A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at .
The tension in the wire should be ⁴ N (or 20589 N).
Explanation
This question applies the formula for speed of transverse waves on a string, v = , which is derived from the context showing that wave speed depends on tension T and mass per unit length μ. The student must first calculate the linear mass density μ from the given mass and length, then rearrange the formula to solve for tension T. The required wave speed is given as 343 m s⁻¹, equal to the speed of sound in dry air at 20°C.
Solution Steps
Step 1: Calculate mass per unit length (μ) μ = mass/length = 2.10 kg / 12.0 m = 0.175 kg m⁻¹
Step 2: Use the speed formula for transverse waves The speed of wave on the wire is given by v = Rearranging: T = μ × v²
Step 3: Substitute values and calculate tension T = 0.175 kg m⁻¹ × (343 m s⁻¹)² T = T = 20588.575 N ⁴ N