Question

The linear mass density of a stretched string is $\mu$ and tension is T. The wave velocity is:

• A. $\sqrt{T/\mu}$
• B. $\sqrt{\mu/T}$
• C. $T/\mu$
• D. $\mu/T$

Hint:

To increase the speed of a wave in a string, you can either increase the tension ($T$) by stretching it tighter, or use a lighter string (decrease $\mu$).

Answer 1

The Correct Option is A

Step 1: Identify the physical quantities involved.
A transverse wave on a stretched string depends on the restoring force provided by the tension \( T \) and the inertia per unit length, given by the linear mass density \( \mu \).
Step 2: Check dimensional consistency.
Tension has units of force (\( \text{kg m s}^{-2} \)) and \( \mu \) has units of \( \text{kg m}^{-1} \), so \( T/\mu \) has units of \( \text{m}^2\text{s}^{-2} \), which is velocity squared.
Step 3: Take the square root to get velocity.
\[ v = \sqrt{\frac{T}{\mu}} \]
A higher tension speeds up the wave, and a heavier string per unit length slows it down, both consistent with this formula.
\[ \boxed{\sqrt{T/\mu}} \]