Physics lab Report

Experiment 5: Newton’s 2nd Law of Motion

This experiment was selected to demonstrate Newton’s second law by measuring the relationship between the acceleration, mass, and the net force on an object. The experiment was split into two main cases. In the first case, mass was removed from the falling object and placed on the glider, meaning the mass of the system remained constant. In the second case the mass of the glider was increased. In both cases, the time it took the glider to move a certain distance was measured to calculate the acceleration and in turn, the experimental force.

Introduction and Theory

Newton’s second law states that the mass and acceleration of an object is directly proportional to the net force. Equation 1 is the equation commonly used to describe this law.

(1)

The purpose of this experiment was to demonstrate this law and its described relationship.

Photogates were set up at the 20cm and 70cm mark on the track and a start position of the glider was decided to be at the 95 cm mark. A mass was placed on a string, attached to the glider, and passed over a pulley. The experiment was split into two cases, each with four trials with varying masses. The masses used in the experiment can be found in the first four columns of Table 1. During each trial of the experiment, the glider is pulled by the mass on the string, providing a force. The time the glider took to pass through each photogate is measured. These times paired with the width of the glider, gives two velocities using Equation 2, where d is the width of the glider at 28mm with an uncertainty of 0.2mm. Using these velocities, the time the glider took to pass both photogates, and Equation 3, an experimental acceleration can be determined.

(2)

(3)

Theoretically, the net force applied to the system should be the mass of the hanging weight (m) times the acceleration due to gravity (g), where g = 9.81 m/s2, as seen in Equation 4. A theoretical acceleration can then be found by rearranging Equation 1, resulting in Equation 5.

(4)

(5)

The uncertainty (σ) of a calculation can be found using Equations 6 and 7, while the uncertainty of a set of numbers can be found with the standard deviation, as seen in Equation 8 where is the mean of the data set.

(6)

(7)

(8)

Using the same masses as the experiment, the theoretical acceleration and uncertainty can be found in Table 1, where σM = 0.002 kg.

Table 1: Theoretical Acceleration

Case	Trial	m (kg)	M (kg)	mg (N)	a (m/s2) (Eq. 4)	σa (m/s2) (Eq. 6)
1	1	0.02	0.185519	0.1962	0.954656	0.013454
	2	0.015	0.190519	0.14715	0.715992	0.0119687
	3	0.01	0.195519	0.0981	0.477328	0.0107832
	4	0.005	0.200519	0.04905	0.238664	0.0100048
2	1	0.005	0.200519	0.04905	0.238664	0.0100048
	2	0.005	0.195519	0.04905	0.244615	0.0102682
	3	0.005	0.190519	0.04905	0.250871	0.0105462
	4	0.005	0.185519	0.04905	0.257455	0.01084

The acceleration can be compared to the mass because Equation 5 is a linear equation, , where in Case 2 and in Case 1 , and . Therefore, when the data is plotted, it should appear to be linear.

The percent difference and percent uncertainty of acceleration can be calculated using Equation 9 and 10 respectively.

(9)

(10)

Data and Analysis

Each trial was performed a total of three times to gather an average time. To perform a trial, the glider is brought to the 95cm mark and let go. Table 2 shows the times collected and the mass of the glider and the hanging mass for each trial.

Case	Trial	m (g)	M (kg)	Δt1a (s)	Δt1b (s)	Δt1c (s)	Δt2a (s)	Δt2b (s)	Δt2c (s)	Δt3a (s)	Δt3b (s)	Δt3c (s)
1	1	20	0.1855	0.0351	0.0341	0.0335	0.0204	0.0194	0.0189	0.5376	0.5384	0.539
	2	15	0.1905	0.0407	0.0403	0.0406	0.0231	0.023	0.0233	0.6276	0.6223	0.6226
	3	10	0.1955	0.0515	0.0505	0.0483	0.0294	0.0286	0.0269	0.7841	0.783	0.7867
	4	5	0.2005	0.0694	0.0677	0.074	0.0385	0.0373	0.042	1.1211	1.1207	1.1206
2	1	5	0.2005	0.0755	0.0749	0.0729	0.043	0.0429	0.0419	1.1336	1.136	1.1061
	2	5	0.1955	0.0746	0.075	0.0751	0.0425	0.0431	0.0403	1.1328	1.1363	1.1364
	3	5	0.1905	0.0699	0.071	0.0648	0.0403	0.0407	0.038	1.0718	1.0772	1.07
	4	5	0.1855	0.0687	0.0688	0.0682	0.0394	0.0396	0.0395	1.0471	1.0492	1.0472

Table 2: Raw Data Collected

The average of Δt1, Δt2, and Δt3 were then found and placed in Table 3, along with the uncertainty found using Equation 7. Using Equation 4, the force due to gravity was calculated.

Case	Trial	Δt1avg (s)	Δt2avg (s)	Δt3avg (s)	σΔt1avg (ms)	σΔt2avg (ms)	σΔt3avg (ms)	mg (N)
1	1	0.03423	0.01957	0.53833	0.65996	0.62361	0.573488	0.1962
	2	0.04053	0.02313	0.62417	0.16997	0.12472	2.430821	0.14715
	3	0.05010	0.02830	0.78460	1.33663	1.04243	1.551344	0.0981
	4	0.07037	0.03927	1.12080	2.66124	1.99388	0.216025	0.04905
2	1	0.07443	0.04260	1.12523	1.11155	0.49665	13.56474	0.04905
	2	0.07490	0.04197	1.13517	0.21602	1.20370	1.673984	0.04905
	3	0.06857	0.03967	1.07300	2.70103	1.18977	3.059412	0.04905
	4	0.06857	0.03950	1.04783	0.26247	0.08165	0.967241	0.04905

Table 3: Average Time and Gravitational Force for Each Trial

Using this information and Equation 2 the velocity of the glider at each photogate was calculated and placed in Table 4 with its uncertainty using Equation 7. Using Equation 3, the acceleration was calculated, and its uncertainty was calculated using Equations 6 and 7.

Table 4: Velocity of Glider at each Photogate

Case	Trial	v1 (m/s)	v2 (m/s)	σv1 (m/s)	σv2 (m/s)	a (m/s^2)	σa (m/s^2)
1	1	0.81792	1.43101	0.01682	0.04674	1.13886	0.09228
	2	0.69079	1.21037	0.00572	0.01083	0.83245	0.01989
	3	0.55888	0.98940	0.01544	0.03712	0.54871	0.05125
	4	0.39792	0.71307	0.01532	0.03656	0.28119	0.03537
2	1	0.37618	0.65728	0.00623	0.00899	0.24982	0.01017
	2	0.37383	0.66720	0.00288	0.01972	0.25843	0.01756
	3	0.40836	0.70588	0.01635	0.02176	0.27728	0.02538
	4	0.40836	0.70886	0.00331	0.00527	0.28678	0.00595

Using the Data collected from Tables 2, 3 and 4, two plots can be constructed. One plot is the acceleration (Table 4) vs the force due to gravity (Table 3) of the data in case 1 and can be seen in Figure 1.

Figure 1: Acceleration of Glider vs Applied Force

The second plot is the acceleration (Table 4) vs the inverse mass sum (Table 1 for m and M) of the data in case 2 and can be seen in Figure 2.

Figure 2: Acceleration vs Inverse Total Mass

The regression analysis of the data set used in Figure 1 can be seen in Table 5. The regression analysis of the data set used in Figure 2 can be seen in Table 6.

Table 5: Regression Analysis of Case 1

Table 6: Regression Analysis of Case 2

Lastly, the percent difference and percent uncertainty in the acceleration was calculated using Equation 9 and 10 where the theoretical value is the “a (m/s2)” found in Table 1 and the experimental value is the “a (m/s2)” found in Table 2. Table 7 shows the results.

Table 7: Percent Difference and Uncertainty

%Diff a (%)	%UNC a (%)
19.29579091	0.081026094
16.26470224	0.023896311
14.95424952	0.093408166
17.81816385	0.12578702
4.672707552	0.04072023
5.648738714	0.067953018
10.52674088	0.091536772
11.39102508	0.020732074

Results and Conclusions

The first warning sign that the experiment did not successfully demonstrate the principle of Newton’s second law is Table 7. The percent difference was consistently much greater than the percent uncertainty. This means that the theoretical values of acceleration were not within the experimental range. Even when factoring in the theoretical uncertainty, the percent difference was still significantly larger than the percent uncertainty.

The results still follow the expected linear trend however. In case 1, the slope of the plotted line was expected to be about equal to , which was equal to 4.866 kg-1 throughout case 1. The slope of the line can be seen in Table 5 as the “X Variable Coefficient”, which is 5.824. Though the experimental value is about 20% more than the expected value, it still shows that the mass of a system has a linear correlation to the acceleration of the system. In case 2, the slope was expected to be the force due to gravity (Table 3). The slope of the experimental values is 0.1016 (Table 6), double the expected 0.0491. Even when factoring in the uncertainty (Standard Deviation), the expected slopes do not fall within the experimental range. This said, the plotted line still shows the expected linear relationship between the applied force and the acceleration of the system.

The y-intercepts of both cases are much more reasonable. In the regression tables (Tables 5 and 6), The “Intercept Coefficient” is the y-intercept of that case. When factoring in the uncertainty (Standard Deviation) of the case 1’s y-intercept, the expected value of 0 lands within the range given in Table 5. The expected value of the y-intercept does not fall within the range given in Table 6.

Taking these results in account, the experiment did not accurately portray Newton’s second law. Very few of the expected values fell within the experimental range. The failure of this experiment could come from a variety of reasons. The easiest to point out are air resistance, friction from the pulley, string, and track, and slight inaccuracies in measurement.