c) The linear density of a vibrating string is 1.3 × 10⁻⁴ kg/m. A transverse wave is propagating on the string and is described by the equation y(x, t) = 0.021 sin(30t − x), where x and y are in metres and t is in seconds. Calculate the tension in the string.

Introduction

We are given the wave function of a transverse wave traveling along a string. The wave equation is:

y(x, t) = 0.021 sin(30t − x)

We are also given:

Linear mass density (μ) = 1.3 × 10⁻⁴ kg/m

The goal is to determine the tension (T) in the string.

Step 1: Identify Wave Parameters

The general wave equation is:

y(x, t) = A sin(ωt − kx)

From this, we can identify:

Angular frequency (ω) = 30 rad/s
Wave number (k) = 1 rad/m

Step 2: Calculate Wave Speed

Wave speed (v) is related to angular frequency and wave number:

v = ω / k = 30 / 1 = 30 m/s

Step 3: Use Formula for Tension

Wave speed on a string is also given by:

v = √(T / μ)

Rewriting for Tension:

T = μ × v² = (1.3 × 10⁻⁴) × (30²)

T = 1.3 × 10⁻⁴ × 900 = 0.117 N

Final Answer

Tension (T): 0.117 Newtons

Conclusion

Using the wave function and known linear density, we calculated the tension in the string. Understanding this relationship is vital for applications in musical instruments, engineering, and wave mechanics.