Laboratory #2: Vibrating String

Objectives:

  1. To find the frequency of various modes of vibration of a string.
  2. To determine how the frequency depends on the length of the string and the tension.
  3. See if the dependencies in 2) agree with our formula.

Equipment:

  1. PASCO software with Lab2.
  2. String with weight pan attached.
  3. Mechanical driver.
  4. Three 0.5 kg weights.
  5. Yard stick.
  6. Wooden block, with small weight, to change string length.

Setup:

 

From class we know that:
,
where T = Mg is the tension, U is the mass of a one meter length of string, L is the length of the string, N is the overtone number and g = 9.8 m/sec2. So,

 

.

 

Instructions:

Part A

  1. Place the mechanical driver very close to one end of the string. The vibrator should barely touch the string.
  2. Measure the length of the string from the actual node to the wooden block. The actual node is about 1 cm from the driver towards the post.
  3. For each of three masses, find the first three resonant frequencies. Start at 100 Hz and increase in steps of 10 Hz and then steps of 1 Hz. You will know when you are at a resonant frequency when you can see a stable pattern of nodes and anti-nodes and a large amplitude. The number of nodes will tell you which mode you are at. Record the mass - be sure to include the mass of the weight pan, if used.
  4. Now choose one mass.
  5. Choose three different string lengths by changing the position of the wooden block. For each string length, find the first three resonant frequencies

 

Results:

Table 1

String length =

 

Mass, M

Fundamental, f1

Second harmonic

Third Harmonic

1

 

 

 

 

2

 

 

 

 

3

 

 

 

 

 

 

 

 

 

Table 2

Mass =

 

Length, L

Fundamental, f1

Second harmonic

Third Harmonic

1

 

 

 

 

2

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

Analysis:

Consider four possible dependencies:

.
We want to see which one works for our data. For each table, fill in the values and circle the column which gives a constant value.

First, check the fundamental frequency, f1 vs. M, from Table 1:

f

M

f× M

f/M

f× Ö M

f/Ö M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Now, check f1 vs. L, from Table 2:

f

L

f× L

f/L

f× Ö L

f/Ö L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Finally, check f vs. N for any one row from Table 1 or 2:

f

N

f× N

f/N

f× Ö N

f/Ö N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Questions:

  1. Do the dependencies found in the previous part agree with our formula for the resonant frequency of a string? Show reasoning for each dependency.

 

 

 

 

  1. What in the mass of one meter of string? Show work.