Question

A uniform rope of mass 0.1 kg and length 2.5 m hangs from ceiling. The speed of transverse wave in the rope at upper end and at a point 0.5 m distance from lower end will be

• A. $5m{s}^{-1},2.24m{s}^{-1}$
• B. $10m{s}^{-1},3.23m{s}^{-1}$
• C. $7.5m{s}^{-1},1.2m{s}^{-1}$
• D. None of the above

Answer 1

Answer: (A)

Mass per unit length $m=\frac{M}{L}$
Tension at the point T = mass of x metre of rope $\times$ g
$=mxg$
where x is the distance of point from lower end.
Velocity of transverse wave along tihe string
$v=\sqrt{\frac{T}{m}}=\sqrt{x\frac{mg}{m}}=\sqrt{xg}$
Given, $x=0.5$ and $g=10m{s}^{-2}$
$v=\sqrt{0.5\times 10}=\sqrt{5}=2.24m{s}^{-1}$
At upper end, the velocity is given by
$v=\sqrt{rg}=\sqrt{2.5\times 10}=5m{s}^{-1}$