A small block of mass m is kept on a bigger block of mass M which is attached to a vertical spring of spring constant k as shown in the figure. The system oscilates verticaly. a.Find the resultant force on the smaller block when it is displaced through a distance x above its equilibrium position. b. find the normal force on the smaller blok at this position. When is this force smallest smaller block at this position. When is this force smallest in magnitude? c. What can be the maximum amplitude with which the two blocks may oscillate together?

Explanation

(a)
When displacement about equilibrium position is x, acceleration of block `a=omega^x`, towards equilibrium position
`omega=(2pi)/(T)=(2pi)/(2pisqrt((M+m)/(k)))=sqrt((k)/(M+m))`
Upper block:

Resultant force on m: `ma=momega^2x=(mkx)/(M+m)`
(b) `mg-N=ma`
`N=m(g-a)=m(g-omega^2x)`
`=m[g-(kx)/(M+m)]`
N is minimum when x is maximum i.e., `x=A` i.e., at the highest point.
(c) N will be minimum when `x=A` (amplitude)
If blocks move together, `N_(min){=m[g-(kA)/(M+m)]}ge0`
`gge(kA)/(M+m)`
`Ale((M+m)g)/(k)`
`A_(max)=((M+m)g)/(k)`