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
\[= \frac{10}{40} = \frac{1}{4}  \left( gm/cm \right)\]
Spring constant, k = 160 N/m

\[Deflection,   x = 1  cm\] 

\[                                     = 0 . 01  m\] 

\[Tension,   T =   kx = 160   \times 0 . 01\] 

\[ \Rightarrow T = 1 . 6  N = 16 \times  {10}^4   dyn\] 

\[Now,   \] 

\[v = \sqrt{\left( \frac{T}{m} \right)} = \sqrt{\left( \frac{16 \times {10}^4}{\frac{1}{4}} \right)}\] 

\[ \Rightarrow v = 8 \times  {10}^2   cm/s = 800  \text{ cm}/s\]
∴ Time taken by the pulse to reach the spring,
\[t = \frac{40}{800} = \frac{1}{20} = 0 . 05  s\]