A unifrom rope of mass 0.1kg and length 2.45 m hangs from a rigid support. The time taken by the transverse wave formed in the rope to travel through the full length of the rope is (Assume g=`=9.8 m//s^2`)

Options

A. 1 s
B. 2 s
C. 3 s
D. 4 s

Answer

A

Explanation

As the rope has mass and it is suspended vertically, tension in it will be different at different points. For a point at a distance x from the free end, tension will be due to the weight of the rope below it. If M is the mass of rope of length L, the mass of length x of the rope will be `(M//L)x`.
`therefore T=[(M)/(L)x]g`
and `v=sqrt((T)/(mu))=sqrt((Mgx)/(L(M//L)))=sqrt(gx)" "...(i)`
The tension and the velocity of the wave is different at different points. If at point x the wave travels a distance dx in time dt,
`v=(dx)/(dt)orsqrt(gx)=(dx)/(dt)` (Using (i))
or `intdt=int(dx)/(sqrt(gx))ort=(1)/(sqrt(g))int_(0)^(L)x^(-1//2)dx`
or `t=2sqrt(((L)/(g)))`
As `L=2.45m therefore t=2sqrt((2.45)/(9.8))=1s`