A body of mass m falls from a height h on to the pan of a spring balance. The masses of the pan and spring are negligible. The spring constant of the spring is k. Having stuck to the pan the body starts performing harmonic oscillations in the vertical direction. Find the amplitude and energy of oscillation.

Explanation

Suppose by falling down through a height h, the mass m compresses the spring balance by a length x.
`x = (mg)/(k) , omega = sqrt((k)/(m))`
Velocity at Q v= `sqrt(2gh)`
`because " " v = omega sqrt(A^(2)- x^(2))`
`sqrt(2gh)= sqrt((k)/(m))sqrt(A^(2)-((mg)/(k))^(2))`
` implies A = (mg)/(k)sqrt(1+(2kh)/(mg))`
Energy of oscilation
`= (1)/(2)kA^(2)= (1)/(2)k((mg)/(k))^(2)(1+(2kh)/(mg))= mgh + (mg)^(2)/(2k)`