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Question

Two wires A and B of radius 2r and r respectively are joined and a force F is applied at the end. The length of each wire is L. Their Youngs modulus are Y and 2Y respectively. Find the net elongation.

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Solution

Given.

Length of both wires, L

Radius of both wires, rA=2rB=2r

Young modulus of A and B, 2YA=YB=2Y

Total extension of joined wire, ΔLnet

Area of B=πr2=A

Area of A =π(2r)2=4πr2=4A

Both wires join in series, so tension in both is equal

T=YAAAΔLAL=YBABΔLBL

Net extension in joined wire.

ΔLnet=ΔLA+ΔLB=TLYAAA+TLYBAB=TLY(4A)+TL(2Y)A

ΔLnet=0.75TLYA

Net extension is 0.75TLYπr2


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