Screw Gauge (or Micrometer Screw)

In a complete rotation, the spindle of a screw gauge advances by $0.5 \mathrm{mm}.$ There are $50$ divisions on circular scale. The main scale has $0.5 \mathrm{mm}$ marks (is graduated to $0.5 \mathrm{mm}$ or has least count $=0.5 \mathrm{mm}$). Read the screw gauge shown below in figure $\left(a\right)$ and $\left(b\right)$.

Figure (a)	Figure (b)

In the determination of Young's modulus $\left(Y=\frac{4MLg}{\mathrm{\pi }l{d}^2}\right)$ by using Searle's method, a wire of length $L=2 \mathrm{m}$ and diameter $d=0.5 \mathrm{mm}$ is used. For a load $M=2.5 \mathrm{kg}$, an extension $l=0.25 \mathrm{mm}$ in the length of the wire is observed. Quantities $d$ and $l$ are measured using a screw gauge and a micrometre, respectively. They have the same pitch of $0.5 \mathrm{mm}$. The number of divisions on their circular scale is $100$. The contributions to the maximum probable error of the $Y$ measurement.

The density of a solid ball is to be determined in an experiment, the diameter of the ball is measured with a screw gauge whose pitch is $0.5 \mathrm{mm}$ and there are $50$ divisions on the circular scale. The reading on the main scale is $2.5 \mathrm{mm}$ and that on the circular scale is $20$ divisions. If the measured mass of the ball has a relative error of $2\%$, the relative percentage error in the density is,

A screw gauge with a pitch of $0.5 \mathrm{mm}$ and a circular scale with $50$ divisions is used to measure the thickness of a thin sheet of aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the ${45}^{\mathrm{th}}$ division coincides with the main scale line and that the zero of the main scale is barely visible. What is the thickness of the sheet if the main scale reading is $0.5 \mathrm{mm}$ and the ${25}^{\mathrm{th}}$ division coincides within the main scale line?