Question

A uniform thin rope of length $12m$ and mass $6kg$hangs vertically from a rigid support and a block of mass $2kg$ is attached to its free end. A transverse short wave train of wavelength $6cm$ is produced at the lower end of the rope. What is the wavelength of the wave train (in $cm$) when it reaches the top of the rope?

• A. $12$
• B. $3$
• C. $9$
• D. $6$

Answer 1

The correct option is A

$12$

Step 1. Given data:

Length of rope, $l=12m$, mass of rope,$m=6kg$

Mass of block,$M=2kg$

Wavelength at lower end $\lambda =6cm$

Step 2. Finding the Wavelength at top end $\lambda \text{'}$:

$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

$\lambda =\frac{c}{f}=c2\pi \sqrt{\frac{m}{k}}$

$\lambda \mathrm{\alpha }\sqrt{\mathrm{m}}$

Mass that hangs at lower end is just of block i.e.$m=2kg$

Mass that hangs at top end is just of block i.e.$m\text{'}=(2+6)kg=8kg$

$\frac{\lambda }{\lambda \text{'}}=\sqrt{\frac{m}{m\text{'}}}=\sqrt{\frac{2}{8}}$

$\frac{6}{\lambda \text{'}}=\frac{1}{2}$

$\lambda \text{'}=12cm$

Hence, option A is correct