In the figure, a string, tied to a sinusoidal oscillator at P and running over a
ID: 1327769 • Letter: I
Question
In the figure, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation L = 1.4 m, linear density ? = 1.9 g/m, and the oscillator frequency f = 190 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q.
(a) What mass m allows the oscillator to set up the fourth harmonic on the string?
(b) What standing wave mode, if any, can be set up if m = 4 kg (Give 0 if the mass cannot set up a standing wave)?
Explanation / Answer
a) speed of standing wave=v=sqrt(tension/mass per unit length)
here tension in the string=weight of the block=m*g=m*9.8
mass per unit length=linear mass density=1.9 g/m=0.0019 kg/m
to set up fourth harmonic,
the length of the string will be equal to 2*wavelength of the wave.
hence wavelength=L/2=0.7 m
as we know,
wavelength*frequency=speed of the wave
==>0.7*190=sqrt(m*9.8/0.0019)
==> m=3.4295 kg
hence 3.4295 kg mass will allow to set up fourth harmonic.
part b:
with mass m=4 kg, speed of wave=sqrt(m*g/0.0019)
=143.6369 m/s
as we know, wavelength*frequency=speed
let n th harmonic is set up in the string
then length of the string=(n/2)*wavelength
==> wavelength=(2/n)*1.4=2.8/n m
using wavelength*frequency=speed
we get
(2.8/n)*190=143.6369
==> n=3.7
hence no standing wave can be set up with m=4 kg.
(as value of n has to be an even integer to set up a standing wave)