A uniform rope of length $20 \mathrm{m}$ and mass $8 \mathrm{kg}$ hangs vertically from a rigid support. A block of mass $2 \mathrm{kg}$ is attached to the free end of the rope. A transverse pulse of wavelength $0.06 \mathrm{m}$ 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 string oscillates according to the equation,
$y^{\text{'}}=\left(0.80 \mathrm{cm}\right)\sin \left[\left(\frac{\mathrm{\pi }}{3} {\mathrm{cm}}^{-1}\right)x\right]\cos \left[\left(40\mathrm{\pi } {\mathrm{s}}^{-1}\right)t\right]$
What is the

$\left(\mathrm{a}\right)$ amplitude

$\left(\mathrm{b}\right)$ speed of the two waves (identical except the direction of travel) whose superposition gives this oscillation?

$\left(\mathrm{c}\right)$ What is the distance between nodes?

$\left(\mathrm{d}\right)$ What is the transverse speed of a particle of the string at the position $x=2.1 \mathrm{cm}$ when $t=0.50 \mathrm{s}$?

The vibration of a string of length $60 \mathrm{cm}$ is represented by the equation,

$y=3 \cos \left(\frac{\pi x}{\text{20}}\right)\cos \left(72\pi t\right)$ where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\sec$.

(i) Write down the component waves whose superposition gives the above wave.

(ii) Where are the nodes and antinodes located along the string.

(iii) What is the velocity of the particle of the string at the position $x=5 \mathrm{cm}$ and $t=0.25 \sec$.