Question

A chain hangs over a nail with 2.0m on one side and 6.0m on other side .if the force of friction is equal to the weight of 1.0m of the chain . Calculate the time required for the chain to slide of the nail.

Answer 1

mass / length of chain = μ
friction force = weight of 1.0 m of the chain = μ * 1.0 * g = μ g
Let y(t) be the length of the chain on the heavier side. Let L be the total length of the chain. The length of chain on the other side is (L- y(t).

Each part of the chain has the same displacement, speed and acceleration (magnitude).

Equation of motion:  
   μ y(t) g - μ (L- y(t)] g - μ g = L μ a = L μ d²y(t)/dt²
   d²y(t)/dt² = 2g/L  y(t) - (L+1)g/L 
                  = 2g/L * [y - (L+1)/2 ]

   y(t) = (L+1)/2 + A exp(√(2g/L) t)  
   Given  L = 8 m ,  y(t=0sec) = 6 m
    y(0) = 9/2  +  A exp(√2.5 t) = 6 m   =>  A = 3/2 m
    y(t) = 4.5 +  1.5 exp(1.581 t)

Now to find  time t when  y = 8 m.
    y(t) = 4.5 + 1.5 exp(1.581 t) = 8
         => t = 0.536 sec