Question 304: Units , Dimensions and Error Analysis

Question

A steel wire of diameter $$0.5$$ $$mm$$ and Young's modulus $$2 	imes {10^{11}}\,\,N{m^{ - 2}}$$ carries a load of mass $$M.$$ The length of the wire with the load is $$1.0$$ $$m.A$$ vernier scale with $$10$$ divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count $$1.0$$ $$mm$$ , is attached. The $$10$$ divisions of the vernier scale correspond to $$9$$ divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by  $$1.2$$ $$kg,$$ the vernier scale division which coincides with a main scale division is _____________. Take $$g = 10\,m\,{s^{ - 2}}.$$  and $$\pi  = 3.2.$$

Accepted Answer

We know that $$\Delta L = {W \over {(YA/L)}}$$
where W is weight or load = mg = 1.2 $$	imes$$ 10 = 12 kg m s
$$-$$2
, Y is Young's modulus = 2 $$	imes$$ 10
11
N m
$$-$$2
, L is length of wire with load = 1.0 m, A is area of steel wire = $$ = \pi {r^2} = {\pi  \over 4}{d^2} = {\pi  \over 4} 	imes {(0.5 	imes {10^{ - 3}})^2}$$
Therefore,
$$\Delta L = {{1.2 	imes 10} \over {2 	imes {{10}^{11}} 	imes {\pi  \over 4}{{(0.5 	imes {{10}^{ - 3}})}^2} 	imes {1 \over {1.0\,m}}}}$$
$$ = {{1.2 	imes 10 	imes 4} \over {2 	imes {{10}^{11}} 	imes \pi  	imes {{(0.5)}^2} 	imes {{10}^{ - 6}}}}$$
$$ \Rightarrow \Delta L = 0.3 	imes {10^{ - 3}}$$ m = 0.3 mm
Now, least count of vernier scale $$ = \left( {1 - {9 \over {10}}} ight)$$ mm = 0.1 mm
Therefore, Vernier reading $$ = {{\Delta L} \over {least\,count}}$$
Vernier reading $$ = {{0.3\,mm} \over {0.1\,mm}} = 3$$
Therefore, 3
rd
vernier scale division coincides with the main scale division.

Units , Dimensions and Error Analysis

Integer / Numeric Answer Type