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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?
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Solution 1
a) Mass of the wire, m = 3.5 × 10–2 kg
Linear mass density, `mu = m/l = 4.0 xx 10^(-2) kg m^(-1)`
Frequency of vibration, ν = 45 Hz
:. Length of the wire, `l = m/mu = (3.5xx10^(-2))/(4.0xx10^(-2)) = 0.875 m`
The wavelength of the stationary wave (λ) is related to the length of the wire by the relation:
`lambda = (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 as:
v = νλ= 45 × 1.75 = 78.75 m/s
b) The tension produced in the string is given by the relation:
T = v2µ
= (78.75)2 × 4.0 × 10–2 = 248.06 N
Solution 2
Here n= 45 Hz, M = `3.5 xx 10^(-2)` kg
Mass per unit length = `m = 4.0 xx 10^(-2) kg m^(-1)`
`:. l = M/m = (3.5xx10^(-2))/(4.0xx10^(-2)) = 7/8`
As `l/2 = lambda = 7/8 :. lambda = 7/4 m = 1.75 m`
a) The speed of the transverse wave,` v = vlambda = 45 xx 1.75 = 78.75 "m/s"`
b) As ` v = sqrt(T/m)`
`:. T = v^2xx m = (78.75)^2xx 4.0 xx 10^(-2) = 248.06 N`
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