Question

The equation of a simple harmonic wave produced in a string under tension $0.4\text{ N}$ is given by $y = 4 \sin (3x + 60t)\text{ m}$. The mass per unit length of the string is

• A. $10^{-3}\text{ kg}\cdot\text{m}^{-1}$
• B. $10^{-5}\text{ kg}\cdot\text{m}^{-1}$
• C. $10^{-3}\text{ g}\cdot\text{cm}^{-1}$
• D. $10^{-5}\text{ g}\cdot\text{cm}^{-1}$

Hint:

Always verify your units before selecting your option! Options (A) and (C) share the same numerical factor ($10^{-3}$), but option (A) uses standard SI units (kg/m) while option (C) uses CGS units (g/cm). Since our inputs were in Newtons and meters, the output must be in SI units.

Answer 1

The Correct Option is A

Step 1: What is given.
A wave on a string under tension $T = 0.4$ N has the equation $y = 4\sin(3x + 60t)$ m. We must find the mass per unit length of the string.
Step 2: Read off the wave numbers.
Compare with the standard form $y = A\sin(kx + \omega t)$. So the wave number $k = 3$ rad/m and the angular frequency $\omega = 60$ rad/s.
Step 3: Find the wave speed.
The speed of the wave is \[ v = \frac{\omega}{k} = \frac{60}{3} = 20\ \text{m/s} \]
Step 4: Speed on a string.
The wave speed also depends on tension and mass per length $m$: \[ v = \sqrt{\frac{T}{m}} \]
Step 5: Rearrange for m.
Square both sides and solve: \[ m = \frac{T}{v^2} \]
Step 6: Put in the numbers.
\[ m = \frac{0.4}{20^2} = \frac{0.4}{400} = 10^{-3}\ \text{kg}\cdot\text{m}^{-1} \] This is option (1). \[ \boxed{m = 10^{-3}\ \text{kg}\cdot\text{m}^{-1}} \]