ cm} \).
Radius of bob:
Radius = \( \frac{1.04}{2} = 0.52 \text{ cm} = 52 \times 10^{-2} \text{ cm} \).
Step 4: Final Answer:
The radius is \( 52 \times 10^{-2} \) cm. The numerical value is 52.