Lab 2 - Determining g on an Incline (1)

Determining Gravity on an Incline Introduction The objective of this experiment is to both conceptualize and analyze the variation between the force of gravity in its theoretical state and an observed metric of gravity values as they pertain to the motion of an object down an inclined surface until it reaches its terminal position. Under ideal conditions, the measurement of gravity on an incline would be equal to 9.8m/s² and be the sole acting force on acceleration; however, during testing, there are other forces that may impede or amplify the acceleration of the object as it approaches its terminal position (Cano, 2019). Thus, although the expected acceleration of the object should be approximately equal to that of the literature value for gravity during free fall, there may be slight differences which can be compared. In this experiment, the core objective would be not only to study this difference between the observed and theoretical value for gravity but to account for differences that occur as well as their possible cause. To account for this discrepancy, the moving object's acceleration and velocity will be measured at predetermined intervals of time. Due to the inclined nature of the surface in question, the angle of inclination of the surface it travels along would be the primary contributor to acceleration. In tandem with the use of multiple trials at varying angles on inclination, an observed value for gravity will be established and further contrasted with the literature value. Theory Throughout history, numerous scientists have devoted their life’s work towards the pursuit of understanding the ability of a force to exert work on a given object at a continuous pace, over a constant period of time. Any push or pull that is exerting pressure on an object and affecting its velocity is referred to as a force. (Cook, 1965). Forces are employed in practically all computations to form the foundation for what may be called modern physics for kinematics,

and few are as influential as the force of gravity, which was initially postulated by Galileo in the 17th century and proved by Newton later that century. (Cano, 2019). Gravity is the natural force exerted by a planet or other entity that will attract objects toward its center and is classified as an acceleration vector (Cook, 1965). Gravity can be measured to act on an object at a rate of -9.8m/s², Figure 1 illustrates the observed relationship between the forces acting on the object and its direction. (Cano, 2019). Figure 1. Idealistic motion diagram for the movement of an object along an inclined plane. The idealistic model demonstrates the movement of an object down an incline in a vacuum with no external forces acting on the object in motion. As a result, it will possess a g value equivalent to a perfect -9.8m/s². The force of gravity can integrate with other forces of motion to produce greater or less acceleration when an object is suspended in the air or moving down an incline toward the surface or ground level. This notion further implies that although the object may be subject to change, the force in itself will remain constant. This well-known process is also what is examined in this experiment, allowing the value of free-fall acceleration, g, to be calculated using the slope of the acceleration vs. sine of the track angle graph, as well as several physics derivations. Additionally, the outcome can be contrasted with the 9.8 m/s 2 norm for the free-fall acceleration g. One of the primary physics concepts investigated in this experiment includes the incorporation of trigonometry in order to find the angle as well as the mathematical correlation

dispersed and far less accurate. Moreover, in order to effectively calculate the standard deviation, the formula for acceleration is rather beneficial. This formula uses the variables V, for velocity, and T for time. The formula for determining the average acceleration: A avg = �� �� The formula for determining the standard deviation: ( −µ ) σ = Σ �� �� �� 2 Finally, as stated in the Laboratory Guide - Physics the measured values should be presented in the format of, µ ± σ , µ being the mean, and σ being the standard deviation. The uncertainty displays the best estimate of how far the experimental quantity may be compared to its “true value” in this case, in contrast with the 9.8 m/s 2 norm for the free-fall acceleration g. A broad replication of the experiment is necessary. This is because repeating an experiment enables researchers to compute the variance of the data, which may contain any remaining inaccuracies, as well as to enhance the accuracy and trust of the results. Procedure Given that there were four steps in this experiment's method, it was fairly cohesive. Using the menu bar in the upper left corner, which included a sign for the Logger pro interface, the experiment's first step was to validate that the device and system were correctly configured. The motion sensor was also set, and the cart was ready to be installed. The understanding of the system, the acquisition of data, and the types of data that were being acquired were all examined during this phase. The cart was positioned at the top of the incline in the second step, and the application began collecting data by hitting the "collect" button while concurrently displaying two graphs. In order to produce a decent run, this step was modified and performed numerous

times, resulting in a constant slope on the velocity vs. time graph during the cart's motion. The perfect run was selected, recorded, and step two was performed two more times as the experiment called for 3 runs per angle. In order to compute for g on the inclined plane, measurements were taken into account in the fourth stage. Moreover, the incline's length and both endpoints for height were documented. Thereafter, steps 2-3 were repeated four additional times for new inclinations and angles, and step 4 was performed four more times, documenting each changed height. Results and Calculations The following calculations and illustrations were carried out with the intention of simplifying and organizing all recorded data with the intent of displaying the true effects as well as the value of gravity on the inclined surface when acting on a rapidly moving object. Firstly, using the sample calculations below, the values of the angle of inclination were calculated using the inverse sine function and the differences in height. Angle 1 Angle 2

The table below depicts all data which will serve as a central focus for the remainder of the lab, each value of which was determined independently. The angle of inclination was calculated using the series of subtraction and trigonometric identities listed above to isolate and solve for the unknown angle θ . Average acceleration was determined by first omitting any “push point” and “trail end” values from the data of each respective trial run and determining the average acceleration of each independent trial. From there, the average of these newly tabulated values was determined and appears as the average acceleration values listed above. Since the smallest unit of measurement used in the calculations was 1mm, the corresponding uncertainty value associated with that metric of measurement would be 0.5mm and this value is constant across all tests. Lastly, alongside the overall average acceleration being tabulated, the standard deviation of the entire experiment was calculated using the same values as well as employing the formula for standard deviation. This value was 0.203 and will be used in Figure 2 and Figure 3. The standard deviation values listed in the chart below were calculated by comparing the individual standard deviation of all acceleration values tabulated at each respective angle using the same formula. [See theory section for standard deviation formula]. Angle of Incline Average Acceleration Uncertainty (mm) ± Standard Deviation (m/s²) sin(9.4 ° ) 1.877m/s² 0.5mm ± 0.14 sin(8.4 ° ) 1.815m/s² 0.5mm ± 0.124 sin(7.7 ° ) 1.594m/s² 0.5mm ± 0.076 sin(10.21 ° ) 1.923m/s² 0.5mm ± 0.073 sin(6.9 ° ) 1.371m/s² 0.5mm ± 0.360 Table 1: Table Displaying the Average Acceleration, Uncertainty and Standard Deviation of all Values Alongside their Corresponding Angle of Inclination.

Figure 2: Figure displaying the Relationship Between the Average Acceleration of all Trial Runs and the Sin of Each Angle Tested in Accordance to a Consistent Standard Error. The graph above depicts the linear relationship between the average acceleration of all trials conducted in a simplified form where the expected average acceleration of the same object 2 at any inclination can be determined using the equation of the line. Although the �� value may not show a perfectly linear relationship, it provides an adequate range in which an expected value can be found.

sin(90) as an angle of inclination, as it will resemble that of free-fall conditions. This is followed by using the aforementioned gravity formula as there are no listed mass values for the object in question, using the average velocity over a defined angle is a means of obtaining the effect of this force. �� 0.482m/s² �� = 0. 164( ������ (90)) + 0. 318 ⇒ �� �� = �� = �� �� ������ (90) ⇒ �� = 0. 482 �� / �� 2 ������ θ ⇒ �� = 0.482 Discussion and Conclusion The purpose of this experiment was to determine the experiment value of gravity and compare it against the theoretical value of acceleration displayed in literature as an object moves down an inclined surface. The results yielded a significant decrease in acceleration when compared to that of free fall acceleration measuring at 9.8m/ �� as the observed value was 2 approximately only 0.482m/ �� . Although the effect of gravity on an inclined surface will be 2 significantly less than that of free fall motion, the disparity between the two values should likely not be this large. This disparity could be indicative of a single or several eros accumulating throughout the testing process and could have stemmed from one of many sources. Throughout the experiment, there were a few challenges faced, such as the effort of avoiding having the cart touch the sensor. However, it was noted that the difficulty grew as the incline's height increased. In such situations, the cart would come into close contact with the motion sensor, resulting in insufficient data and repeated trials. References Cano Porras, D., Zeilig, G., Doniger, G. M., Bahat, Y., Inzelberg, R., & Plotnik, M.

(2019;2020;). Seeing gravity: Gait adaptations to visual and physical inclines - A virtual reality study. Frontiers in Neuroscience , 13, 1308-1308. https://doi.org/10.3389/fnins.2019.01308 Cook, A. H. (1965). A new absolute determination of the acceleration due to gravity at the national physical laboratory. Nature (London), 208(5007), 279-279. https://doi.org/10.1038/208279a0 Pearson, G. (2021). 1 Laboratory Guide - Physics. Appendix Angle 1 Trial Values Run 1: Time (s) Run 1: Acceleration Run 2 (m/s²) (m/s 0.05 0.1 0.15 0.2 0.25 - 2.7513 0.3 0.35 0.4 0.45 0.5

0.9 1.490 0.95 1.170 1 1.238240586 - 1.05 1.377 1.1 1.017672531 - Average of Each Run Average of the Averages 1.815 Standard Deviation of Values 0.076 Angle 3 Trial Values Run 1: Time (s) Run 1: Acceleration Run 2 (m/s²) (m/s 0.05 - 0.1 - 1.285847716 0.15 - 2.192976852 0.2 0.25 0.3 0.35 0.4 0.45 0. 0.5 2 0.55 3 0.6 0.65 2 0.7 1. 0.75 1.006768519 1.1059 0.8 1.676571296 1.0138 0.85 2.026240741 - - 0.9 1.712512191 - - 0.95 1.352468056 - - 1 1.192613117 - - 1.05 1.245439352 - - 1.1 1.307475772 - - 1.15 1.136293364 - -

Average of Each Run 1.65 Average of the 1.593 Averages Standard Deviation of Values 0.124101006 Angle 4 Trial Values Run 2: Time (s) Run 1: Acceleration Run 2: Acceleration Run 3: Acceleration (m/s²) (m/s²) (m/s²) 0.05 - 1.086886543 3.523531019 0.1 1.152929923 1.204702809 3.281287269 0.15 1.399551157 1.341733426 2.745794398 0.2 1.548195077 1.450530062 2.13504267 0.25 1.560750571 1.547903951 1.791407485 0.3 1.573237253 1.618261296 1.598337654 0.35 1.623004012 1.633378704 1.570442438 0.4 1.641741975 1.601407716 1.612205864 0.45 1.630096914 1.474847068 1.648623148 0.5 1.59468534 0.972309722 1.646558796 0.55 1.425990741 - 1.585051698 0.6 0.657469599 - 1.314939198 Average of Each Run 1.437059324 1.39319613 2.03776847 Average of the Averages 1.622674641 Standard Deviation of Values 0.360150191

0.35 0.661068981 1.48119892 1.603366204 0.4 1.399312963 1.029476389 2.098704784 0.45 1.821446451 0.971304012 1.748876543 0.5 1.591191821 1.269788117 1.142645216 0.55 1.388779475 1.692980247 1.022542284 0.6 1.271481944 1.631790741 1.087489969 0.65 1.061182716 1.16090679 0.931922531 0.7 0.924882562 0.806843981 0.769579784 0.75 0.879255093 0.794034414 0.69785679 0.8 0.792499383 0.949284259 - 0.85 0.657363735 1.022118827 - 0.9 - 0.97453287 - 0.95 - - - Average of Each Run 1.457041021 1.444950084 1.208336068

Average of the Averages 1.370109058 Standard Deviation of Values 0.140229893