stretched string is observed to vibrate with frequency 30 Hz in its fundamental mode

when the supports are 60 cm apart. The amplitude at the antinode is 3 cm. The string has

a mass of 30g. Calculate the speed of propagation of the wave and the tension in the

first we need the two equations that define the speed of a wave in a string:

 

v=wf (where w is the wavelength and f is the frequency)

v=SQRT(Ft/d) (where Ft is the tension, and d is the linear density (mass per unit length))

 

first to solve for the speed, we will use v=wf, since we have both unknowns needed. the fundamental frequency of a string is when the middle is an antinode and the two ends are nodes, meaning that the length is one have of an entire wavelength.

L=w/2

w=2L

so:

 

v=wf

=2Lf

=2(0.6)(30)

=36m/s

 

for the next one, we can set the two equations equal to each other:

 

wf=SQRT(Ft/d)

 

now, the linear density is the mass of the string divided by the length of the string (we could of course use the speed we found in the first part, but in more complicated questions its best to use the original variables as much as possible to avoid rounding error). so:

wf=SQRT(Ft/(m/L))

2Lf=SQRT(Ft*L/m)

4L^2f^2=Ft*L/m

4Lf^2=Ft/m

Ft=4Lf^2m

now we just plug and chug:

Ft=4(0.6)(30)^2(0.030)

=64.8N