Two full turns of the circular scale of a screw gauge cover a distance of 1 mm on its main scale. The total number of divisions on the circular scale is 50. Further, it is found that the screw gauge has a zero error of $$-0.03$$ mm while measuring the diameter of a thin wire, a student notes the main scale reading of 3 mm and the number of circular scale divisions in line with the main scale as 35. The diameter of the wire is

Solution & Explanation

1. Calculate the Pitch and Least Count (LC) of the Screw Gauge

The pitch of a screw gauge is defined as the distance advanced by the spindle per complete rotation of the circular scale. We are given that 2 full turns cover a distance of $$1 \,\, \text{mm}$$:

$$\text{Pitch} = \frac{\text{Distance covered on main scale}}{\text{Number of full rotations}} = \frac{1 \,\, \text{mm}}{2} = 0.5 \,\, \text{mm}$$

The Least Count (LC) is the smallest measurement that can be accurately taken using the instrument. It is calculated by dividing the pitch by the total number of divisions on the circular scale ($$50$$):

$$\text{Least Count (LC)} = \frac{\text{Pitch}}{\text{Total number of circular divisions}} = \frac{0.5 \,\, \text{mm}}{50} = 0.01 \,\, \text{mm}$$

2. Determine the Observed Reading

The total observed reading before adjusting for any systematic tool misalignment is calculated using the main scale reading (MSR) and the circular scale reading (CSR) matching line:

• Main Scale Reading ($$\text{MSR}$$): $$3 \,\, \text{mm}$$
• Circular Scale Reading ($$\text{CSR}$$): $$35 \,\, \text{divisions}$$

$$\text{Observed Reading} = \text{MSR} + (\text{CSR} \times \text{LC})$$

$$\text{Observed Reading} = 3 \,\, \text{mm} + (35 \times 0.01 \,\, \text{mm}) = 3 + 0.35 = 3.35 \,\, \text{mm}$$

3. Correct for Zero Error to Find True Diameter

The instrument is given to have a built-in zero error of $$-0.03 \,\, \text{mm}$$. To find the true, accurate thickness parameter, we must subtract this zero error from our raw observed calculation:

$$\text{True Diameter} = \text{Observed Reading} - (\text{Zero Error})$$

$$\text{True Diameter} = 3.35 \,\, \text{mm} - (-0.03 \,\, \text{mm})$$

$$\text{True Diameter} = 3.35 \,\, \text{mm} + 0.03 \,\, \text{mm} = 3.38 \,\, \text{mm}$$

Concept Check: Because the tool has a negative zero error, it sits slightly behind the true zero index mark when completely closed. As a result, it consistently under-reports measurements by $$0.03 \,\, \text{mm}$$. We mathematically add this value back to our observed measurement to calculate the correct structural thickness.

Correct Option Key: Option D ($$3.38 \,\, \text{mm}$$)