_PCS Lab Report 2

Physics Lab Two - Determining g Using an Incline Course Number PCS120 Course Title Physics 1 Semester/Year Fall 2023 Instructor Ms. Antimirova TA Name Nima Student Name Student ID Signature Nick D’Elia 510264069 N.D Andrew Moore 501118154 A.M Lab/Tutorial Report No. Lab 2 Report Title Determining g using an incline Physics Lab 2 Section No. 172 Group No. 129 Submission Date 10/12/2023 Due Date 10/16/2023

Introduction: In this Physics Lab, we will be re-visiting Galileo’s famous experiment in which he was able to extrapolate a value for the acceleration of gravity. We will be attempting to replicate the very same experiment, but with today’s modern technology. During the 17th century, Galileo attempted to determine the acceleration due to gravity by taking a small component of acceleration due to gravity; by observing how the object moved on several different inclined planes. We will do the same today, however; we will be utilizing motion sensors to capture the motion of the object, and using the given velocity values, we will apply linear regression to further deduce the acceleration of gravity at a free-fall. Procedure: In this lab, our goal is to find an approximate value of gravity by following similar strides to Galileo in the 17th century. To begin our procedure, we are equipped with Graphical Analysis (Software), a LabQuest Mini, a Motion Sensor, a Vernier Cart, an Angle finder, an Aluminium Track with feet, a Retort Stand with clamp, and a Simple graphing program. We began by setting up the aluminum track at 20 degrees using the angle finder to begin with our experiment. We will run five trial runs for each angle, in which the cart starts from rest and descends down the track. We will be evaluating the displacement and velocity of the cart against time, and will be exported into Graphical Analysis. The angles in which this experiment will explore are: 20 degrees, 17 degrees, 14 degrees, 10 degrees, and 7 degrees. We will record five trail runs for each angle in order to capture a general average, assuming there will be inconsistencies. With all 25 trail runs, having each group of 5 runs belonging to a different corresponding angle, we will place our data into a scatter plot and using linear regression, will find the line of best. We plan to find the RMSE of the data, and then further extrapolate the acceleration due to gravity by evaluating the angle at which sin( θ) is equal to 1, 90 degrees, or at a state of free fall. To conclude, we will find just how inaccurate our lab results will be by using the percent error equation. Results and Calculations: After performing 5 consecutive trials for each angle, the data gathered in the table represents the angle of acceleration sin (θ) along with its corresponding slope of acceleration. It also displays the expected acceleration calculated from sin( θ) x 9.8. *SEE TABLE 1 FOR THE LISTED DATA*

Table 1 *Highest incline → Lowest incline* Trials Angle of Acceleration Sin( θ) Acceleration (m/s^2) Expected Acceleration (m/s^2) 1 Sin(20 o ) = 0.342 3.895 3.352 2 Sin(20 o ) = 0.342 3.708 3.352 3 Sin(20 o ) = 0.342 3.378 3.352 4 Sin(20 o ) = 0.342 3.054 3.352 5 Sin(20 o ) = 0.342 4.176 3.352 6 Sin(17 o ) = 0.292 2.752 2.867 7 Sin(17 o ) = 0.292 2.801 2.867 8 Sin(17 o ) = 0.292 2.897 2.867 9 Sin(17 o ) = 0.292 2.812 2.867 10 Sin(17 o ) = 0.292 2.949 2.867 11 Sin(14 o ) = 0.242 2.151 2.371 12 Sin(14 o ) = 0.242 2.154 2.371 13 Sin(14 o ) = 0.242 2.2 2.371 14 Sin(14 o ) = 0.242 2.152 2.371 15 Sin(14 o ) = 0.242 2.079 2.371 16 Sin(10 o ) = 0.174 1.347 1.708 17 Sin(10 o ) = 0.174 1.21 1.708 18 Sin(10 o ) = 0.174 1.488 1.708 19 Sin(10 o ) = 0.174 1.324 1.708 20 Sin(10 o ) = 0.174 1.43 1.708 21 Sin(7 o ) = 0.122 0.7211 1.194 22 Sin(7 o ) = 0.122 0.8978 1.194 23 Sin(7 o ) = 0.122 0.7663 1.194 24 Sin(7 o ) = 0.122 0.7065 1.194 25 Sin(7 o ) = 0.122 0.7946 1.194

Table 1: The data gathered from the experiments and expected results utilizing theoretical and experimental practices. From the information given, we can deduce that the acceleration of the object increases when the angle of incline is increased. As the angle of the incline goes to 90 degrees, the acceleration of the object approaches free fall, or gravity. So far, our lab results are a little shaky due to natural inconsistencies in the lab, but the general line of direction for our lab is approaching the likes of gravity. To further extrapolate the relationship between the angle of incline and acceleration, we will attempt to find the Root Mean Square Error, or RMSE, of our calculations in order to determine how much error we are working with. We will be using our data’s RMSE to find how much our recorded data varies from the actual predetermined values. RMSE: From this data, we were then able to further quantify it by calculating the RMSE . To calculate RMSE we must first identify each of our variables. N = number of data points or experiments. y i = observed value (recorded accelerations) y i fit = the predicted value according to the linear regression equation. (Line of best fit) RMSE = = 0. 046597 RMSE = 0.2159 Overall, to quantify the RMSE, we utilized a line of best fit which ultimately represents the linear regression. Dissecting the RMSE formula, y i is our observed acceleration gathered from the experiment while y i fit is the value of acceleration corresponding to the linear regression, the actual value. To elucidate, we inserted f(sin(20 o )) into our line of best fit f(x) = 12.5x - 0.811, which gave us a value of 3.464m/s 2 . We performed this same calculation for each of the tested angles, taking the differences, squaring them, square rooting and dividing the summation of all the values collectively.

Discussion: 1. A) Systemic mistakes made throughout the experiment could be one reason why our linear fit has an offset such as a y-intercept. For instance, any measurement-related errors could lead to a non-zero y-intercept. As common knowledge, the y-intercept should always equal zero because there is no actual acceleration at an angle of 0 degrees. B) Due to a variety of variables including poor timing, poor judgment, and other pressures, our linear fit may have a different slope than the theoretical relationship. C) There are several factors that could explain why the acceleration we measured may not match the expected value at the assessed point, where a = freefall. The first factor involves external forces that exert a negative influence on the car's motion, such as friction and air resistance. These forces reduce the measured acceleration, causing a disparity between the measured and expected values. The second factor contributing to the unequal accelerations is the actual rolling of the car. If the car's wheels don't roll perfectly due to issues with the axle or wheels, it can also lead to a deviation from the expected acceleration. The third factor affecting the consistency of accelerations is the imperfection in the ramp angle. Because this experiment was conducted by humans, human errors will inevitably occur. In this particular case, we had to estimate the ramp angle by visually assessing it with an angle finder, which might not have been precisely as we anticipated. 2. There are plenty of potential experiments that would incorporate our modern technology to find gravity. However, we came up with a simple, and effective experiment that would be potentially accurate in finding the acceleration of gravity at a free fall, would be to just simply evaluate the acceleration of an object during a free fall. The object would begin at rest, then would have a little trap door that would cause the object to fall onto the ground. The higher the object, the better as there would be a larger time frame to see how the velocity changes. Preferably, getting the object to fall for a second or two would be ideal. We could set up motion lasers from above to track the change in velocity, and two lasers, one to the side and the start and one at the bottom to record the time interval of which the object is falling. Using the lasers to track the movement of the object, we would then import the recorded data into Graphical Analysis, where we could get a visual for the graphs of the object's displacement vs. time, velocity vs. time, and even acceleration vs. time, but that’s too easy! If we just had the velocity vs. time graph, we could follow a similar process in which we did in Lab 1, where we collect all the data (y) for each respective time interval (x), put it in a scatter plot, and then find the line of best fit for the data. The slope of the line will, presumably, be the acceleration due to gravity.

Conclusion: In summary, this experiment provides insights into the connection between the incline’s angle and the resulting acceleration. Through conducting 25 trials with varying angles, the collected data unmistakably illustrates that as the angle increases, the acceleration also increases. Conversely, when the angle is reduced, the acceleration decreases. Furthermore, through the results of this experiment, we were able to quantify the RMSE, percent error, and measure free fall. The RMSE value we calculated is 0.2159, indicating that the data results are accurate and exert less variability. The percent error value of 19.2% is evidence that our RMSE value is correct. A low RMSE value will result in a low percent error value. A high RMSE value demonstrates that there is more variation in observed values compared to the line of best fit, indicating that there are more errors. Looking at all of our values, some of our observations were closer to theoretical values and some were further. For example, the slope of the theoretical relationship is 9.8 while our slope was 12.5, which is much more and may be due to external factors during the experiment such as timing, surface, materials, friction, etc. Factors as such can alter our values and cause a bigger or smaller slope since the experiment is not a perfect simulation.