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 ($$3.38 \,\, \text{mm}$$)