COMEDK · Physics · 15. Oscillations

A wire of negligible mass having uniform area of cross section 'A' and young modulus ' Y ' is used to suspend a point mass ' m '. The point mass executes simple harmonic motion in a vertical plane with a period ' T ', then the length of the wire is :

A \(L=\dfrac{T Y^2 A}{4 \pi m^2}\)
B \(L=\dfrac{T^2 Y A}{4 \pi^2 m}\)
C \(L=\dfrac{T^2 Y A}{4 \pi m^2}\)
D \(L=\dfrac{T Y^2 A}{4 \pi^2 m}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(L=\dfrac{T^2 Y A}{4 \pi^2 m}\)

Step-by-step Solution

Detailed explanation

The wire acts as a spring with a spring constant \(k\) given by the formula \(k = \dfrac{YA}{L}\), where \(Y\) is the Young modulus, \(A\) is the area of cross-section, and \(L\) is the length of the wire. The time period \(T\) of a mass \(m\) executing simple harmonic motion…

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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