Let us consider the following diagram in whcih a block of mass M is being supported by a uniform rope of mass 1 kg and length 10 metre. Apulse is created at the bottom of the rope and it reaches the top. In column I the value of M in kg is given and in column II time (in sec) after which the pulse reaches the top is given. Match them:

Answer

A, B, C, D

Explanation

Tension in the string at point A is `T = Mg + (m/L x)g =(M + 1/10 x)g, v= sqrt(T/mu)`
`(dx)/(dt) = sqrt((M +x//10)/(1//10)) = sqrt((10 M +x)g)`
`int(0)^(1)dt = int_(0)^(L=10) (dx)/sqrt((10 M + x)g)`
`t = 2/sqrt(10) [(10 M + 10)^(1//2)] =2[(M +1)^(1//2) - M^(1//2)]`...........(i)
a) `M = 1kg, t= 2 [sqrt(2)-1]`
`M = 2kg, t = 2[sqrt(3)- sqrt(2)], M = 3kg, t=2 [2-sqrt(3)], M = 4kg,t=2[sqrt(5)-2]`