A body of mass m falls from a height h onto the pan of a spring balance. The masses of the pan and spring are negligible. The force constant of the spring is k. The body sticks to the pan and oscillates simple harmonically. The amplitude of oscillation is

Options

A. `(mg)/(k)`
B. `(mg)/(k)sqrt(1+(2hk)/(mg))`
C. `(mg)/(k)(1+sqrt(1+(2hk)/(mg)))`
D. `(mg)/(k)sqrt(1+((2hk)/(mg))-1)`

Answer

B

Explanation

Energy conservation between (1) and (2)
`mg(h+x)=1/2kx^2`
`kx^2-2mgx-2mgh=0`
`x=(2mg+-sqrt((-2mg)^2+4k*2mgh))/(2k)`
`=(mg)/(k)[1+sqrt(1+(2kh)/(mg))]`
x: maximum compression of spring
In equilibrium, `mg=kx_0`
`x_0=(mg)/(k)`
Amplitude `=x-x_0=(mg)/(k)sqrt(1+(2kh)/(mg))`