The speed of a transverse wave passing through a string of length 50 cm and mass 10 g is $60 \mathrm{~ms}^{-1}$. The area of cross-section of the wire is $2.0 \mathrm{~mm}^{2}$ and its Young's modulus is $1.2 \times 10^{11} \mathrm{Nm}^{-2}$. The extension of the wire over its natural length due to its tension will be $x \times 10^{-5} \mathrm{~m}$. The value of $x$ is ______.

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Elasticity & SHM

The speed of a transverse wave passing through a string of length 50 cm and mass 10 g is

60 ms^{- 1}

. The area of cross-section of the wire is

2.0 mm^{2}

and its Young's modulus is

1.2 \times 1 0^{11} Nm^{- 2}

. The extension of the wire over its natural length due to its tension will be

x \times 1 0^{- 5} m

. The value of

x

is ______.

Answer: 15

The wave speed on a string is

v = \frac{T}{μ}

, where

T

is tension and

μ

is linear mass density.

Linear mass density:

μ = \frac{m}{ℓ} = \frac{10 \times 1 0 ^{- 3}}{0.5} = 0.02

kg/m

From wave speed:

60 = \frac{T}{0.02}

Solving:

T = 6 0^{2} \times 0.02 = 72

N

Using Young's modulus:

Δ ℓ = \frac{F ℓ}{A Y}

Δ ℓ = \frac{72 \times 0.5}{2 \times 1 0 ^{- 6} \times 1.2 \times 1 0 ^{11}} = \frac{36}{2.4 \times 1 0 ^{5}} = 15 \times 1 0^{- 5} m

Therefore,

x = 15

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Integer answer