A body of mass `m` is attached by an inelastic string to a suspended spring of spring constant `k`. Both the string and the spring have negligible mass and the string is inextensible and of length `L`. Initially, the mass `m` is at rest.
The largest amplitude `A_(max)`, for which the string will remain taut throughout the motion is

Options

A. `(mg)/(2k)`
B. `(mg)/(k)`
C. `(2mg)/(3k)`
D. `L`

Answer

B

Explanation

In the position of equilibrium when the tension in the string `=mg` the extension in the spring is `x_0=(mg)/(k)`. If the amplitude exceeds this value, then in its upward motion, the body will rise to a height at which the string will become loose and `T=0` Hence maximum amplitude is `x_m=(mg)/(k)`.
It also follows from the fact that for `Tge0` the maximum downwards acceleration that the body can have is `g`. Since `f(t)=(g-T(t))/(m)`, `f(t)leg`. Hence maximum amplitude is `A_m=(g)/(omega^2)=(mg)/(k)`