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

Correct answer

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

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]}^{2}Y}

∆ L = \dfrac{Mg}{\piY}\left[- \dfrac{1}{{\left[\dfrac{2Rx}{L} + R\right]}_{0}^{L}} \times \dfrac{L}{2R}\right]

= \dfrac{MgL}{3\pi{R}^{2}Y}

{L}_{f} = L + \DeltaL = L[ 1 + \dfrac{mg}{3\pi{R}^{2} Y} ]

Practice Mechanical Properties of Solids on Quantrex Academy →

More from Mechanical Properties of Solids

Figure represents the extension ( Δ l ) of a wire of length 1 meter, suspended from the… JEE Main 2026 A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the… JEE Main 2026 The two wires A and B of equal cross-section but of different materials are joined… JEE Main 2026 A copper wire of length 3 m is stretched by 3 mm by applying an external force. The… JEE Main 2026 The Young's modulus of steel wire of radius r and length L is Y . If the radius r and… JEE Main 2026 Two wires as shown in the figure below, made of steel and have breaking stress of 12 ×… JEE Main 2026 A cube has side length 5 cm and modulus of rigidity 10^5 N/m ^2 . The displacement… JEE Main 2026 A metal string A is suspended from a rigid support and its free end is attached to a… JEE Main 2026 Full Mechanical Properties of Solids list All JEE Main PYQs