Question

A uniform rope of length $12m$ and mass $6kg$ hangs vertically from a rigid support. A block of mass $2kg$ is attached to the free end of the rope. A transverse pulse of wavelength $0.06m$ is produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top of the rope?

• A. $0.06m$
• B. $0.12m$
• C. $0.09m$
• D. None of these

Answer 1

Answer: (B)

$0.12m$

Given, $l=12m$

$m_1=6kg$

$m_2=2kg$

wavelength at lower end of the rope $={\lambda }_1=0.06m$

Let, the wavelength at the top of the rope be$={\lambda }_2$

As we know that,

$velocity\propto \sqrt{T}$

$\therefore \lambda \propto \sqrt{T}$

$T_1$ (Tension at lower end) $=m_{2}g=20N$

$T_2$ (Tension at upper end) $=(m_1+m_2)g=(6+2)g=80N$

Now,

$\frac{{\lambda }_1}{{\lambda }_2}=\sqrt{\frac{T_1}{T_2}}=\sqrt{\frac{20}{80}}$

$\Rightarrow {\lambda }_2={\lambda }_2\sqrt{\frac{80}{20}}=0.06\times 2=0.12m$

Thus the wavelength of the pulse when it reaches the top of the rope is $0.12m$