Ncert Exercises & Solutions

15.14 A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is $3.5 \times 10^{- 2}$ kg and its linear mass density is $4.0 \times 10^{- 2}$ kg/m. What is:

(a) the speed of a transverse wave on the string, and
(b) the tension in the string?

It is given in the question that :

Mass of the wire, m = 3.5 \times 10^{- 2} kg

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

Frequency of vibration, ν = 45 Hz

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

The wavelength of the stationary wave : λ = \frac{2 l}{n}

where, n = number of nodes in the wire

For fundamental node, n = 1 -

λ = 2 l \Rightarrow λ = 2 \times 0.875 = 1.75 m

The speed of the transverse wave in the string is given as :

v = νλ = 45 \times 1.75 = 78.75 m / s

The tension produced in the string :

T = v^{2} μ = {(78.75)}^{2} \times 4.0 \times 10^{- 2} = 248.06 N

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