Question

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

• A. $5\text{m/s},2.24\text{m/s}$
• B. $10\text{m/s},3.23\text{m/s}$
• C. $7.5\text{m/s},1.2\text{m/s}$
• D. $2.25\text{m/s},5\text{m/s}$

Answer 1

The correct option is A $5\text{m/s},2.24\text{m/s}$

Velocity of wave in the string $v=\sqrt{\frac{T}{\mu }},$
Mass per unit length of the string
$\mu =\frac{0.1}{2.5}$
At topmost point
$T=0.1\times 10=1\text{N},$

Velocity at upper point $v=\sqrt{1\times 25}$
$v=5\text{m/s}$
Now, at $0.5\text{m}$ distance from lower point
Tension
$\Rightarrow T=\frac{1}{2.5}\times 0.5=\frac{1}{5}\text{N}$
Velocity of wave is
$v=\sqrt{\frac{1}{5}\times \frac{25}{1}}=\sqrt{5}=2.24\text{m/s}$.