If transverse waves on a string travel with a velocity of 20 m/s when the string is under a tension of 30 N,what tension will be required for the waves to travel with a velocity of 40 m/s on the same string?
A)7.5 N
B)15 N
C)42 N
D)60 N
E)120 N

Question

Quizplus · Accepted Answer

The velocity of a wave on a string is given by the formula $v = \sqrt{\frac{T}{\mu}}$, where $v$ is the velocity, $T$ is the tension in the string, and $\mu$ is the mass per unit length of the string. If the velocity doubles, the equation can be set up as $40 = \sqrt{\frac{T_2}{\mu}}$ and $20 = \sqrt{\frac{T_1}{\mu}}$. Squaring both sides of each equation and setting them in a ratio eliminates $\mu$, leading to $\frac{T_2}{T_1} = \left(\frac{40}{20}\right)^2 = 4$. Given that $T_1 = 30$ N, solving for $T_2$ gives $T_2 = 4 \times 30$ N = 120 N.

If Transverse Waves on a String Travel with a Velocity