Question

A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 × 10^–2 kg and its linear mass density is 4.0 × 10^–2 kg m^–1. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string?

Answer 1

Mass of the wire, m= = 3.5 × 10^–2 kg 

Linear mass density, \(μ =\frac{m}{l}=4.0×10^{-2}\,kg\,m^{-1}\)

Frequency of vibration, ν = 45 Hz

∴Length of the wire, \(l=\frac{m}{μ}=\frac{3.5×10^{-2}}{4.0×10^{-2}}=0.875\,\,m\)

The wavelength of the stationary wave (λ) is related to the length of the wire by the relation:

\(λ=\frac{2l}{n}\)

Where n= Number of nodes in the wire

For fundamental node, n = 1:

λ = 2l

λ = 2 × 0.875 = 1.75 m

The speed of the transverse wave in the string is given

v = νλ= 45 × 1.75 = 78.75 m/s

The tension produced in the string is given by the relation:

T = v ² µ

= (78.75)² × 4.0 × 10^–2 = 248.06 N