Question

a 2m string clamped at both ends with a mass of 87 kg has an applied tension of 120n while it is plucked. what is the frequency of the wave if it produces a stamding wave with 3 antinodes

Answer 1

Let L = length of string = 2 m

Let M = mass of the string = 87 kg

Let T = tension on the string = 120 N

Let n = no. of loops on the string = no. of anti-nodes on the string = 3

Let λ = wavelength of wave on the string = 2.(L/n)

Let v = speed of wave on the string = [T / (M/L)]^1/2

Let f = frequency of wave on the string = v / λ = ?

Wavelength of wave on the string, λ, is given by:

λ = 2 x (L/n)

Put L = 2 m and n =3 in the above equation and we get:

λ = 2 x (2 m / 3) = 1.33 m

λ = 1.33 m---------------------------(Eqn 1)

Speed of wave, v, is given by:

v = [T / (M/L)]^1/2

Put T = 120 N, M = 87 kg and L = 2 m in the above equation and we get:

v = [120 N / (87 kg / 2 m)]^1/2 = 1.66 m/s

v = 1.66 m/s-----------------------(Eqn 2)

Frequency of wave, f, is given by:

f = v / λ

Put v = 1.66 m/s and λ = 1.33 m from equations (2) and (1) in the above equation and we get:

f = 1.66 m/s / 1.33 m = 1.248 Hz

f = 1.248 Hz

f = 1.25 Hz--------------------------(answer)

(answer is rounded to three significant figures)

Answer: Frequency of the wave on the string is given by, f = 1.25 Hz