Experiment 3: speed of transverse waves

the objective for this experiment is to find a relationship between lambda and f in a long spring.
To do so we create a wave in fundamental form from the same spring but each time we change the length.
So each time λ = 2L where L is the distance between the two nodes.

Here is the data collected:

λ , m	T, s	1/T = ƒ, Hz	λ*ƒ, m/s
2.85 ± 0.05	0.423  ± 0.005	2.367 ± 0.005	6.75 ± 0.05
4.90 ± 0.05	0.754 ± 0.005	1.326± 0.005	6.50 ± 0.05
5.52 ± 0.05	0.871 ± 0.005	1.149± 0.005	6.34 ± 0.05
6.62 ± 0.05	1.027 ± 0.005	0.974± 0.005	6.45 ± 0.05

the average of the last column is 6.51 ± 0.17 and since the units are m/s we can assume that it is the value for speed. (λ*ƒ, m *1/s so m/s)
also to back up the theory the graph of λ vs T should look linear. The graph is as follow:

y = 6.4537x


the fitted line is linear and passes through zero and the slop is 6.4537 m/s which is very close to the average of λ*ƒ values. 
As for uncertainty the error in the experiment comes from the non homogeneous shape of the spring and also us not being able to create a perfect fundamental loop since we were creating it with hand.  Also another problem was the exact measuring of time for ten oscillation.
   

Experiment 2 : Fluid dynamics

In this experiment the goal is to obtain the amount of time required for 200 ml of water to transfer from a large bucket to a 200 ml marked beaker underneath with two different method. First is to use the following setup and then theoretically.

A. 

Trials	1	2	3	4	5	6
Time to empty (tactual)	12.2 ± 0.1	12.1 ± 0.1	11.4 ± 0.1	11.6 ± 0.1	11.5 ± 0.1	12.1 ± 0.1

S = √((S(x_i –x_avg)⁄n-1) = 0.4

t_average = 11.9 ± 0.4

                                   Theoretically

d_hole  = 0.5 ± 0.05cm = 0.005 ± 0.0005 m

h = 4.5 ± 0.05cm = 0.045 ± 0.0005 m

Area_hole = ¼πd² = 0.196 × 10^-4 ± 0.0025 × 10^-4 m

g = 9.81 m/s²

V = 2.1 × 10² ± 0.15 × 10² ml = 0.21 ± 0.015 L = 0.00021 ± 0.000015 m³

Beside all the errors in measuring different quantities, since the poured water made the top of the water in the 200 ml beaker turbulent every time we stop the flow the actual amount of water was higher than 200 ml, which is why we choose 210 ml with uncertainty of 15 ml.  This is the reason behind the large uncertainty for the final time.

t_theoretical= V⁄(A√2gh) = 11.4 ± 0.9 s

Error = 1 - t_theoretical ⁄t_average = 0.042 or 4.2%

Both values agree within their uncertainties. The huge uncertainty in the theoretical time is due to the big uncertainty in volume and also for the fact that the area is of the hole that let the water pass through is not exactly circle so the actual value should be smaller which makes the t_theoretical bigger and closer to experimental value.

If we rearrange the  t_theoretical= V⁄(A√2gh) to solve for d we get:

d = √ ((4V)/(πt_actual√2gh)) = 4.8 × 10^-3 m

Error for d = 0.038 or 3.8 %

Experiment 1 : Fluid Statics

In this experiment the goal is to measure the buoyant force acting on a cylinder three different ways and see how they compare.


                                 A. Underwater Weighing Method 
         In this method using a force probe, first we we hang the cylinder from a string and weigh it in the air and then we weight the cylinder while completely submerged in water.
 W_air = 0.229 ± 0.006 N

 W_sub = 0.165 ± 0.006 N

In both cases T is the value that the force probe reads or W_sub and W_air 

F_b = W_air - W_sub = 0.229 - 0.165 = 0.064 ± 0.0012 N

                                  B. Displaced Fluid Method

         For this method we obtain the F_b value by measuring the volume of the overflow water and calculating its weight (W_f) since W_f,  according to Archimede’s principle, is the F_b . To do so we weight the 500 ml beaker empty then place a 200 ml beaker full of water inside the bigger beaker and then submerge the cylinder. Afterwards we weight the beaker with the overflow water in it. 

W₂ = 0.162 ± 0.0005 N

W_beaker = 0.0962 ± 0.0005 N

W₂ = W_beaker + W_f

F_b = W_f = W₂  - W_beaker = 0.162 - 0.0962 = 0.0657 ± 0.0005 N

                                            C. Volume of object method 

         We calculate the volume of the cylinder from its height and diameter measurements and since the volume of the displaced water is the same as the volume of the cylinder by using the density of water we obtain W_f .he first method 

h = 0.0470 ± 0.0005 m   

d = 0.0140 ± 0.0005 m

V = ¼πd²h= 7.24 × 10^-6 m³

W_f = ρ_wVg = 0.0710 ± 0.006 N

the uncertainty that I calculated was by dividing by two the difference of the highest and  lowest possible value for W_f.

Conclusion:

The sources of error lies in slight increase of volume because of the submerged string with the cylinder for the first method and not being able to fill the beaker in a way that submerging the slightest volume lead to water overflow for the second method and possible error in reading the values for height and diameter of the cylinder with the third method. Also with the third method we are assuming that the cylinder does not have any imperfections, which most likely is not true. All these error might make the calculations inaccurate. However since all obtained F_bs calculated from the three methods agree within the same boundary of their uncertainties we can trust the result of all three method and say they are the same within their error estimation.

Out of the three methods the first one seems to be the most accurate since the only thing that might cause error is the extra volume of the string submerged with the cylinder in the water. Even though the force probe might not be calibrated but it does not matter since we are measuring the difference of the two read values. Unlike the other two methods the area of error for this method is not very significant.

In part A even if the cylinder touches the bottom the buoyant force remains the same since it only is calculated from the displaced water which does not change with the object submerging in different depths.