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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
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