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A block of weight 100N is suspended by copper and steel wires of same cross sectional area 0.5cm2 and length 3m and 1m respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30 and 60 respectively. If elongation in copper wire is (ΔlC) and elongation in steel wire is (ΔLS), then the ratio ΔlCΔlS is _____. (Young’s modulus for copper and steel are 1×1011Nm2 and 2×1011Nm2, respectively)
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Answer
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Hint: Here we apply the Hooke’s law for stress and strain to find the ratio ΔlCΔlS.
The Hooke’s law says that the displacement or scale of the deformation is directly proportional to the deforming force or load for relatively minor deformations of an object.
Hooke’s law also states that the strain of the material beyond the elastic boundary of that material is equal to the stress applied. The atoms and molecules contract as the elastic structures are strained before tension is added and they revert to their original state when the tension is withdrawn.

Complete step by step answer:
Given,
Young’s modulus for copper, YC=1×1011Nm2
Young’s modulus for steel, YS=2×1011Nm2
Length of copper wire, LC=3m
Length of steel wire, LS=1m
Elongation in copper wire is (ΔLC)
Elongation in steel wire is (ΔlS)

The angles subtended by copper and steel wires with ceiling are 30 and 60.

The cross-sectional area 0.5cm2 is the same for both copper and steel wires.
Let T1 and T2 be the tensional force for steel and copper wire.
Let A be the cross-sectional area.
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According to Hooke’s law
Stress = Y × strain
FA=Y×ΔllΔl=FlAY
ΔlC=T2×lCA×YC ...... (i)
Force F is equal to the torsional force T2 for copper wire
Also,
ΔlS=T1×lSA×YS ...... (ii)
Force F is also equal to the torsional force T1 for steel wire
Equation (i) divided by equation (ii), we get-
ΔlCΔlS=T2lC×YSYCT1×lS=YSyC×lClS×T2T1

ΔlCΔlS=2×10111011×31×T2T1 ...... (iii)
Now applying resultant of horizontal component
Fx=0T2cos30T1cos60=0T2T1=13

So, from equation (iii), we get-
ΔlCΔlS=2×10111011×31×T2T1=23×13=2
Hence,
If elongation in copper wire is (ΔlC) and elongation in steel wire is (ΔlS), then the ratio ΔlCΔlS is 2.

Note:
Here we have to pay attention that the area is the same for both the wires, so the area gets cancelled. Also the horizontal component will be negative for steel wire.