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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?
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Solution
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,
\[t = \frac{40}{800} = \frac{1}{20} = 0 . 05 s\]
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