Two wires of the same material (Young’s modulus Y) and same length L but radii

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 in the system is

Options:

A . 3w2L4π R2Y

B . 3w2L8π R2Y

C . 5w2L8π R2Y

D . w2Lπ R2Y

Answer: Option C
:
C

K_{1} = \frac{y π (2 R)^{2}}{L}

K_{2} = \frac{y π (R)^{2}}{L}

Equivalent

\frac{1}{K_{1}} + \frac{1}{K_{2}} = \frac{1}{4 y π R^{2}} + \frac{L}{y π R^{2}}

Since,

K_{1} x_{1} = K_{2} x_{2} = w

Elastic potential energy of the system
U=

\frac{1}{2} k_{1} x_{1} + \frac{1}{2} k_{2} x_{2}^{2}

= \frac{1}{2} k_{1} (\frac{W}{k_{1}})^{2} + \frac{1}{2} k_{2} (\frac{W}{k_{2}})^{2}

= \frac{1}{2} W^{2} + (\frac{1}{k_{1}} + \frac{1}{k_{2}})

= \frac{1}{2} W^{2} + (\frac{5 L}{4 y π R^{2}})

(\frac{5 w^{2} L}{8 π y R^{2}})

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