Question

One end of a 75-cm long horizontal wire is fixed and the other end passes over a pulley that supports a 450-N weight. The linear density of the wire is 0.00500 kg/m. A vibrating tuning fork placed near the wire causes the wire to vibrate in its 1st overtone. The vibrating tuning fork is then placed near an adjustable pipe closed at one end and causes the air column in the pipe to vibrate in its first overtone.

(a) what is the length of the pipe?

(b) if the pipe is open at both ends, what must be the length of the pipe for the same tuning fork to make the air column vibrate in its first overtone? The velocity of sound in air is 340 m/s.

Answer 1

The fundamental mode or first harmonic for stationary waves in a string is given by

T is the tension in the string

u is the mass per unit length

L is the total length of the string

L = 75 cm = 0.75 m

T = W = 450 N

The first overtone is second harmonic, so the frequency of the first overtone is

a) The first overtone in a closed pipe is given by third harmonic because even harmonics are absent in it

v is the speed of sound

The first overtone frequency is same as the tuning fork frequency as the tuning fork frequency resonates with the first overtone

L = 0.6375 m

L = 63.75 cm

b) for open pipe first overtone is given by

The first overtone frequency is same as the tuning fork frequency as the tuning fork frequency resonates with the first overtone

L = 0.85 m

L = 85 cm

L = 85 cm