Experiment 4

Lab 4
Standing Waves

The purpose of this lab is to observe and understand the relationship between a standing wave's Force, amount of harmonic motions, applied frequency, its velocity, and wavelength altogether in one big picture.
The equipment used in this lab were:
A wave generator, various hanging mass, a string with relatively uniform μ, pulley, meter stick.

The setup is as follows:

First, the string will be extended across a flat surface.

One end is tied onto a fixed rod; a pulley will be placed with the string running through it. 

Tied to the pulley-end is a hanging mass that serves as external force. The wave generator is setup under the vicinity of one end of the string which is tied onto a fixed surface. 

The wave generator will be set on a certain frequency until a fundamental loop is achieved (2 nodes and 1 anti-node). Such frequency will be recorded. 

The frequency will be gradually increased until the number of nodes and antinodes increased by exactly one, then the frequency will be recorded; this will be repeated until the frequency reaches no longer than 100 Hz. 

Standing wave can be expressed in a simplified form of

y(x,t) = 2A (sin (kx)) cos (ωt)

Nodes will occur whenever sin (kx) = 0, or when kx is multiples of 0, π, 2π... (kx = nπ), with n integers.

k is 2π/λ. x is the actual length of string. Therefore, 

λ = 2L / n

Using v = f λ, 

f = v n/2L

Observations and Data:

There are 2 parts of this experiments. The difference between the two are the mass of hanging mass and length of string used.

(The steps of measured uncertainty is measured on the last part of this blog)

Mass string = 3.08 * 10^-3 kg +/- 0.000005  

Length string = 2.00 m +/- 0.005

μ = mass / length = 1.18*10^-3 kg / m +/- 3.42 * 10^-4

First Part

Mass of object ~ 0.200 kg +/- 0.0005 

Length of string used (length of string within two rigid exposures) : 1.40 +/- 0.005 m

Second Part

Mass of object~ 0.050 kg +/- 0.0005

Length of exposed string: 1.87 m +/- 0.005 

Analysis (questions)

a. Find λ and value of n:

The harmonics of Part 1 looks like the picture below:

There are 6 experiments. They will be called, respectively, experiment 1-6. 

Value of λ and n for:

Experiment 1 = 2.80 m +/- 0.005, n = 1

Experiment 2 = 1.40 m +/- 0.005, n = 2

Experiment 3 = 0.93 m +/- 0.005, n = 3

Experiment 4 = 0.70 m +/- 0.005, n = 4

Experiment 5 = 0.56 m +/- 0.005, n = 5

Experiment 6 = 0.47 m +/- 0.005, n = 6

Part 2 is shown below:

There are 7 experiments. Part 2 did not start from fundamental, but 4 nodes. The values are:

Experiment 1 = 1.25 m +/- 0.005, n = 3

Experiment 2 = 0.94 m +/- 0.005, n = 4

Experiment 3 = 0.75 m +/- 0.005, n = 5

Experiment 4 = 0.62 m +/- 0.005, n = 6

Experiment 5 = 0.53 m +/- 0.005, n = 7

Experiment 6 = 0.47 m +/- 0.005, n = 8

Experiment 7 = 0.42 m +/- 0.005, n = 9

b. Plot f vs 1/λ and compare the slope with theoretical v = (T / μ)^(0.5)

x axis = 1/λ

y axis = frequency

(Slope is velocity)

Part 1 

Slope = 39.6 (m/s)

Tension = Force of gravity; 

Tension = 2 N

v = (T / μ) ^ (0.5)

v = 42.2 m/s

Δv = 2.6 m/s (6.6% difference - insignificant)

c. Repeat above for part 2 of experiment

Part 2

Slope = 22.2 (m/s)

While v = (T / μ) ^ (0.5) = 21.0 m/s

Δv = 1.2 m/s (5.4% difference - insignificant)

d. Ratios? Case 2's wave speed is approximately half of the case 1's. 

The calculated wave speed is obtained by measuring the squareroot of tension over mu. This implies that v^2 is proportional to tension (mu is constant and can be neglected). The mass difference is in the multiples of 4 (case 1 uses 200g mass whilst case 2 50g, 4 times difference). 

It is proportional, since 4 is a square of 2, and their speed is different by multiples of 2.

e. Are measured f squal to nf_1, whereas n is number of harmonics?

It is f_1. The first trial in case 1, the fundamental motion (with only 2 nodes and 1 anti-node), by using 

f_n = nf_1

and

f_1 = v/2L,

if n is 1, then

f_n = f_1, 

whilst

v/2L will return a value of 16.5.

n=1 then, implies that it is a fundamental harmonic motion.

f. Ratio of frequency of the second harmonic for case 1 compared to case 2?

In the first case, the frequencies increase by roughly  16 Hz.

In the second case, they are increasing by, roughly, 5 Hz, throughout all harmonics.

Error calculations:

Uncertainties can be expressed as

((∂F/ ∂x * uX)^2 + (∂F / ∂y * uY)^2...)^(1/2) 

Whereas m = mass and L = length,

μ (m, L) = m/L

∂μ/∂m = 1/L * um = 3.57*10^-6

∂μ/∂L  = ln(L) * m * uL = 3.46 * 10^-4

uμ then, is 3.42 * 10^-4

μ then, is 1.18 * 10^-3 kg / m +/- 3.42 * 10^-4

Possible cause of errors and other observations:

During the lab, we realized that a the string had to be relatively long in order for (any) harmonics to occur. Our very first experiment (preceding the ones listed above) had an L value of 1.1 m, and we were almost unable to get a steady fundamental nodes.

An inevitable, major source of error is, as we noticed, that the "taut and rigid" surface which one end of string is tied on is not really absolutely taut and rigid. In fact, it wiggled and shook rather violently throughout the entire experiments. We exhausted all methods trying to eliminate/ damp the unwanted oscillations (such as using our hand on the shaking area) without avail. This caused involuntary nodes and antinodes to occur throughout the entire experiment, slightly altering the harmonic sequences. 

END.

(To be continued in Lab 5...!)