Standing Waves

The goal for this experiment is to investigate and have a basic understanding of force driven standing waves.

A standing wave is a transverse wave traveling along a medium, in this case a string, and then reflected back and returns interfering with other waves. When interfering, at resonance, a standing wave is created with nodes and antinodes in the wave pattern.

Equations:

-Transverse wave in the positive x direction.

y1 = A sin(kx - wt),

where k = 2*pi/lambda & w = 2*pi*f

-Transverse wave reflected from fixed end

y2 = A sin(kx + wt)

-At resonance, the two waves combine and become the some of both waves.

                                y = y1 + y2 = (2A sin (kx))*cos(wt)

 Other equations:

-Wavelength for each harmonic of vibrating string.

Lambda = 2L/n

-If frequency is known, the wave speed can be determined with fundamental wave velocity equation.

v = f*lambda

-Substituting both.

f = vn/2L

-velocity of wave traveling on a string.

v = sqrt(T/mu),

where T is tension & and mu is mass per unity length.

Experiment:

We used mechanism that drives a wave on one side of the string, and the other side of the string was pulled by a hanging mass creating tension. We adjusted the frequency of the driving mechanism to find each harmonic and measured the wavelength.

For case 1 we had a string of length 140 cm and a hanging mass of 200 g.

-Data:

16.5 Hz, 1 loop,  2 nodes, lambda 140 cm

32.5 Hz, 2 loops, 3 nodes, lambda 67 cm

45.6 Hz, 3 loops, 4 nodes, lambda 45 cm

56.7 Hz, 4 loops, 5 nodes, lambda 37 cm

74.6 Hz, 5 loops, 6 nodes, lambda 28 cm

87.4 Hz, 6 loops, 7 nodes, lambda 23 cm

For case 2 we had a string of length 186.5 cm and a hanging mass of 50 g.

Fluid Dynamics

In this experiment we filled a bucket with water and recorded the time elapsed as the water drained from a small hole at the bottom of the bucket. Then, calculated the theoretical time elapsed and compared both results.

Experiment: The bucket was filled with water 2.3 inches above the small hole, the hole had a diameter of approximately 0.6 centimeters. We ran six trials and recorded the time it took for 473 mL (16 ounces) of water to flow out of the bucket.




Volume emptied (V) = 16 ounces = 473 mL
 Height of water (h) = 2.3 in = 0.192 feet
 Area of drain hole (A) = pi*(.3cm)^2 = .28 cm^2 = 0.0003014 ft^2
 Acceleration due to gravity (g) = 32 ft/s^2

Trial #   time (t_actual)
1            25.06 s
2            24.85 s
3            25.24 s
4            25.36 s
5            25.17 s
6            26.09 s

The average of the six trials was 25.30 seconds. We then calculated the theoretical time by using the Flow rate equation, Continuity equation and Torricellis Theorem, which relates the speed of fluid flowing out an opening to the height of fluid above the opening.

R = V/t = Av  &  v = sqr(2gh)  ----->  V/t = A[sqr(2gh)]  ----> t(theoretical) = V/A[sqr(2gh)]    

                                                         

                                                         t (theoretical) = 15.14 s

Calculate Error:    |theoretical - actual|/theoretical x 100 = % error

                              |25.3s - 15.14s|/15.14 x 100 = 67% error

Comparing our measured value with the theoretical value gave us a 67% error. This does not agree within uncertainty. We then assumed that the measured diameter was inaccurate and decided to solve for the actual diameter using the same equations and the data recorded.

    

     v = sqr(2*32*0.192) = 3.505 ft/s,      R = V/t = (0.0160 ft^3)/25.30s = 6.324*10^-4 ft^3/s

                          V/t = A[sqr(2gh)] --- >  6.324*10^-4 ft^3/s = pi*r^2*3.505

                           r = 0.0076 ft = 0.28 cm, diameter = 0.56 cm

                          |0.6cm - 0.56cm|/0.6 x 100 = 6.67 % error

Here we see that our diameter should had been 0.56 cm. To check our result we compared both the calculated and measured diameters, this gave us a percent error of 6.67% which is within our uncertainty. This proves that our measurements were correct but there is still an unknown source that altered our results drastically.

There may be a few reasons why our result for our time had a big error. It is possible that we did not measured the height of the water correctly since we did not have a good angle reading the measurement while looking into the bucket. The increments on the beaker were not small enough to read exactly 473 mL, it was an approximate. Having better measuring equipment would had definitely lowered the error on our results.

 

Fluid Statics

Our goal for this experiment is to measure the buoyant force acting on an object using three methods and then compare the results.

Method 1: Underwater Weighing Method

The underwater weighing experiment consist of measuring the weight of a metal cylinder and then submerging the cylinder in water by holding the cylinder with a string that is attached to a force probe. We then drew a free body diagram of the cylinder submerged in water showing the forces acting on the cylinder, as shown on the left. There are three forces acting on the cylinder; Tension, Buoyant Force and Weight.
Mass of the cylinder was measured by placing the cylinder on the weighing scale and calculated the weight by multiplying the result by 9.81m/s^2 (gravity).
Mass (cylinder) = 110.62 g
Weight = mg = 1.085 +/- 0.00005N

Tension was measured by using a Force Probe that was attached to a string that held the cylinder in submerged water. The force probe measured a tension force of .725 +/- .05N.

Using the expression created by the free body diagram, we then calculated the Buoyant force acting on the cylinder.
   B = mg - T = 1.085 N - .725 N = 0.359076 N

Uncertainty:
There is uncertainty in our result because the equipment does not give exact measurements, but we can attempt to calculate our accuracy for our final results. We can find the uncertainty for our result using a propagation technique as shown above. Since the buoyant force depends on the mass and tension, we had to consider their uncertainty for each value. This gave us an uncertainty of  0.05 N.
                                                            B = 0.3591 +/- 0.05 N

Method 2: Displaced Fluid Method

For this experiment we measured the mass of an empty beaker. We then used another beaker and filled it with water up to the top, almost over flowing the beaker. Using the same cylinder from the last experiment, we lowered it into the cylinder with water and had the water over flow into the empty beaker. The beaker with displaced water was measured again and then calculated the mass of the water alone.
Beaker m = 142.99 +/- 0.05
Beaker + water m = 178.20 g +/- 0.05
Water m = 0.03521 kg
Weight (water) = 0.345058 N

According the Archimede's principle, when a body is immersed in water the water exerts an upward force equal to the weight of the water displaced.
Buoyant Force (B) = 0.345058 N






Uncertainty:
Using the same propagation technique as shown above. Since the buoyant force depends on the mass of beaker and mass of beaker with water, we had to consider their uncertainty for each value. This gave us an uncertainty of 0.0006929 N.
                                                    B = 0.345058 +/- 0.0006929


Method 3: Volume of Object Method

For this final method, the volume of the cylinder was calculated by measuring the height and diameter with a ruler. the volume is then calculated using the expression:
V = pi * r^2 *h
diameter (d) = 0.023 +/- 0.0005 m
height (h) = 0.074 +/- 0.0005 m
V = 3.0745*10^-5 m^3

We then write an expression for the weight of the displaced water using the Volume and density of the water as shown on the left. Acrchimede's principle states that the weight of the displaced water is equal to the buoyant force.
B = 0.301301 N

Uncertainty:
Again, we calculated the uncertainty with the same propagation technique. Since the buoyant force depends on the volume of the cylinder and the volume depends on the diameter and height, we had to consider their uncertainty for each value. This gave us an uncertainty of  0.0132 N.
                                                       B = 0.301301 +/- 0.0132 N

Our measurements were recorded as accurate and precise as possible, but we were limited with the equipment that was used. The displaced fluid method shows better promise because its uncertainty proved to have a smaller uncertainty. Possibly because we only used one instrument, weighing scale, for our experiment. Where as in the underwater weighing  method two instruments were used to measure, weighing scale and force probe, each with its own uncertatinty. The volume of object method had a high uncertainty because the rulers used to measure the heigt and diameter did not have small enough increments for a better accuracy.

We also considered other reasons that our experiment could alter our results. For example, in the underwater weighing method if the cylinder was touching the bottom of the container it would have given us inaccurate results. There would be a force pushing up from the bottom of the container acting on the cylinder giving us a higher buoyant force and a smaller tension force. This would had given us a completely wrong answer. For this reason we made sure the cylinder did not tough the bottom.