A CHAIN OF MAAS M AND LENGHT L is heaped on a smooth ground

1 find the vertical force required to pull the chain with constant accn A and velocity v as the function of the lenght x pulled up..

2 find the w.d by the driving force and power delivered by external agent

3 what is the normal reaction offered by the grount also n.r as a function of distance raised

4 find accn of C.M.

plz give easy expressions with full explanation calculus is weak xi standard i approve

let mass per unit length is k ... k = M/L

now , after some time let x part of chain is above the ground

mass of part above ground = kx

let constant force applied by external source = F then

F - (kx)g = (kx)a            (a is accleration)

F = (kx)g + kx(a)       ....................1

from above eq , a = [F - (kx)g]/kx

       a = [F/kx - g]         ................2

now ,  a = dv/dt

multiply both sides by dx

   a(dx)= dv(dx/dt)                   {dx/dt = v }

   adx = vdv = v²/2           ............3

from eq 2 & 3

  [F/kx - g ]dx = v²/2

  integrating both sides

 [Flnx/k - gx] = v² 

 v = { [Flnx/k - gx ]}^1/2               ...................4

workdone = F.dx

F =(kxg + kxa)

k = M/L          (mass per unit length)

W= k(g+a)xdx

W = k(g+a)x²/2                  (work performed by external agent)

when x part  is above the ground then L-x part is on ground ,

mass of this (L-x) part = k(L-x)

normal reaction will be only due to this part

N = k(L-x)g