Question

The speed of a wave on a string is 150 ms\(^{-1}\) when the tension is 120 N. The percentage increase in the tension in order to raise the wave speed by 20\% is

• A. 44
• B. 40
• C. 22
• D. 20

Hint:

The speed of a wave on a string is proportional to the square root of the tension. Use this relationship to find the change in tension when the wave speed changes.

Answer 1

The Correct Option is A

The speed of a wave on a string is given by: \[ v = \sqrt{\frac{T}{\mu}} \] where \( T \) is the tension and \( \mu \) is the mass per unit length of the string. The wave speed increases by 20\%, so the new speed is: \[ v' = 1.2v \] The ratio of the new speed to the original speed is: \[ \frac{v'}{v} = \frac{1.2v}{v} = 1.2 \] The speed is proportional to the square root of the tension, so we have: \[ \frac{v'}{v} = \sqrt{\frac{T'}{T}} = 1.2 \] Squaring both sides: \[ 1.44 = \frac{T'}{T} \] Thus, the new tension \( T' \) is: \[ T' = 1.44T = 1.44 \times 120 = 172.8 \, \text{N} \] The percentage increase in tension is: \[ \text{Percentage increase} = \frac{T' - T}{T} \times 100 = \frac{172.8 - 120}{120} \times 100 = 44\% \] Thus, the percentage increase in tension is 44\%.