Question

Answer the following : 

1. You are given a thread and a metre scale. How will you estimate the diameter of the thread ? 
2. A screw gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Do you think it is possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular scale ? 
3. The mean diameter of a thin brass rod is to be measured by vernier calipers. Why is a set of 100 measurements of the diameter expected to yield a more reliable estimate than a set of 5 measurements only ?

Answer 1

(a) Estimation of the diameter of a thread using a thread and a metre scale

Since the diameter of a thread is very small, it cannot be measured accurately by placing it directly against a metre scale.

Method:

• Wind the thread closely around a pencil or a cylindrical rod, making a large number of turns (say N turns), ensuring there are no gaps between adjacent turns.
• Measure the total length L occupied by these N turns using the metre scale.
• The diameter of the thread is then given by:
  Diameter = L / N

Taking many turns increases accuracy by reducing the error in measurement.

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(b) Increasing accuracy of a screw gauge by increasing circular scale divisions

A screw gauge has:

• Pitch = 1.0 mm
• Number of circular scale divisions = 200

Least count = Pitch / Number of divisions
= 1.0 / 200 = 0.005 mm

Although increasing the number of divisions reduces the least count, the accuracy cannot be increased arbitrarily.

This is because:

• Mechanical limitations such as backlash error and wear of the screw
• Manufacturing imperfections
• Elastic deformation and friction

Thus, beyond a certain limit, increasing divisions does not improve practical accuracy.

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(c) Why 100 measurements give a more reliable mean diameter than 5 measurements

Each individual measurement is affected by random errors.

When a large number of readings are taken:

• Random errors tend to cancel out
• The mean value approaches the true value

Statistically, the uncertainty in the mean decreases as:

Uncertainty ∝ 1 / √N

Therefore, a set of 100 measurements provides a much more reliable estimate of the mean diameter than only 5 measurements.

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Final Summary:

• (a) Diameter is found by winding the thread multiple times and dividing total length by number of turns.
• (b) Accuracy of a screw gauge cannot be increased indefinitely due to mechanical limitations.
• (c) Larger number of measurements reduces random errors and improves reliability.