Determining Normal Modes of Vibration

Name:

Lab Partners:

When entering numeric data, use exponentials: ie., 1.6 * 10^-19 = 1.6E-19.

Procedure

1. Measure the room air temperature and record the frequency stamped on the tuning fork (should be between 250 and 525 Hz):

   T = C

   n_fork = Hz

2. Begin with the piston very near the end of the tube. Do not touch the tube between the open end and the piston. Strike the tuning fork against the heel of your shoe (NOT any hard surface.) With the tuning fork vibrating near the tube end, slowly pull the piston out until the sound reaches a maximum amplitude. Now move the piston more quickly back and forth across this position until you narrow down the exact location of the node. Measure the distance from the end of the tube to the end of the piston:

   d₁ = m

   This distance is an odd integer (n) times one quarter wavelength of some harmonic (m) of the tuning fork. We will have to determine m and n in each case.
3. Repeat step 2 five more times, each time pulling the piston further from the open end of the tube:

   d₂ = m

   d₃ = m

   d₄ = m

   d₅ = m

   d₆ = m

Analysis

1. Look up the speed of sound at the recorded air temperature (in, for example, the Handbook of Chemistry and Physics in the Library):

   c = m/s

2. Divide the speed of sound by four times the distance for each maximum:

   x_i = c / (4 d_i)

   x₁ = Hz

   x₂ = Hz

   x₃ = Hz

   x₄ = Hz

   x₅ = Hz

   x₆ = Hz

   x_i is the fundamental frequency (normal mode) for a tube of length d_i which is open at one end.
3. Find odd integers n_i and integers m_i (m_i <= 10) which satisfy the following relationship for each trial:

   n_i x_i = m_i n_fork

   The equality should hold to within a few percent; compute the relative error:

   Dn_{i, rel} = 100% * | m_i n_fork - n_i x_i | / ( m_i n_fork )

   m₁ = , n₁ =

   Dn_{1, rel} =

   m₂ = , n₂ =

   Dn_{2, rel} =

   m₃ = , n₃ =

   Dn_{3, rel} =

   m₄ = , n₄ =

   Dn_{4, rel} =

   m₅ = , n₅ =

   Dn_{5, rel} =

   m₆ = , n₆ =

   Dn_{6, rel} =