A sand bag of mass m is hanging from a light spring of stiffness k. Find the elongation of the spring. If we pull the sand bag down by an additional distance x and release it, find its acceleration and maximum velocity of block.

Explanation

Since `F=-kx hat(j)` and downward force =`+mg hat(j)`
`Sigma F_(y)=0`
`mg= k Delta x`
`Delta x =(mg)/(K)`
If we apply addition force to pull it down by distance x, then FBD
Upward forces are =`k(x + Delta x)`
Downward force =`F+mg`
If we remove from F then `Sigma F_(y)=ma_(y)`
`K(x+Delta x)-mg=ma`

Maximum acceleration,
`a = (k(Delta x +x))/(m)-g`
`a =(k Delta x)/(m) +(k x)/(m) -g =(k x)/( m)`
Maximum velocity: `F=kx`
`ma=kx implies (vdv)/(dx)=(kx)/(m)implies (v^2)/(2)=(kx^(2))/(2m)`
`v=sqrt((k)/(m)x)`.