The equation of a wave on a string of linear mass density 0.04 kgm⁻¹ is given by

The tension in the string is

Concept:

Wave Equation and Tension in a String:

• The wave equation for a string is given by: y = A sin( 2π ( t / T - x / λ ) )
• Where:
  • A = Amplitude of the wave
  • T = Period of the wave
  • λ = Wavelength of the wave
  • t = Time
  • x = Position along the string
• For a string under tension, the tension (T) is related to the wave speed and linear mass density (μ) by the formula: T = μ v² = μ ( ω / k )²
• Where:
  • μ = Linear mass density (kg/m)
  • v = Wave speed (m/s)
  • ω = Angular frequency (rad/s)
  • k = Wave number (rad/m)

Calculation:

Given,

Amplitude, A = 0.02 m

Linear mass density, μ = 0.04 kg/m

The wave number (k) is given by: k = 2π / λ = 2π / 0.50 = 2π / 0.50 rad/m

The angular frequency (ω) is given by: ω = 2π / T = 2π / 0.004 rad/s

The tension in the string is:

T = μ ( ω / k )² = 0.04 × ( (2π / 0.004) / (2π / 0.50) )² = 6.25 N

∴ The tension in the string is 6.25 N.