MHT CET · Physics · Oscillations
A mass M attached to a horizontal spring executes S.H.M. of amplitude \(\mathrm{A}_{1}\). When
the mass M passes through its mean position, then a smaller mass \(\mathrm{m}\) is placed over
it and both of them move together with amplitude \(\mathrm{A}_{2}\). The ratio of \(\left(\frac{\mathrm{A}_{1}}{\mathrm{~A}_{2}}\right)\) is
- A \(\frac{\mathrm{M}+\mathrm{m}}{\mathrm{M}}\)
- B \(\left(\frac{\mathrm{M}}{\mathrm{M}+\mathrm{m}}\right) \frac{1}{2}\)
- C \(\left(\frac{M+m}{M}\right) \frac{1}{2}\)
- D \(\frac{\mathrm{M}}{\mathrm{M}+\mathrm{m}}\)
Answer & Solution
Correct Answer
(C) \(\left(\frac{M+m}{M}\right) \frac{1}{2}\)
Step-by-step Solution
Detailed explanation
\(M V=(M+m) V^{\prime}\)
\(\frac{1}{2}(M+m) V^{\prime 2}=\frac{1}{2} k A_{2}^{2}\)
\(\frac{1}{2} M V^{2}=\frac{1}{2} k A_{1}^{2}\)
\(\frac{M+m}{M} \cdot \frac{V^{\prime 2}}{V^{2}}=\frac{A_{2}^{2}}{A_{1}^{2}}\)
\(\frac{M+m}{M} \cdot\left(\frac{M}{M+m}\right)^{2}=\frac{A_{2}^{2}}{A_{1}^{2}}\)
\(\frac{A_{1}^{2}}{A_{2}^{2}}=\frac{M+m}{M}\)
\(\therefore \frac{A_{1}}{A_{2}}=\left(\frac{M+m}{M}\right)^{\frac{1}{2}}\)
\(\frac{1}{2}(M+m) V^{\prime 2}=\frac{1}{2} k A_{2}^{2}\)
\(\frac{1}{2} M V^{2}=\frac{1}{2} k A_{1}^{2}\)
\(\frac{M+m}{M} \cdot \frac{V^{\prime 2}}{V^{2}}=\frac{A_{2}^{2}}{A_{1}^{2}}\)
\(\frac{M+m}{M} \cdot\left(\frac{M}{M+m}\right)^{2}=\frac{A_{2}^{2}}{A_{1}^{2}}\)
\(\frac{A_{1}^{2}}{A_{2}^{2}}=\frac{M+m}{M}\)
\(\therefore \frac{A_{1}}{A_{2}}=\left(\frac{M+m}{M}\right)^{\frac{1}{2}}\)
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