Question

During Searle’s experiment, zero of the Vernier scale lies between $3.20 \times {10^{ - 2}}m$ and $3.25 \times {10^{ - 2}}m$of the main scale. The${20^{th}}$ division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of $2kg$is applied to the wire, the zero of the Vernier scale still lies between $3.20 \times {10^{ - 2}}m$and $3.25 \times {10^{ - 2}}m$of the main scale but now the ${45^{th}}$ division of Vernier scale coincides with one of the main scale divisions. The length of the thin metallic wire is $2m$ and its cross sectional area is $8 \times {10^{ - 2}}{m^2}$. The least count of the Vernier scale is $1.0 \times {10^{ - 5}}m.$ The maximum percentage error in the Young’s modulus of the wire is

Answer 1

Hint: The Young’s modulus or elastic modules is in essence the stiffness of material. It is low, easily a material that can bend or stretch going modules is strictly proportional to stress in an elastic material inversely proportional to strain.

Complete step by step answer:
Young’s Modulus
$Y = \left( {\dfrac{{FL}}{{Al}}} \right)$
$\dfrac{{\Delta Y}}{Y} = \dfrac{{\Delta l}}{l}$ . . . . (1)
Where $\Delta l$ is least count of Vernier scale
And value of $\Delta l$ is $ = {10^{ - 5}}$
$l$ is change in length and its value
 is $\left[ {{{45}^{th}} - {{20}^{th}}} \right] \times {10^{ - 5}}$
to find out the percentage we multiply both side by $100\% $
we get, $\dfrac{{\Delta Y}}{Y} \times 100\% = \dfrac{{\Delta l}}{l} \times 100\% $
putting the value of $\Delta l$and $l$ we get,
$\dfrac{{\Delta Y}}{Y} \times 100\% = \dfrac{{\Delta l}}{l} \times 100\% $
$ = \dfrac{{{{10}^{ - 5}}}}{{25 \times {{10}^{ - 5}}}} \times 100\% $
$ = 4\% $

The maximum percentage of error in young's modulus of a wire is $4\% .$

Note:
Young modules are a useful property that is important for almost everything around us like buildings, bridges, vehicles and more. Stress is a quantity that describes the magnitude of force that causes deformation; it is defined as force per unit area when force pull is an object that causes its elongation like the stretching of an elastic band. The quantity that describes this deformation is called strain. Is given as a fractional change in either length or volume or geometry. Therefore, strain is a dimensionless unit.