Question

A standing wave pattern is created on a string with mass density μ = 3 × 10^-4 kg/m. A wave generator with frequency f = 63 Hz is attached to one end of the string and the other end goes over a pulley and is connected to a mass (ignore the weight of the string between the pulley and mass). The distance between the generator and pulley is L = 0.68 m. Initially the 3rd harmonic wave pattern is formed.

1) What is the wavelength of the wave?

2) What is the speed of the wave?

3) What is the tension in the string?

4) What is the mass hanging on the end of the string?

5) Keeping the frequency fixed at f = 63 Hz, what is the maximum mass that can be used to still create a coherent standing wave pattern?

Answer 1

1)

Initially the 3rd harmonic wave pattern is formed.

And L =0.68m

So 3λ/2 = 0.68 m

λ = (2)(0.68m)/3 = ​​​​​​0.453 m

λ = 0.453 m

2)

Given data

f=63Hz

​​​​speed of the wave is v = fλ

So v =(63 Hz) (0.453 m) = 28.56 m/s

V=28.56 m/s

3)

We know that

V =(T/μ) ^1/2

T = V²μ = (28.56)(28.56)(3 × 10^-4 kg/m)

T= 0.2447 N

4)

We know that

T =mg

m = T/g = (0.2447N) /(9.8 m/s²​​​​​​) = 0.025 kg = 25 g

Mass m= 25 g

5)

For maximum mass

λ/2 = 0.68 m

λ = 1.36 m

V=fλ = (63)(1.36) = 85.68 m/s

V= 85.68 m/s

T =V²μ = (85.68)(85.68)(3 × 10^-4) = 2.2023 N

T=Mg

M= T/g = (2.2023/9.8) = 0.2247 kg

M =224.7 g

Maximum mass is = 224.7 g

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