Question

A chain of length L and mass per unit length rho is piled on a horizontal surface. One end of the chain is lifted vertically with constant velocity by a force P.

Answer 1

Answer:

P as a function of height x of the end above the surface will be : [(rho)(gx+v²)]

Explanation:

• f = m(dv/dt) + v(relative)(dm/dt) : to pull the chain

(dv/dt)=0

f =(rho)v²

force by gravity on chain : (rho)gx

P = (rho)[ gx + v² ]

P is the sum of the force to pull the chain

and force to counter force by gravity.

Answer 2

The work done by the force P is (ρL²g)/2 + ρLv²

Explanation:

The exiting force is given as:

F = V.dm/dt + m.dv/dt

Since, the velocity is constant. dv/dt = 0

The force P as function given as:

P = ρgx + ρv²

Now, the change in kinetic energy is given as:

ΔK.E = wp + wg

wp = $\int_{0}^{L}$ ρgx.dx + $\int_{0}^{L}$ (ρ.dx)v²

∴ wp = (ρL²g)/2 + ρLv²