Question

A string fixed at both ends has a length of L = 2.20 m, and a mass of m=0.0022 kg. The tension in the string is fixed at 14.30 N. You will find the frequencies of various standing wave patterns. (a) What is the linear density of the string? Keep 5 decimal places.

Answer 1

Answer:

a) f₀ = 3250 Hz ,  f₁ = 6500 Hz, b)   μ = 0.00100 kg/m

Explanation:

a) For standing waves to occur, waves of equal frequency that travel in opposite directions must be added

In the points where it is fixed there are nodes and in bellies or anti nodes in another symmetrical point

Fundamental wave:   node at the ends, a belly (antinode in the center

                                 L = 1 / 2 λ

Extreme node, two bellies   L = 2/2 λ

For more bellies      L = n / 2 λ

Therefore the equation for the wavelength is

                λ = 2 L / n                 n = 1, 2, 3,…

Let's calculate

    λ₁ = 2/1 (2.20)

    λ₁ = 4.4 m

    λ₂ = 2/2 2.20

    λ₂ = 2.20 m

Having the wavelength we can use the relationship of wave speed and wavelength and frequency

     v = λ f

     v = √ T /μ

     v = √ 14.30 / 0.001

     v = 14300 m / s

     f = v / λ

     f₀ = 14300 / 4.4

     f₀ = 3250 Hz

     f₁ = 14300 / 2.2

     f₁ = 6500 Hz

b) linear density is defined as mass per unit length

      μ = m / l

     μ = 0.0022 / 2.20

      μ = 0.001 kg/m