Speed of a transverse wave on a straight wire (mass 6.0 g, length 60 cm and area of cross-section $1.0\,\text{mm}^2$) is $90\,\text{ms}^{-1}$. If the Young's modulus of wire is $16 \times 10^{11}\,\text{Nm}^{-2}$, the extension of wire over its natural length is:

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Elasticity & SHM

Speed of a transverse wave on a straight wire (mass 6.0 g, length 60 cm and area of cross-section

1.0 mm^{2}

) is

90 ms^{- 1}

. If the Young's modulus of wire is

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

, the extension of wire over its natural length is:

Answer: (A)

The wave speed on the wire,

v = T / μ

, fixes the tension as

T = μ v^{2}

with linear density

μ = m / ℓ

. Young's modulus relates this tension to the extension through

\frac{T}{A} = Y \frac{Δ ℓ}{ℓ}

, so

Δ ℓ = \frac{T ℓ}{A Y} = \frac{μ v ^{2} ℓ}{A Y}

. With

μ = \frac{6.0 \times 1 0 ^{- 3}}{0.6} = 0.01 kg/m

v = 90 m/s

ℓ = 0.6 m

A = 1.0 \times 1 0^{- 6} m^{2}

and

Y = 16 \times 1 0^{11} N/m^{2}

, this gives

Δ ℓ = \frac{0.01 \times 8100 \times 0.6}{1 0 ^{- 6} \times 16 \times 1 0 ^{11}} = 0.03 mm

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Single correct option