Question

Consider a 3 m long string, clamped at both ends. The string has a mass of 60 g and waves travel at v = 100 m/s. The third harmonic is excited on the string. a) Draw a picture for third harmonic on the string. b) What is the wavelength for the third harmonic? c) What is the frequency of the third harmonic? d) What is the maximum acceleration off any point on the string if the amplitude of the motion for the third harmonic is 0.2 cm?

Answer 1

a) λ = 2 m

, c) f = 50 Hz

Step-by-step explanation:

When a string is fixed at the ends the wave is reflected at each end, giving rise to a standing wave.

Since we extract them are fixed we have nodes at these points, the wavelength in the string is

fundamental λ = 2L

2nd harmonic λ= 2L / 2

3 harmonica λ= 2L / 3

a and b) from aui we can find the wavelength

λ = 2 3/3

λ = 2 m

c) the speed of the wave is related to the frequency and wavelength

v = λ f

f = v / λ

f = 100/2

f = 50 Hz

d) the acceleration can be found with the equations

a = d²y / dt²

the standing wave equation is

y = 2A sin kx cos wt

a = -2A w² sin kx cos wt

the acceleration is maximum when the cosine is ±1

A = 2A w² sin kx

the oscillatory part indicates that the wave moves, if we make this maximum vine, they relate it to

a = 2A w²

w = 2πf

A = 0.2 cm = 0.002 m

a = 2 0.002 (2π 50)²

a = 98.7 m / s