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JEE Physics 2023 Question with Solution

A steel wire with mass per unit length 7.0×10−3 kg m−17.0 \times 10^{-3} \, {kg\, m}^{-1} is under tension of 70 N70 \, \text{N}. The speed of transverse waves in the wire will be:

  • A

    200π200 \pi m/s

  • B

    100100 m/s

  • C

    1010 m/s

  • D

    5050 m/s

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: tension T=70 NT = 70 \, \text{N} and mass per unit length μ=7.0×10−3 kg m−1\mu = 7.0 \times 10^{-3} \, {kg\, m}^{-1}.

Find: the speed of transverse waves in the wire.

For a stretched wire, the wave speed is given by

v=Tμv = \sqrt{\frac{T}{\mu}}

Substituting the values,

v=707.0×10−3=100 m/sv = \sqrt{\frac{70}{7.0 \times 10^{-3}}} = 100 \, \text{m/s}

Therefore, the speed of transverse waves is 100 m/s100 \, \text{m/s}. The correct option is B.

Approach Solution - 2

Given: T=70 NT = 70 \, \text{N} and μ=7.0×10−3 kg m−1\mu = 7.0 \times 10^{-3} \, {kg\, m}^{-1}.

Find: wave speed in the stretched steel wire.

The velocity of transverse waves in a stretched string is given by

v=Tμv = \sqrt{\frac{T}{\mu}}

Substituting the values,

v=707.0×10−3=100 m/sv = \sqrt{\frac{70}{7.0 \times 10^{-3}}} = 100 \, \text{m/s}

Hence, the correct option is B.

Common mistakes

  • Using a direct proportionality like v∝T/μv \propto T/\mu instead of the square-root relation is incorrect. The correct formula is v=T/μv = \sqrt{T/\mu}.

  • Ignoring the power of 10−310^{-3} in the linear mass density gives a wrong numerical value. Always substitute μ=7.0×10−3\mu = 7.0 \times 10^{-3} carefully.

  • Confusing mass per unit length with total mass is wrong because the formula requires linear density μ\mu, not the full mass of the wire.

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