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: Understanding the Question:
We are given the mathematical wave function for a wave propagating along a string under a mechanical tension $T = 0.4\text{ N}$. We need to compute the mass per unit length (linear mass density, $m$) of the string.

Step 2: Key Formula or Approach:
1.

Wave Speed from Wave Function: Compare the given function to the standard form $y = A \sin(kx + \omega t)$ to find the wave number $k$ and angular frequency $\omega$. The wave velocity is: $$v = \frac{\omega}{k}$$ 2.

Wave Speed on a Taut String: The velocity is physically governed by tension and linear density: $$v = \sqrt{\frac{T}{m}} \implies m = \frac{T}{v^2}$$

Step 3: Detailed Explanation:
From the given wave equation $y = 4 \sin (3x + 60t)$, identify the kinematic coefficients: Wave number, $k = 3\text{ rad/m}$ Angular frequency, $\omega = 60\text{ rad/s}$ Calculate the propagation speed ($v$) of the wave: $$v = \frac{\omega}{k} = \frac{60}{3} = 20\text{ m/s}$$ Now, relate this speed to the tension expression to solve for the linear density $m$: $$v = \sqrt{\frac{T}{m}} \implies v^2 = \frac{T}{m} \implies m = \frac{T}{v^2}$$ Substitute the given values ($T = 0.4\text{ N}$ and $v = 20\text{ m/s}$) into the equation: $$m = \frac{0.4}{20^2} = \frac{0.4}{400} = \frac{4}{4000} = \frac{1}{1000} = 10^{-3}\text{ kg}\cdot\text{m}^{-1}$$

Step 4: Final Answer:
The mass per unit length of the string is $10^{-3}\text{ kg}\cdot\text{m}^{-1}$, which matches option (A).