Lab Quiz: Marshmallow Lab

In this lab we placed marshmallows into the microwave and warmed them up. When we warmed up the marshmallows we observed which parts of the marshmallows rose. 

We used this to determine the wavelength of the wave. The length of the part of the marshmallow that rose was 0.12m which corresponds to the wavelength. We use this formula v = f/λ we can solve this equation for the frequency and the resulting equation is v/λ = f. The wave travels at the speed of light which is 3.00 * 10^8 m/s. Once we plug in the velocity of the wave and the wavelength it results in 2.5 GHz. The smallest dimensions of the microwave must be 0.085m by 0.085m. This is assuming that the wave travels along the diagonal of the microwave where the diagonal is 0.12m. The minimum height of the microwave corresponds to the amplitude of the wave. 

We also microwaved one cup of water for 30 seconds. To find the total energy in the water we used the equation E_T = MCΔT. Where E is the energy in Joules. M is the mass of the water, C is the specific heat of water, and ΔT is the change in temperature of the water in the reaction. M of the water was 100g. C is 4.186 J/g*°C . The change in temperature was 34°C. This results in the total energy is 14.232 kJ. 


We then determined the amount of energy per photon  by using the equation E_p=hf, where h is Planck's constant and f is the frequency. This value was 1.659 * 10^-24 J. We can determine the number of photons this way by equating nE_p=E_T. Where n is the number of photons and we can the equation for n which results in
n=E_T/E_p which results in 8.59*10^27 photons. 


We can find the pressure by using this equation P=power/Ac, where A is the area and c is the speed of light. Power can be rewritten as dU/dt. Once we plug in these values  our pressure comes out to be 3.87 * 10^-7 Pa. 
We can determine the pressure exerted per photon by dividing this value by the number of photons. 
P/n = 4.51 * 10^-35 Pa/photon.

Experiment 4: Standing Waves

Experiment 4: Standing Waves

In this lab the goal was to observe standing waves while they were in resonance. We would vary the conditions to observe different wavelengths(λ). The wavelength would change when the frequency changed as we kept the mass and the length of the string to remain constant. We did this for two trials. In the second trial we varied the mass and then repeated the same as we did for the first part of the experiment

To begin this experiment we had mass hanging from a string and our string was connected to a frequency generator. After setting up our apparatus we collected various data from our two trials. We began collecting data once our string reached its fundamental frequency. We began collecting data until we hit a maximum of 100 Hz.

 String mass per until length is 1.25g/m

Part 1

Mass 198.7g ± 0.05

Frequency (Hz)	15.52 ± 0.005	31.52 ± 0.005	47.41 ± 0.005	62.21 ± 0.005	78.76 ± 0.005	91.38 ± 0.005
Wavelength λ  (m)	2.52 ± 0.005	1.242 ± 0.005	0.82 ± 0.005	0.62 ± 0.005	0.51 ± 0.005	0.41 ± 0.005
Δx (m)	1.26 ± 0.005	0.621 ± 0.005	0.41 ± 0.005	0.31 ± 0.005	0.255 ± 0.005	0.205 ± 0.005
Number of Nodes	2	3	4	5	6	7

Part 2

Mass 50.0g

Frequency (Hz)	8.21 ± 0.005	16.62 ± 0.005	25.34 ± 0.005	34.02 ± 0.005	46.06 ± 0.005	63.83 ± 0.005
Wavelength λ (m)	2.52 ± 0.005	1.242 ± 0.005	0.82 ± 0.005	0.62 ± 0.005	0.51 ± 0.005	0.41 ± 0.005
Δx (m)	1.26 ± 0.005	0.621 ± 0.005	0.41 ± 0.005	0.31 ± 0.005	0.255 ± 0.005	0.205 ± 0.005
Number of Nodes	2	3	4	5	6	7

In the data table the frequency represents the frequency the generator was turned to. The wavelength was how long the wave was. Δx represents half of one wavelength. The number nodes corresponds to how many nodes were produced. 

In the first part of the experiment we used 198.7g of mass attached to our pulley. 

In the second part of the experiment we only used 50.0 g of mass attached to our pulley.

The amount of nodes were produced at lower frequencies in the second part of the experiment because the mass was less.

Data Analysis 

Here are plots of each part of the experiment. These graphs are plotted as Frequency vs. 1/λ. The slope of the line is equal to the wave speed. The wave speed can be calculated with this formula 

V= sqrt(T/μ) where T is the tension and μ is the mass per unit length. 

The theoretical velocity using this formula in Part 1 is 39.47 m/s.

the theoretical velocity in Part 2 is 19.80 m/s.

The ratio of the waves speeds from part 1 is 3.86% where the ratio is Δv/Experimental.

This error is not significant.

The ratio of the wave speeds from part 2 is 25.8%. However on part 2 the error is significant. I think this error may have occurred because the length of the string might have changed when we were
the mass.

The calculated theoretical frequencies for part 1 are

15.66 Hz, 31.32 Hz, 46.99 Hz, 62.65 Hz, 78.31 Hz, 93.98 Hz. 

The values are not equal but the percent error for all the calculations is insignificant because they are all under 5%. 

The calculated theoretical frequencies for part 2 are

7.85 Hz, 15.7 Hz, 23.6 Hz, 31.4 Hz, 39.3 Hz, 47.1 Hz

The ratios of the frequencies is Δf/Experimental.

Part 1

0.90%, 0.63%, 0.89%, 0.71%, 0.70%, 2.85%

Part 2

4.45%, 5.73%, 7.20%, 8.28%, 15.20%, 35.46%

Conclusion

The ratios from each part are different, there is much more error in the second because the wavelength was changed and was not remeasured. There is not a pattern here, the ratios from each part vary too much for one to be noticed.

The uncertainties were determined by taking half of the next increment of measure. The main source of error that changed was that the length of the string was changed in the second part of the experiment and was not noted which caused a massive amount of percent error.

Experiment 2: Fluid Dynamics

In this experiment we had a bucket full of water and had to determine the size of a hole that was drilled into the bottom of the bucket. We would not find the size by simply measuring it, but instead we would determine its size by using Bernoulli's equation.

We had the bucket filled up 3 inches high with water and timed how long it took for 16 oz. of water to run out of the bucket. We did this repeated this process 6 times.

Trial 1: 40.68 +/- 0.30s Trial 2: 41.16 +/- 0.30s Trial 3: 41.77 +/- 0.30s 

Trial 4: 41.65 +/- 0.30s Trial 5: 42.94 +/- 0.30s Trial 6: 42.70s +/- 0.30s

We had two people with stopwatches time our experiment and averaged their times. The biggest variation between their times was 0.30s so we will use that as our uncertainty.

The volume emptied theoretically  was 16oz/99.883 oz/ft^3 which is 0.016ft^3 which is the same as 453.07mL. The area of the drain hole is πr^2 = π(0.0025m)^2 = 1.963 * 10^-5m^2.
the height of the water 3.0 in. which is 0.0762m. The theoretical time it would take for it to empty is expressed by this equation.         t = V/A(2gh)^1/2 when the values are put in we get that 19.71s

Conclusion

Percent Error: Trial 1: 51.5% Trial 2: 52.1% Trial 3: 52.8% Trial 4: 52.7% Trial 5: 54.1% Trial 6: 52.70%
.051 +/- .005
We had a drastic percent error through all of our trials. This was because when we measured our water in the bucket, we filled the bucket up to 3 in. We should have filled the bucket up 3 in. above the hole. This would speed the rate of the water flowing out of the bucket and would have given us a more accurate reading.

Using the formula above and solving for the radius we get r=(V/πt(2gh)^1/2) ^1/2
When we input our numbers we can determine that r = 0.00168m +/- 0.0008m,
so d is .00336m +/- 0.0008m
The given value for the diameter was 0.005m

Our percent error from calculating the radius was 48.83%. Our error is from reusing the same water from each experiment along with some of the water spilling. It is also from filling up the bucket to the incorrect volume.

Experiment 1: Fluid Statics

In this lab we measured the buoyant force of the water by using three different methods. We used an underwater weighing method. We also used a displaced fluid method, and we also found the buoyant force by using the volume of the weight.

Underwater Weighing Method

We began by using a force sensor to determine the force of the weight which was 1.099 +/- 0.0005 N.(Picture 1) We then placed the weight in to a graduated cylinder that was filled with water. The force of the weight was lower when it was in the water and it was 0.710 +/- 0.0005 N. Through this method we can determine that the buoyancy force was 0.389 +/- 0.0010 N.

Displaced Fluid Method

We began this method by measuring the weight of the beaker which was 0.217 +/- 0.0005 kg. Next, we had a graduated cylinder filled with water and then inserted our weight slowly. As we did this the displaced water spilled over into our beaker. After the weight was fully inserted we measured the beaker again with the displaced water and it weighed 0.256 +/- 0.0005 kg. By subtracting the two measurements we can determine that the water weighed 0.039 +/- 0.0010 kg. The weight of the water that was displaced is

 0.383 +/- 0.0010 N which is the buoyancy force according to Archimedes' principle. 

Volume of Object Method

We found the volume of the of weight which was in the shape of a cylinder. The volume of a cylinder is 

V = πr^2h so we measured the diameter and the height of the cylinder to determine the volume which was 3.78 * 10^-4 m^3. This would be the volume of water displaced if it were to be inserted into any container with water in it.  The buoyant force of water should be W_f = ρgV where ρ is the density of the water and g is gravity. When using ρ to be 1000kg/m^3 and g to be 9.81 m/s^2 we can determine that the buoyancy force of the water is 0.371 +/- 0.005 N. 

Conclusion

The first method yielded a result of 0.389 +/- 0.0010 N. The second method recorded was 0.383 +/- 0.0010 N. The third method resulted in 0.371 +/- 0.005 N. The first method seems the most accurate because of the experimental method but the uncertainty value is the same as the second method. This was based off of the tools that were used for measurement. The third method has the highest because the tools were the most inaccurate.

I think that the most accurate method is underwater weighing method because the main source of error would be to allow the the weight to touch the ground. If this error is avoided then this method is accurate. The other two methods can have more sources of error by spilling water outside of the beaker or not collecting all of the water that spills because it can spill onto the graduated cylinder. The other method can have error by inaccurate measurements.

If the weight touches the bottom of the graduated cylinder then it incurs a normal force from the surface it is resting on. Therefore, the normal force is now acting on the weight along with the buoyancy force of the water which would lower the value calculated for the buoyancy force.