A 50cm long hollow glass cylinder is open at both ends and is suspended in air.

Question

A 50cm long hollow glass cylinder is open at both ends and is suspended in air. A source of sound that produces a pure frequency is placed close to one end of the tube. The frequency of the sound source is slowly increased from a very low value. The tube is first observed to resonate when the source frequency is 320Hz.

- Calculate the speed of the sound in the tube.

- The frequency of the source is slowly increased until resonance occurs again FOR THE THIRD TIME. Calculate the frequency at which this happens.

- Calculate the frequency when resonance occurs for the FIFTH time.

- The end of the cylinder opposite the end where the sound source is positioned is tightly sealed with a rigid metal plate. Calculate the first THREE resonant frequencies(lowest to highest).

- If the apparatus is now placed in helium gas instead of air, do the resonant frequencies referred to in previous parts of this question increase or decrease? Give your reasoning.

Explanation / Answer

length of the cylinder, l=50cm

frequencies for open end pipe is, fn=n*(v/2l), n=1,2,3,4 ........

frequencies for cosed end pipe is, fn=n*(v/4l), n=1,3,5, 7........

a)

open end:

resonate frequency f=320 Hz

frequency, fn=n*(v/2l), n=1,2,3,4 ........

320=1*v/(2*0.5)

==> v=320 m/sec

speed of the sound v=320 m/sec

b)

frequency, fn=n*(v/2l), n=1,2,3,4 ........

fn=n*f1

now,

resonating frequencies, fn=n*f1

f1=320

f2=2*320=640 Hz

f3=3*320=960 hZ

c)

resonance frequency,

fn=n*f1

if n=5

f5=n*f1

f5=5*320

f5=1600 Hz

d)

closed end:

fn=n*(v/4l), n=1,3,5, 7........

if n=1

f1=1*(320/4*0.5)

f1=40 Hz,

and

if n=2, ---> f2=n*f1 ---> f2=2*40=80 Hz

if n=3, ---> f3=n*f1 ---> f3=3*40=120 Hz

if n=4, ---> f4=4*f1 ---> f4=4*40=160 Hz

e)

if the apparatus is placed in helium gas,

density of air, rho_air =1.225 kg/m^3

density of Helium gas, rho_He=0.164 kg/m^3

hence,

speed of the sound decreases,

and

all resonant frequencies in the above cases will decreases

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