A uniformly tapering conical wire is made from a material of...

A uniformly tapering conical wire is made from a material of Young's modulus

Y

and has a normal, unextended length

L

. The radii, at the upper and lower ends of this conical wire, have values

R

and

3R

, respectively. The upper end of the wire is fixed to a rigid support and a mass

M

is suspended from its lower end. The equilibrium extended length, of this wire, would equal: A uniformly tapering conical wire is made from a material of Young's modulus Y and has a normal, unextended length L .

A uniformly tapering conical wire is made from a material of Young's modulus Y and has a normal, unextended length L .

Options

A $L$ \left( $1 + \dfrac{2}{9}\dfrac{Mg}{\piYR^{2}}\right)$
B $L$ \left( $1 + \dfrac{1}{9}\dfrac{Mg}{\piYR^{2}}\right)$
C $L$ \left( $1 + \dfrac{1}{3}\dfrac{Mg}{\piYR^{2}}\right)$
D $L$ \left( $1 + \dfrac{2}{3}\dfrac{Mg}{\piYR^{2}}\right)$

Correct answer

C. $L$ \left( $1 + \dfrac{1}{3}\dfrac{Mg}{\piYR^{2}}\right)$

Step-by-step solution

r = \dfrac{2R}{L}x + R

For a small section

\dfrac{F}{A} = Y\dfrac{dl}{dx}

⟹{∫}_{0}^{L}dL = {∫}_{0}^{L}\dfrac{Mgdx}{\pi\left[\dfrac{2R}{L}x \times R\right]2Y}

∆ L = \dfrac{Mg}{\piY} \left[-

\dfrac{1}{{ \left[\dfrac{2Rx}{L} +

R\r

right]}_{0}^{L}}

\times

\dfrac{L}

{2R}\right]

= \dfrac{MgL}{3\piR^{2}Y}

{L}_{f} = L +

\Delta

L = L[ 1 + \dfrac{mg}{3\piR^{2} Y} ]

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