Question

The diameter of a wire of length $50\text{cm}$ is measured with the help of screw gauge. The main scale reading is $1\text{mm}$ and circular scale reading is 25. Pitch of the screw gauge is $1\text{mm}$ and the total number of divisions on the circular scale is 100. This wire is used in an experiment of determination for the Young's modulus of the material of a wire by Scarle's method. The following data are available: elongation in the wire $l=0.125\text{cm}$ under the tension of $50\text{N}$, least count for measuring normal length of wire is $0.01\text{cm}$ and for elongation in the wire is $0.001\text{cm}$. The maximum percentage error in calculating the value of Young's mudulus(Y) is $n\times 0.484$. Find $n$. (assuming that the force is measured very accurately, $Y=Fl/A\Delta l$ where the symbols have their usual meaning)

Answer 1

Least count of screw gauge $=\frac{1}{100}\text{mm}=0.01\text{mm}$
Diameter of the wire $=(1+25\times 0.01)\text{mm}=0.125\text{cm}$
Since $Y=\frac{4Tl}{\pi d^2\delta l}$,
$\frac{\Delta Y}{Y}=\frac{\Delta l}{l}+\frac{2\Delta d}{d}+\frac{\Delta \left(\delta l\right)}{\delta l}$
$=\frac{0.01}{50}+\frac{2\times 0.001}{0.125}+\frac{0.001}{0.125}=0.0242$
Percentage error $=\frac{\Delta Y}{Y}\times 100=2.42$
$=5\times 0.484$