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. $5\, ms^{-1},2.24\, ms^{-1}$
• B. $10\, ms^{-1},3.23\, ms^{-1}$
• C. $7.5\, ms^{-1},1.2\, ms^{-1}$
• 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 $g = 10\, ms^{-2}$
$ v= \sqrt{0.5 \times 10} =\sqrt5 = 2.24\, ms^{-1}$
At upper end, the velocity is given by
$ v =\sqrt {rg} = \sqrt{2.5 \times 10} = 5\, ms^{-1}$