Question

A mass `M` attached to a horizontal spring executes `SHM` with an amplitude `A_(1)`. When mass `M` passes through its mean position a smaller mass `m` is placed over it and both of them move togther with amplitude `A_(2)`. Ratio of `((A_(1))/(A_(2)))` is:
A. `(M+m)/(M)`
B. `((M)/(M+m))^(1//2)`
C. `((M+m)/(M))^(1//2)`
D. `(M)/(M+m)`

Answer 1

Correct Answer - C
When mass `M` attached to a horizontal spring executes S.H.M., its frequency of oscillation is
`v_(1)=(1)/(2pi)sqrt((k)/(M))` …(i)
When a smaller mass `m` is placed over mass `M` , the frequency of oscillation is
`v_(2)=(1)/(2pi)sqrt((k)/(M+m))` ...(ii)
According to law of conservation of linear momentum
`M_(1)upsilon_(1)=(M+m)upsilon_(2)`
or `MA_(1)omega_(1)=(M+m)A_(2)omega_(2)`
or `MA_(1)2piv_(1)=(M+m)A_(2)2piv_(2)`
`:. (A_(1))/(A_(2))=(M+m)/(M)xx(v_(2))/(v_(1))=((M+m))/(M)xxsqrt((M)/(M+m))`
`=sqrt((M+m)/(M))`