Figure shows a rope of variable mass whose linear mass density is given by , `lambda = Ax` , where A is + ve constant and x is distance from left end . It b is placed on a smooth horizontal surface . A force F is acting on it as shown .
(i) Find total mass of the rope .
(ii) Find tension at point P.

Explanation

(i) Consider a differential element of thickness dx at a distance from left end . Let its mass be dm .
Now, `dm = lambda dx = Axdx`
Total mass, ` M int dm = A overset (L)underset (0) int xdx`
` M = (AL^(2))/2 `
(ii) Net acceleration of the rope is
` a = F/M = (2F)/(AL^(2))`
For system (1)
` T = m_(1)a = (Al_(2))/2 xx (2F)/(AL^(2))`
` T = m_(1)a = (Al_(2))/2 xx (2F)/(AL^(2)) " "[ :'" Here " m_(1) = overset (i) underset (0) int lambda dx = overset (i) underset (0) int Axdx]`
` rArr T = F (l^(2)) /(L^(2)) `