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Question

Two wires of the same material (Young's modulus Y) and same length L but radii R and 2R respectively are joined end to end and a weight w is suspended from the combination as shown in the figure. The elastic potential energy stored in the system is



A
3w2L4πR2Y
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B
3w2L8πR2Y
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C
5w2L8πR2Y
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D
w2LπR2Y
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Solution

The correct option is C 5w2L8πR2Y
As we know, using spring-rod analogy, 
k=YAL
where, k= Spring constant
Y= Young's modulus
So, k1=Yπ(2R)2L,k2=Yπ(R)2L
Equivalent spring constant  =1k1+1k2=L4YπR2+LYπR2=5L4YπR2

Since block is in equilibrium, hence
k1x1=k2x2=w
[for springs in series. force acting is the same]
Elastic potential energy of the system
U=12k1x21+12k2x22
U=12k1(wk1)2+12k2(wk2)2
U=12w2{1k1+1k2}=12w2(5L4YπR2)
U=5w2L8πYR2

Physics

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