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Concepts Of Physics MCQ Edition [Volume 1]

Exam: Concepts Of Physics MCQ Edition [Volume 1]
Subject: Physics
Chapter: Some Mechanical Properties of Matter

Concepts Of Physics MCQ Edition [Volume 1] · Physics · Some Mechanical Properties of Matter

As a metal wire is stretched by a load, the fractional change in...

As a metal wire is stretched by a load, the fractional change in its volume

\dfrac{\Delta V}{V}

is proportional to

Options

A $\dfrac{l}{\Delta l}$
B\left(\dfrac{\Delta l}{l}\right)^2
C $\sqrt{\dfrac{\Delta l}{l}}$
D $\dfrac{\Delta l}{l}$

Correct answer

D. $\dfrac{\Delta l}{l}$

Step-by-step solution

Volume of a cylindrical wire is

V = \pi r^2 l

Differentiating to find the fractional change in volume:

\dfrac{\Delta V}{V} = 2\dfrac{\Delta r}{r} + \dfrac{\Delta l}{l}

Poisson's ratio

\sigma

is defined as

\sigma = -\dfrac{\Delta r / r}{\Delta l / l}

\Rightarrow \dfrac{\Delta r}{r} = -\sigma \dfrac{\Delta l}{l}

Substituting this into the expression for fractional change in volume:

\dfrac{\Delta V}{V} = -2\sigma \dfrac{\Delta l}{l} + \dfrac{\Delta l}{l} = (1 - 2\sigma)\dfrac{\Delta l}{l}

Since

\sigma

is a constant for a given material,

(1 - 2\sigma)

is constant.

Therefore,

\dfrac{\Delta V}{V} \propto \dfrac{\Delta l}{l}

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