Concave and Convex Mirrors

Statement of Purpose
To explore the image formed by both concave and convex mirrors

Procedure

Convex Mirror

 

Nearby image

The image appears at the center of the mirror about the same size, maybe smaller and upright.

Distant Image

At a further distance, the image decreased in size.

 The observations that were made agree with phase diagrams listed here.

Sample calculation

 

 

Convex mirror

Distant image

 The image here is focused at the center of the mirror which appears smaller and inverted.

Nearby Image
 

At a closer distance the image increases in size and is still inverted.

The phase diagram shows that if the object distance decreases and the image distance increases the height of the object will increase, perfectly describing our observation.

sample calculation

Conclusions 
The purpose of the mirrors has been observed. That is why convex mirrors are used in small markets for a lower magnification, hence wider view of the market, whereas concave mirrors are handy for putting on make-up because the higher magnification.

Introduction to Reflection and Refraction

Statement of Purpose

To observe the behavior of light refracted and reflected and to measure the index of refraction of a semicircular plastic material.

Procedure

When the light ray hits the normal, the incident angle is 0, and the refraction angle is 0.

 Critical angle

Data

Step 1
 
θ1 θ2 sinθ1 sinθ2
0   0 0.000 0.000
5 3.5   0.087 0.061
10 5      0.174 0.087
15 9.3   0.259 0.162
20 12.6 0.342 0.218
25 16 0.423 0.276
30 17.5 0.500 0.301
35 22.2 0.574 0.378
40 25.5 0.643 0.431
45 28.5 0.707 0.477
50 31.5 0.766 0.522
55 35 0.819 0.574
60 37.5 0.866 0.609


Uncertainties (+/-)
uθ1 uθ2 usinθ1 usinθ2
1 1 0.017 0.017
1 1 0.017 0.017
1  1 0.017 0.017
1 1 0.017 0.017
1 1 0.016 0.017
1 1 0.016 0.017
1 1 0.015 0.017
1 1 0.014 0.016
1  1 0.013 0.016
1 1 0.012 0.015
1 1 0.011 0.015
1  1 0.010 0.014
1  1 0.009 0.014

mθ ≈ .64
sinθ1 vs sinθ2
msinθ ≈ .66

 sinθ1 vs sinθ2

θ1 vs θ2

Step 2


θ1 θ2 sinθ1 sinθ2
0 0 0.000 0.000
5 9.5 0.087 0.165
10 16.5 0.174 0.284
15 24.5 0.259 0.415
20 34 0.342 0.559
25 44 0.423 0.695
30 51.5 0.500 0.783
35 64.5 0.574 0.903
40 80    0.643 0.985


Uncertainties (+/-)
uθ1 uθ2 usinθ1 usinθ2
1 1 0.017 0.017
1 1 0.017 0.017
1         1 0.017 0.017
1 1 0.017 0.016
1 1 0.016 0.014
1 1 0.016 0.013
1         1 0.015 0.011
1 1 0.014 0.008
1         1 0.013 0.003

mθ ≈ 1.6
sinθ1 vs sinθ2
msinθ ≈ 1.5 

sinθ1 vs sinθ2

θ1 vs θ2

Conclusion
In the experiment we notice that we could not go beyond a particular angle of about 40 degrees, and this is known as the critical angle. 

Electromagnetic Radiation

Statement of Purpose
To transmit and receive electromagnetic waves.

Procedure

First Test: Adjust the frequency of the wave generator to affirm that the waves seen by the receiver (oscilloscope) is indeed a electromagnetic wave.

Second test: The behavior of the wave as the antennae moves towards and away from the receiver. The observation to be made is that the amplitude of the near distance waves is proportional to 1/r^2 and the far distance waves is proportional to 1/r.

Third test: Point the antennae parallel to the receiver and observe the amplitude of the wave. Contrast that observation with the amplitude obtained by pointing the antennae perpendicular to the receiver.

Data

Near distance EM wave

1/r^3

Antennae nearby will emit electromagnetic waves that vary as 1/r^3.

1/r

Antennae far away will emit electromagnetic waves that vary as 1/r.

In an ideal case, the theoretical values would fall off as follows:

Distance (m) Vtheoretical P2P
0
0.05                 0.011631
0.1                   0.005687
0.15                 0.003616
0.2                   0.002704
0.25                 0.002234
0.3                   0.001965
0.35           0.001796

Measurement Uncertainty:

Peak to Peak Voltage
ΔV = ±1mV

Quantization uncertainty for z:
Δz = ±.01m

accounting for BNC adapter length:
Δz = ±.05m

Conclusions 
The effect of the electric pulses sent by the antenna creates a electromagnetic wave that exerts forces on free charges within the receiver antenna, producing an oscillating current.

Sound Waves

Statement of Purpose
To analyze the properties of sound as a mechanical wave.

Procedure
Create a sound wave with voice. Then create a sound wave with a tuning fork.

Data

1. Leon sings "aaaaaah", consequently generating pressure variations in the air.

2.  Janice's attempt ranged between arbitrary values of -1 and 1.

3. Attempt with tuning fork.

 

4. When tapped softer than in part 3. the amplitude and the frequency diminishes.

4 waves +- 1
ave. frequency = 4/0.02 = 200Hz
max. frequency = 5/0.02 = 250Hz
min frequency = 3/0.02 = 150Hz

Conclusion

An observation is that in part 4, the frequency of the tuning fork diminishes when tapped softer than part 3. This is interesting because the fork should display similar frequencies as in part 3 since it is a tuning fork, the expectation is that the frequency that it generates should be constant. However, the frequency in this sample within uncertainty values with a maximum frequency of 250Hz affirming the 250Hz value in part 3.

Standing Waves

Statement of Purpose
To analyze the properties of a standing wave.

Procedure

Tie one end of a string to an oscillator and for the first case tie a 200g mass to the other end, and in another case tie a 150g mass. Adjust the frequency of the oscillations until standing waves are identified.

Data

CASE 1:

Harmonic n	frequency +/- 1Hz	fn = nf1 +/- n1Hz	λ  +/- .01m	Δx nodes +/- .005m
1	24	24	3.32	1.66
2	48	48	1.66	0.83
3	72	72	1.11	0.555
4	96	96	0.83	0.425
5	120	120	0.66	0.33
6	144	144	0.55	0.275
7	169	168	0.465	0.233
8	193	192	0.405	0.203
9	217	216	0.365	0.183
10	241	240	0.33	0.165

CASE 2:

Harmonic n	frequency +/- 1Hz	fn = nf1 +/- n1Hz	λ  +/- .01m	Δx nodes +/- .005m
1	12	12	3.32	1.66
2	24	24	1.66	0.83
3	36	36	1.11	0.555
4	48	38	0.83	0.415
5	60	60	0.663	0.332
6	72	72	0.543	0.272
7	84	84	0.475	0.238
8	96	96	0.412	0.206
9	109	108	0.369	0.185
10	121	120	0.334	0.167

Plotting frequency vs 1/λ :

Conclusion

There is a linear growth between the frequency and wavelength which affirms that they are inversely proportional. This relationship is given by v=fλ, where v is the slope of graph.

Mechanical Waves

Procedure

1. Measure the distance between two people.
2. Oscillate spring until 1 cycle is formed.
3. Measure time for 30 periods.
   
   
   
   

   Data

wavelength	trial 1	trial 2	trial 3		Trial 1 period	Trial 2 period	Trial 3 period	Ave period	frequency
2.2	9.8	10.6	8.53	9.6433333333	0.3266666667	0.3533333333	0.2843333333	0.3214444444	3.1109574836
3.2	10.8	11.6	9.38	10.5933333333	0.36	0.3866666667	0.3126666667	0.3531111111	2.8319697923
4.2	11.9	11.3	11.7	11.6333333333	0.3966666667	0.3766666667	0.39	0.3877777778	2.5787965616

 

Conclusion

Fluid Dynamics

In this experiment, we will be testing Bernoulli's equation.

We first measure the diameter of the drilled hole which was .005m. We fill the black bucket with water up to a height of 0.066m or 3 inches. From there, we let 200 ml of water flow out and while that is happening we are measuring the time for that to happen.

	1	2	3	4	5	6
t.actual	12.04	11.58	11.26	11.54	11.19	10.81

ave. time = 11.4s

data:

Volume emptied = 2*10^-4m^3

Area of drain hole = 1.96*10^-5m^2

Height of water = .066m

With Bernoulli's equation we calculate for the theoretical time.

t.theoretical = V/(A(2gh)^(1/2))

 t.theoretical = 8.98s

error analysis:

U.Volume emptied =.0002+.000001

L.Volume emptied =.0002-.000001

U.diameter of drain hole =.005+.001

L.diameter of drain hole =.005-.001

U.Area of drain hole = .0000283

L.Area of drain hole =  .0000126

U.Height of water = .066+.001

L.Height of water = .066-.001

U.theoretical = 14.13s

L.theoretical = 6.13s

 Our attained value was well within the error bounds. This error may be due to the fact that we considered a simple model where the height of the water was held constant which is in fact not true! 

If the diameter of the hole was not measured accurately we can arrange our equation to solve for diameter with our average time value.

d = 2*(V/((pi*t(2gh)^(1/2)))^(1/2)

d = .015m

The percent error can be simple calculated as follows:

d.actual = .005

d.theoretical = .015

(d.actual-d.theoretical)/d.theoretical*100

= 66.67%

Our error is concerning, but by the nature of this experiment some error could not be avoided

 

 

Fluid Statics

There are three methods which we used to calculate the bouyant force of a fluid.

In the first method we measure the weight of the cylinder, and then place the cylinder in the fluid and measure the tension with LoggerPro. The difference between the two forces gives us the bouyant force of the liquid.

Bouyant force = Weight - Tension

data:

 mass = .11237 g

Tension = .73 N

Weight = 1.1 N

Bouyant Force = 0.37 N

In the second method, we measured the displacement of the fluid when the cylinder is dropped. Archimedes principle states that the displacement of a fluid equals the bouyant force of the fluid. 

data:

beaker = .14718 kg

beaker + water = .17872 kg

water = .0316 kg

Water displacement = Bouyant force = .31N

The third method, in my opinion is more elegant uses the fact that the volume of the cylinder will determine the displacement of the water. Since the density of the water is uniform we can simple find the displacement of the water.

water displacement = (density of water)(volume of cylinder)(9.81)

data:

height = .0771m

diameter = .0253m

Volume = 3.876 * 10^-5 m^3

Water displacement = Bouyant force = (1000 kg/m^3)(3.876*10^5m^3)(9.8m/s)

Bouyant force = .38N

error analysis:

we could have an upper bound and a lower bound for possible values since no experiment is 100 percent certain. 

In the first case:

U. mass = .11237+.00005

L. mass = .11237-.00005

U. Tension = .73 +.05

L. Tension = .73-.05

U. Weight =  1.1N

L. Weight 1.1N

U. Bouyant force = U.Weight - L. Tension = .42N

L. Bouyant force = L. Weight - U.Weight = .32N

In the second case:

U. beaker = .14718+.00005

L. beaker = .14718-.00005

U. beaker+water =.17872+.00005

L. beaker+water =.17872-.00005

U.water=U. beaker+water-L. beaker = .03164

L.water=L. beaker+water-U. beaker = .03144

U. Bouyant force = .310N

L. Bouyant force =  .308N

In the third case:

U.height = .0771 +.001

L.height = .0771-.001

U.diameter = .0253+.001

L.diameter = .0253-.001

U.Volume = .00004243

L.Volume = .00003529

U.Bouyant force = .42N

L.Bouyant force = .35N

I think that the method that was the most accurate was the third method because the value that we attained is well within the error, but also the range of error was not too great. My second choice would be the first method because the value we attained was also well within error. The second method was barely within error which concerned me, but the range of error was significantly lower than the other experiments.

If in the first method, the cylinder had fallen to the bottom of the water container, then the value we attained would be greater than the true value because then there would also be a normal force pushing upwards as well as the bouyant force of the liquid.