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PI Name
DA Name
R Name
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Kaleigh Wright
Lauren Hall
Allison Kidwell
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δM1 F1
δM2
F2
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180
101.3
0.05
993.753
20
51.7
0.05
507.177
F3min
F3max
F3_measured
δF3
57.15
57.5
560.6415
564.075
562.35825
1.71675
339
0.5
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X component
Y component
~
F1
-993.753
1.216996430593E-13
F3x
F3y
F3_calc
δF3
~
F2
476.5904843323
173.464750231483 517.16252 -173.464750231 545.47876878 0.95440509 341.45771 0.048436870741442
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DA 1: Include the two scatter plots that compare the magnitudes and angles. These charts should have titles, axes, labels, units, etc, as shown in Figure 2. ~
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F1x_max
F1x_min
F2x_max
F2x_min
F1y_max
F1y_min
F2y_max
F2y_min
F3x_max
F3x_min
F3y_max
F3y_min
~
-993.263
-994.243
477.05093371652 476.13003 1.21759651E-13 1.216396E-13
173.63234 173.29716
518.11296505185 516.21206628 -173.2972 -173.6323
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F3x
F3y
F3_min
F3_max
F3
δF3
θ3max
θ3min
517.16251566767 -173.464750231
544.524382526562 546.43319 545.478787613 0.9544050866
-18.493932 -18.59081
-18.5423688493472 0.0484368707
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Researcher: Explain in detail how your group determined θ_max and θ_min.
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Large surface area
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θmax
θmin
μs_large max
μs_large mμs_large
δμs
~
23
21 0.424474816209605
0.383864 0.40416942562 0.0203053906
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Medium surface area
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θmax
θmin
μs_Medium max
μs_Mediumμs_Medium
δμs
~
18.5
16 0.334595319502073 0.2867454 0.31067035263 0.0239249669
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small surface area
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θmax
θmin
μs_small max
μs_small mμs_small
δμs
~
15.5
13.5 0.277324544059838 0.2400788 0.25870165157 0.0186228925
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Researcher: Create a careful scale drawing, similar to Figure 1, of the three forces, F_1,F_2, and F_3 that you have selected. Then create a second figure where these forces are rearranged without changing the magnitudes or directions so they follow the “tip-to-tail” method of addition of vectors. Explicitly explain the concept(s) which connect(s) your drawing to Newton’s Second Law.
Experimental Determination of F
3
and θ
3
θ
1
M1 (holder included)
θ
2
M2 (holder included)
The drawings connect to Newton's Second Law of Motion because it states that a force is equal to the mass of an object times the acceleration. The equation for this law is = , where is force, m is mass, and is acceleration. This relates to the drawings because in this experiment, two angle 𝐹⃗
𝑚𝑎⃗
𝐹⃗
𝑎⃗
measurements and masses were known. These values were then used to determine the forces, using gravity as the acceleration. The known forces were determined by multiplying the measured masses by the universal value of gravity on Earth (9.8m/s^2). Acceleration of an object is proportional to the magnitude of the net force. Therefore, the overall force acting on the object can be calculated. The third angle was then calculated by resolving F1 and F2 into the x and y components using F1x=F1Cosθ1 and F1Y=F1Sinθ1 and repeating for Force 2. Then, based on Newton's Second Law for the ring which is motionless, F3x can be theoretically solved for using ΣFx=m*a=0 → ΣFx=F1x+F2x+F3x=0 → F3x(theory)=-F1x-F2x. Similiarly, the F3y(theory) was found. Then, since the x and y components of F3 were known, the magnitude of F3 can be found using Pythagorean's Theorem. Lastly, the direction or angle of F3 was found using ARCTAN(F3y/F3x).
M3_min (holder included)
M3_max (holder included)
θ
3_measured
δθ
3
Calculation of F
3
and θ
3
F
3
theory
θ
3
_calc
δθ
3
θ
3
δθ
3
θ_max and θ_min were determined for the large, medium, and small surface areas of the wooden block. The large surface area of the block was the large face of the block, the medium surface area of the block was the larger of the two sides of the block, and the small surface area is the smallest side of the block. θ_min was determined by placing the block with the respective surface area size on the incline, and raising the incline from zero until the block began to slide. The value of the angle when the block began to slide was recorded. Multiple trials were repeated to clarify the θ_min, but the first signs of the block beginning to slide down the incline at the smallest angle was determined to be the θ_min. The θ_max was determined by the repetition of this process and was recorded as the greatest angle reached before the block began to slide. DA2: One striking feature of Figure 2 is the fact that the “Measured” points have error bars that are much larger than the “Calculated” points. Highlight rows 31 through 37, right-click, and select “Unhide.” Use these additional cells to determine why
the “Measured” results have uncertainties that are so much bigger than the “Calculated” results. Ultimately you want to determine which cells in row 5 control the size of the error bars for each type of result.
The Measured results have larger uncertainties than the Calculated results because the calculated results factor in the minimum and the maxiumum of both the y and x components, hence, causing a smaller standard deviation, and in turn, a smaller error bar. This was done by subracting D5 from E5 and adding D5 to E5, multiplying both values separately to 9.8, then muliplying both by cos θ to get the minimum and maximum x components of the magnitude. The same process was repeated, except using sin θ to get the minimum and maximum y components of the magnitude. This additional process made the standard deviation of the calculated value significantly smaller than the measured value's standard deviation, which only accounted for the overall minimum and maximum. DA: Create a scatter plot to compare μs based on the surfaces in contact. Label all axes. In your caption, comment on whether the three μ measurements agree or not.
PI: Using the charts provided by the DA, explain why the measured and calculated values of F_3 and θ_3 overlap or not. In your explanation, use terms such as overlap and error bars etc. The graph below illustrates the comparison between Surface in Contact and the static friction (μs). We compared the results of Large, Medium, and Small surface area with their corresponding static friction, including the standard deviation for each static friction value. The three measurements do agree, as can be observed in the graph, showing a clear lineation that as the surface area decreases, so does the static friction. From looking at the graphs above, we can tell that there are differences between the measured and calculated values of both F_3 and θ_3; furthermore, there is clearly no overlap shown on either of the graphs. In the "F3 : Measured vs. Calculated" graph there is a gap of about 17 (N), which is assumably the miscalculation. The error bars on the F_3 measured point are slightly bigger than the error bars on the F_3 calculated point, proving that the calculated point is significantly more accurate. However, on the "F3 Angle: Measured vs. Calculated" graph, the error bar on the θ_3 measured is much larger than the error bar on θ_3 calculated, but the gap is much smaller, only being about 2 (°). This proves there is slightly less miscalculation when determining θ_3 measured. The fact that the error bars are smaller for the calculated points on both graphs, is because these points are formulated values and they are fixed, while the measured points are found with vast uncertainty. PI: Using the chart provided by the DA, explain whether μs depends on the surface area of the wooden block. In your explanation, use terms such as overlap and error bars etc.
From looking at the graph, it is proven that the surface area of the wooden block impacts the prevalence of static friction. This being that the greater the surface area, the greater the appearance of static friction (μs), and vice versa. Thus, we can see the overlap between these variables behaviors, because of the trend that shows they are directly related in the graph. In terms of error bars on the graph, the smaller surface area presented a normal, and smaller, range of standard error compared to the medium and large ones. The medium and large surface areas had a slightly wider range of error, and this could be caused by random uncertainty from human error, PI: The formula μ_s≥tanθ relies on the fact that there are exactly three forces involved on the incline plan, which form a triangle in exactly the same way that the Researcher described at the end of part 1. Build upon the Researchers work within the context of static friction to determine WHY μ_s≥tanθ.
For static friction to be able to overcome gravity and the mass of an object, it must be equal to or greater than the maximum angle of a stationary object. So, μ_s≥tanθ can be proven through the relationship between the max angle of the stationary object and static friction. Since the wooden block is at a certain angle, tanθ can be used because it is obtained from sinθ/cosθ. Therefore, μ_s≥tanθ is true when the static friction is found to be the same as tanθ, or greater than it, because the object will then be stationary.
0.8
1
1.2
1.4
1.6
1.8
2
337.5
338
338.5
339
339.5
340
340.5
341
341.5
342
F3 Angle: Measured vs. Calculated
θ3_measured
θ3_calc
Degrees (°)
0.5
1
1.5
531
536
541
546
551
556
561
566
|F3| : Measured vs. Calculated
F3_measured
F3_calc
F (N)
0.8
1
1.2
1.4
1.6
1.8
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Surface In Contact vs. Static Friction (μs) μs_large
μs_Medium
μs_small
Static Friction (μs)
