An open pipe of length l1 and a closed pipe of length l2 both resonate with the same tuning fork in their fundamental mode of vibration. Calculate the value of l1 /l2 ?

1 Answer
Jul 4, 2018

This is what I get

Explanation:

We know that an open pipe is open at both ends and closed pipe is closed at one end. Also the fundamental for an open end pipe needs to be an antinode at both ends, since the air can move at both ends. While the fundamental for a closed pipe has a node at the closed end and an antinode at the open end.

For l_1 being the length of the open pipe , then for its fundamental we have

l_1 = λ_1/2 ......(1)

If n_"1open" is the fundamental frequency and v is the velocity of sound in air, then

n_"1open" = v/λ_1 .....(2)

Using (1) we get

n_"1open" = v/(2l_1) …... (3)

For l_2 being the length of the closed pipe then for its fundamental we have the relationship

l_2=lambda_1/4 .......(4)

Using expression similar to (2) and (4) we get

n_"1close"= v/(4l_2) .........(5)

It is given that both resonate with the same tuning fork in their fundamental mode of vibration.

=> n_"1open"=n_"1close"

Dividing (3) by (5) we get

1=( v/(2l_1))/(v/(4l_2))
=>l_1/l_2=2