Question

A string of length 40 cm and weighing 10 g is attached to a spring at one end and to a fixed wall at the other end. The spring has a spring constant of 160 N m⁻¹ and is stretched by 1⋅0 cm. If a wave pulse is produced on the string near the wall, how much time will it take to reach the spring?

Answer 1

Given,
Length of the string, L = 40 cm
Mass of the string = 10 gm
Mass per unit length$=\frac{10}{40}=\frac{1}{4}\left(\mathrm{gm}/\mathrm{cm}\right)$
Spring constant, k = 160 N/m

$$\begin{gathered} \mathrm{Deflection},x=1\mathrm{cm} \\ =0.01\mathrm{m} \\ \mathrm{Tension},T=kx=160\times 0.01 \\ \Rightarrow T=1.6\mathrm{N}=16\times {10}^4\mathrm{dyn} \\ \mathrm{Now}, \\ v=\sqrt{\left(\frac{T}{m}\right)}=\sqrt{\left(\frac{16\times {10}^4}{{\displaystyle \frac{1}{4}}}\right)} \\ \Rightarrow v=8\times {10}^2\mathrm{cm}/\mathrm{s}=800\mathrm{cm}/\mathrm{s} \end{gathered}$$

∴ Time taken by the pulse to reach the spring, $t=\frac{40}{800}=\frac{1}{20}=0.05\mathrm{s}$