JEE Mains · Physics · STD 11 - 13. oscillations

In the given figure, a mass \(M\) is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is \(k\). The mass oscillates on a frictionless surface with time period \(T\) and amplitude \(A\). When the mass is in equilibrium position, as shown in the figure, another mass \(m\) is gently fixed upon it. The new amplitude of oscillation will be

A \(A \sqrt{\frac{M-m}{M}}\)
B \(A \sqrt{\frac{M}{M+m}}\)
C \(A \sqrt{\frac{M+m}{M}}\)
D \(A \sqrt{\frac{M}{M-m}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(A \sqrt{\frac{M}{M+m}}\)

Step-by-step Solution

Detailed explanation

Momentum of system remains conserved. \(p_{i}=p_{i}\) \(MA \omega=( m + M ) A ^{\prime} \omega^{\prime}\) \(M A \sqrt{\frac{k}{M}}=(m+M) A^{\prime} \sqrt{\frac{k}{m+M}}\) \(A ^{\prime}= A \sqrt{\frac{ M }{ M + m }}\)

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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