_Lab 3 report (PCS211)

TORONTO METROPOLITAN UNIVERSITY FACULTY OF ENGINEERING DEPARTMENT OF PHYSICS Newton’s Second Law PCS 211: Physics: Mechanics | Section 43 | Group 639 Vallis King Hang Luong(501214102) Jal Gandhi (501263806) Course: PCS211 Section: 43 Instructor: Professor Rabello TA name: Professor Diana Ha Date of experiment: Oct 23th, 2023 Date of submission: Oct 30th, 2023

1 Introduction The objective of this lab is to verify the validity of Newton’s Second Law. The objective of this lab is to observe and consider the relationship of the net force, mass and acceleration. The objective of this lab is to analyze the proportional relationship of mass and force using carts being pulled by different masses, Vernier Photogates, pulleys and analysis software (Graphical Analysis App). By observing the relationship between the mass and net force, it is possible to develop an understanding of how the graphical representation correlates with the mathematical interpretation of the same thing

3 Derivation of Acceleration The net force is the sum of all forces. We already touched upon how the friction is negligible and how the y component of Fg cancels out with the normal force, making the remaining force the x component of Fg. In conclusion, the total net force would equal to the x component of the gravitational force. According to Newton's 2nd Law, total net force = mass x acceleration (Knight, 2021 ) 3 or . In conclusion, acceleration can be calculated as: ? = ? · ? (6) ?? = ?𝑔 (m is canceled from both sides) (7) ? = 𝑔 Comprehension of forces acting on the object Figure 2.2.1: free body diagram of the forces acting on the cart The Figure 2.2.1 outlines all forces that act on the cart upon release on the track ( m denotes mass). From the diagram, an perpendicular intersection of the normal force (

4 ) can be seen along with the force opposite to the normal force, the gravitational force. ? ? Notice that the gravitational force and the normal force are equal in magnitude and direction which cancels out the y-component, hence there is no net force in the y-direction for the cart. To conclude, the total net force is pointing in the direction of the x-component with the tension force making up the magnitude. Figure 2.2.2: free body diagram of the forces acting on the mass The Figure 2.2.2 outlines the forces that are acting on the mass ( ) with the net ? ? force being the gravitational force and tension pulling the mass in the opposite direction. Hence, 2 equations can be written out, (8) ? ?𝑒?,1 = 𝑇 = ? ? ? (9) ? ?𝑒?,2 = ? ? 𝑔 − 𝑇 = ? ? ?

6 multiplication/division formula ( Equation 3), the mean formula ( Equation 4) and the standard deviation formula( Equation 5) . (8) △? = △? 2 + △? 2 (9) △? ? = ( △? ? ) 2 + ( △? ? ) 2 (10) µ = Σ𝑥 𝑖 𝑁 (11) σ = ∑(𝑥 𝑖 −µ) 2 𝑁 Procedure Figure 3.1: Experimental Apparatus used Photogate trigger Vernier Photogates (2) Vernier Dynamics Cart 3x weights (increments of 50g) Clamps String Vernier LabQuest mini Pulley Aluminum track Hanger Note: the graphic analysis app was specifically calibrated for this experiment Set Up of the experiment Figure 3.2.1 correctly shows the placement of which the tools were set up.

7 Figure 3.2.1: Drawing depicting the usage of tools The photogates were placed in between the cart and the pulley with a distance of about 60cm from each other and their distance was recorded and fixed for all trials for a more accurate result. The string was placed in between the gates and ensured that the string was long enough for the cart to clear the second photogate. Ensured that the cart was able to pass through the gates before proceeding. The mass of the cart and the weight were both recorded before the start of the experiment. The Graphical Analysis app must be configured with the “Timing Speed Through Gate” mode. Selected “Use object/flag width” using the radio buttons and recorded the length of the plate on top of the cart. The starting position of the glider was recorded and fixed for all trials. The experiment should start with all 3 masses on the cart and none on the hanger. The mass of the string was set to negligible.

9 constant, a line was drawn in the velocity versus time graph in both experiments to find the most constant and linear slope in a certain time interval. Finding the Theoretical Acceleration Finding Theoretical Acceleration. To find the acceleration of the cart, rearranging Equation 10 to find the total acceleration: (12) ? ? 𝑔 (? ? +? ? ) = ? Finding the Theoretical Slope of Acceleration vs Hanger Mass Finding Theoretical Slope. To find the expected slope of the acceleration vs hanger mass graph, using the theoretical acceleration found for each different hanger mass, we will be able to find the average slope of the graph. Figure 4.1: Table showing different expected Accelerations for different hanger mass Hanger Mass Acceleration 0g (51.7𝑔)(9.8 ?/? 2 ) (51.7𝑔 +151.6𝑔+540.7𝑔) = 0. 68 ?/? 2 50.9g (50.9+51.7𝑔)(9.8 ?/? 2 ) (50.9𝑔+51.7𝑔 +540.7𝑔+100.7𝑔) = 1. 35?/? 2 100.9g (100.9𝑔+51.7𝑔)(9.8 ?/? 2 ) (744𝑔) = 2. 01?/? 2

10 151.6g (151.6𝑔+51.7𝑔)(9.8 ?/? 2 ) (744𝑔) = 2. 67?/? 2 1 (13) ? = 2.67?/? 2 −0.68?/? 2 151.6𝑔−0𝑔 = 0. 013 Finding Experimental Acceleration for the cart Figure 4.2.1: Table showing the recorded lengths and masses Plate Length Length between cart Cart mass Weight masses Starting position from gate 1 11.9 cm ± 0.05 60 cm ± 0.05 540.7g ± 0.05 Mass 1: 50.9g ± 0.05 Mass 2: 50.0g ± 0.05 Mass 3: 50.7g ± 0.05 2 cm ± 0.05 Figure 4.2.2: Table showing the acceleration of all the weights Recorded Tests Hanger mass Increased by Velocity 1 (m/s) Velocity 2 (m/s) Acceleration for 3 trials (m/s^2) Average Acceleration ~150g on cart 0g Trial 1: 0.361 Trial 2: 0.360 Trial 3: 0.387 Trial 1: 0.870 Trial 2: 0.869 Trial 3: 0.884 Trial 1: 0.596 Trial 2: 0.596 Trial 3: 0.586 0.593 Standard deviation: 0.68 ~100g on cart 50.9g Trial 1: 0.458 Trial 2: 0.462 Trial 3: 0.486 Trial 1: 1.258 Trial 2: 1.241 Trial 3: 1.210 Trial 1: 1.31 Trial 2: 1.21 Trial 3: 1.15 1.223 Standard deviation: 1.41 ~50g on cart 100.9g Trial 1: 0.548 Trial 2: 0.591 Trial 3: 0.599 Trial 1: 1.539 Trial 2: 1.555 Trial 3: 1.546 Trial 1: 1.978 Trial 2: 1.95 Trial 3: 1.917 1.948 Standard deviation: 2.25 ~0g on cart 151.6g Trial 1: 0.696 Trial 1: 1.788 Trial 1: 2.56 2.63

12 Figure 4.3.1: Velocity vs Time graph for all 4 different weights on Hanger mass Figure 4.3.2: Acceleration vs Hanger Mass graph for all 4 different weights (14) ????𝑒 = ? = 2.63?/? 2 −0.593?/? 2 151.6𝑔 = 0. 0134 Percent Error of Measured Slope (15) δ = 0.0134−0.0131 0.0131 | | | | · 100% = 2. 29% 𝑒???? Comparing our results with the theoretical slope It is important to have the theoretical slope compared with our actual slope. It is found that the experimental slope of the mass versus acceleration graph (0.0134) bears a close resemblance to the theoretical slope which is calculated to be 0.0131. Using the percent

13 error formula to find the percent error, it gives a percent error of 2.29. This is a small error to have and it is concluded that this experiment is accurate. The standard deviation for the experimental accelerations are also small, hence this experiment is precise. Discussion and Conclusion Concrete Values and Conclusion Following the completion of the experiment where 12 trials in total had to be conducted, it is calculated and proven that the acceleration found has a proportional relationship with the mass added to the hanger. This relationship is graphed and proven to be linear with a slope of 0.0134. Our theoretical value for the slope is calculated to be 0.0131. While there is a slight difference when compared to the theoretical value, the standard deviation in the different average accelerations would be more than enough to come close to concluding that both accelerations correspond to a proportional relationship with the mass. Hence, this validates Newton's 2nd Law. If the cart and the hanger had equal masses, the magnitude of the acceleration would be less than the gravitational acceleration as inserting these values into Equation 11 would give us ½ of the gravitational acceleration. This corresponds with the initial prediction and further reinforces Figure 2.2.2 and Figure 2.2.1 ’s credibility. If friction was present, the slope of the measured line would be smaller due to having the friction force act against the tension force created by the mass of the hanger on the pulley, hence having

15 Some calculations were done on paper and calculators, hence there are rounding uncertainties. A lot of random errors might also have occurred. We are not capable of releasing the cart with the exact time interval along with the starting position having minor differences is also a factor of human error in the results of the measurements. An unnoticeable zero error not found on the Graphical Analysis and the gate sensor might also result in a difference in dimensions. The angle of the rope caused by the pulley might also not be 90 degrees which would introduce less acceleration due to the force having a smaller y-component force. Considering these factors, by repeating the measurements and the experiment a few more times, a more accurate result will have been determined and that might lead to a different conclusion. References Toronto Metropolitan University Laboratory 3 “Lab 3 - Newton's 2nd Law ” Accessed: October 23, 2023.