Incline Plane Lab PHYS 253 Kevin Hamilton July 13, 2021 1 Theory Figure 1 We know from our experiences in reality that when we roll a ball down a hill, the ball will roll down the hill and continuously accelerate, or speed up. if unhindered. Now, we will be able to actually make measurements of the acceleration. First, however. we will need to nd the angle at which the inclined plane sits with respect to the horizontal surface. We can do this by measuring the height from the table at two different points along the plane, h] and hg. These two measurements will be separated by some distance. which we call E. If we look at Figure 1. we see two smaller triangles formed within the larger triangle, both of which have the same angle, (9, between the inclined plane and the surface. If we remember some basic trigonometry, we can nd that angle based off of our measurements. OPP 17.2 h] 7 1 ADJ t ( ) Note that we are measuring the angle within the small triangle. but that angle is the same that is labeled to the right side of Figure 1. Lastly, in order to get the actual angle, we take the inverse tangent of both sides. h- n' 9 = tan_1 (gill) (2) This equation gives us a measurement for the angle between the inclined plane and the table, We can then use this measurement to relate our acceleration that we measure to the value of g, the acceleration due to gravity. If we look at Figure 2 on the next page. we recognize that as physicists, we can dene our coordinate systems in a way that is convenient for us to describe the motion of the ball, and so the zedirection lies parallel to the inclined plane. we can then consider our basic kinematic equations for position and velocity. tan(9) : 1103) =1)\" + at (3) 1 . 31H) : $0 + not + Ear; (4) Figure 2: Experimental Setup. The box on the left is the motion sensor. We are also able to define our coordinates such that the x-direction is parallel to the incline. Since we can define our coordinates in a way that helps our measurements, we define the initial position of the ball at x = 0. Then, when we start our ball from rest, the initial velocity vo is also zero. This means that Equation 3 becomes v(t) = at (5) We then consider how gravity is pulling straight down, at an angle relative to the plane. The x-component of this gravity is going to be the cause of the acceleration along the plane, and we define it in our coordinate system as a = g sin(0) (6) where g is value of the acceleration due to gravity, 9.81m/s and 0 is the angle between that we measured between the table and the plane. By solving for g, then we can put that expression into Equation 3 to find the velocity in terms of g. v(t) = g sin(0)t (7) When we make measurements of position and velocity, as we will be during the experiment, then we can plot the data, and we should see a linear function between velocity and time, similar to the basic linear equation y = mx +b, where m is the slope of the line. Comparing y = mx + b, we see that t is our independent variable, and so the slope is related to g by m m = g sin(0) - 9 = sip sin (0) (8) We cant then compare our experimental value of g to the accepted value, 9.81 m/s by using the definition of percent error: Error = [Accepted - Measured x 100% (9) Accepted Experimental Procedure 1. Adjust the height of the track so that the top of the track approximately 20 cm above the surface of the table. Take two height measurements at two different points of the track and record the heights. Be sure to record the distance between the two measurements. These will be your values for l, h1, and h2. 2. Place the cart at the top of the track as close to the motion detector as possible, holding it steady. In the LoggerPro program, press the collect button to begin measuring position and velocity as a function of time. Once the motion sensor begins the clicking sounds, let the cart roll down the incline, making sure to stop it before the cart rolls off the end of the track. 3. In the center of the data, you should see a linear portion on the velocity versus time graph. On that graph, highlight that linear portion. Once highlighted, copy the data into a Colab notebook. As always, one person can do this and share data with the rest of the group, though each person should do their own analysis(i. r 1). Repeat the measurements in Step 2 two more times. moving the data into a notebook . Adust the angle between the track and the table by either raising or lowering the tall end of the track. Record the new heights. hl and h; as well as the distance between those measurements. Repeat Step 2 for this new angle three times. while making sure to record the relevant data in Colah (and share data and the notebook as necessary with your colleagues. again, as long as it is only the data and not the analysis) Adjust angle between the track and the table one more time. Redo Steps 1: 2, and 3 for this nal angle. Once complete. you should have 9 sets of data with columns for time, position= and velocity. Data Analysis . You should have three separate measurements for In. fig, and 1 recorded: one for each angle. Use Equation 2 to calculate each of the three angles that you tested. . For each of the sets of data. generate in Colab a scatter plot for position versus time and a scatter plot for velocity versus time, . On each of the velocity graphs. add a trendline for the data. Print out the values for your trendline t and the R2. and include them in your report. The equation that best ts data is how we nd our experimental result for 9. while the R2 value tells us how well the trendline ts the data. Using your results from Step 1 in the Analysis and the slopes of the data you have tted. nd your 9 experimental values for 9. Use Equation 9 to nd the percent error for each of your trials. While writing your lab report. make sure that you address how your experimental values of 9 compare to the accepted value. Are the values close or are they not? Why are the values that you recorded so close or so far away from 9? What factors could we consider within the experimental setup that would adjust your values of g to make them more accurate? These are the things you should be thinking about and addressing while writing the report. particularly the Discussion