Question

A string of linear mass density $0.5gc{m}^{-1}$ and a total length 30 cm is tied to a fixed wall at one end and to a frictionless ring at the other end (figure 15-E4). The nng can move on a vertical rod. A wave pulse is produced, on the string which moves towards the ring at a speed of $20cm{s}^{-1}$. The pulse is symmetric about its maximum, which is located at a distance of 20 cm from the end joined to the ring. (a) Assuming that the wave is reflected from the ends without loss of energy, find the time taken by the string to regain its shape. (b) The shape of the string changes periodically with time.Find this time period. (c) What is the tension in the string ?

Answer 1

The crest reflects as a crest here as the wave is travelling from denser to rarer medium

$\Rightarrow$ Phase change = 0

(a) To regain shape, after travelled by the wave

S = 20 + 20 = 40 cm

Wave speed, v = 20 m/s

$\Rightarrow Time=\frac{s}{v}=\frac{40}{20}=2sec$

(b) The wave regains its shape, after travelling a period distance

$=2\times 30=60cm$

$\therefore$ Time period $=\frac{60}{20}=3sec$

(c) Frequency, $n=\left(\frac{1}{3}se{c}^{-1}\right)$

$n=\frac{1}{2l}\sqrt{\left(\frac{T}{m}\right)}$

m = mass per unit length = 0.5 gm/cm

$\Rightarrow \frac{1}{3}=\frac{1}{(2\times 30)}\sqrt{\left(\frac{T}{0.5}\right)}$

$\Rightarrow T=400\times 0.5$

$=200dyne$

$=2\times {10}^{-3}$ Newton