MHT CET · Physics · Waves and Sound

A uniform metal wire has length ' \(L\) ', mass ' \(M\) ' and cross-sectional area ' \(A\) '. It is under tension ' \(T\) ' and ' \(V\) ' is the speed of transverse wave along the wire. The density of the wire

A \(\frac{A T}{V^2}\)
B \(\frac{T}{A^2 V}\)
C \(\frac{T}{V^2 A}\)
D \(\frac{V^2}{A^2 T}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{T}{V^2 A}\)

Step-by-step Solution

Detailed explanation

For transverse waves on a wire \(V=\sqrt{\frac{T}{m}}\)
The mass per unit length can be written as \(m=\frac{M}{L}=\frac{A L \rho}{L}=A \rho\)
\(\begin{aligned} & \therefore V=\sqrt{\frac{T}{m}}=\sqrt{\frac{T}{A \rho}} \\ & \therefore V^2=\frac{T}{A \rho} \\ & \therefore A=\frac{T}{V^2 \rho}\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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