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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

A metal wire has a length of l_1 when subjected to a tension T_1 ,...

A metal wire has a length of

l_1

when subjected to a tension

T_1

, and a length of

l_2

when the tension is

T_2

. The natural length of the wire is

Options

A $\sqrt{l_1 l_2}$
B $\dfrac{l_1 T_2 - l_2 T_1}{T_2 - T_1}$
C $\dfrac{l_1 + l_2}{2}$
D $\dfrac{l_1 T_2 + l_2 T_1}{T_2 + T_1}$

Correct answer

B. $\dfrac{l_1 T_2 - l_2 T_1}{T_2 - T_1}$

Step-by-step solution

Let the natural length of the wire be

L

, cross-sectional area be

A

, and Young's modulus be

Y

.

The extension in the wire under tension

T_1

is

\Delta l_1 = l_1 - L

.

From Hooke's law,

Y = \dfrac{T_1 L}{A \Delta l_1} \Rightarrow T_1 = \dfrac{YA}{L}(l_1 - L)

.

Similarly, the extension under tension

T_2

is

\Delta l_2 = l_2 - L

.

T_2 = \dfrac{YA}{L}(l_2 - L)

.

Dividing the two equations, we get:

\dfrac{T_1}{T_2} = \dfrac{l_1 - L}{l_2 - L}

T_1 l_2 - T_1 L = T_2 l_1 - T_2 L

L(T_2 - T_1) = l_1 T_2 - l_2 T_1

L = \dfrac{l_1 T_2 - l_2 T_1}{T_2 - T_1}

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