JEE Main 2019 · Physics · Waves and Sound · medium

A stretched wire in a sonometer, kept under tension by a hanging metal block, vibrates in its fundamental mode at 300 Hz. The block has density 4000…

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A stretched wire in a sonometer, kept under tension by a hanging metal block, vibrates in its fundamental mode at

300\ \mathrm{Hz}

. The block has density

4000\ \mathrm{kg\,m^{-3}}

. When the block is completely immersed in water (density

1000\ \mathrm{kg\,m^{-3}}

) without touching the container, the fundamental frequency of the wire (length unchanged) becomes

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Answer: (A) $150\sqrt3\ \mathrm{Hz}$

The tension is the apparent weight of the block. In air

T = Vρg

with

\rho=4000

; in water the buoyant force

V(1000)g

reduces it to

T' = V(4000-1000)g = \tfrac34 T

. For a wire of fixed length and mass per unit length,

f\propto\sqrt T

, so

f' = 300\sqrt{3/4}=150\sqrt3\ \mathrm{Hz}\approx 260\ \mathrm{Hz}

. Scaling the frequency linearly with tension (

225\ \mathrm{Hz}

) is the usual error.

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