Experiment 2 - The Falling Body

The Falling Body  Purpose/Goal: The purpose of this experiment is to find and calculate the acceleration of the falling bob by using a least squares regression line. The measurements of interest include the distance travelled by the bob in a certain time interval and the additional calculations that will help calculate the acceleration include the calculation of the velocity, and then graphing velocity vs. time on the graph to find the line of best fit. The pertinent results will be the regression line that we can use to calculate the acceleration.  List of Materials: Materials include falling bob apparatus, falling bob, spark tape, electromagnet, and meter stick.  Diagram of Apparatus:  Procedure: First, take the end of the spark tape located at the bottom of the falling bob apparatus and pull it to the top of the apparatus and secure it with the clip. Second, turn the power supply for the apparatus on and attach the falling bob to the top of the apparatus. Note that the bob will only stay attached if the power supply to the magnet is on. Third, when you are getting ready to release the bob turn on the spark generator. A spark occurs every 1/60 th of a second. Let it spark for a few seconds so that the sparks can calibrate to be more accurate to sparking every 1/60 th of a second. Fourth, while holding down on the spark generator, turn the power supply of the magnet off until the falling bob reaches the bottom (to the sandbox). Now, remove the section of the spark tape that has all your data point markings/dots and measure the distances between the n = 0 dot (first dot) and the nth (2, 3, 4, 5, …) dot using the meter stick. Skip the n = 1 dot to increase precision and accuracy. Calculate the time between each dot (n th dot * 1/60 sec), again excluding the n = 1 dot. Graph a position vs time graphs and velocity vs time graphs. Find the line of best fit for the velocity vs time graph and identify the slope of the graph as the acceleration/earth’s gravity.  Data: Dot Number (n) Time (t(s)) Meter Stick Reading (cm) Fall Distance (y(cm)) Velocity (v(cm/s)) 0 0 0 0 0 3 4 5 2 0.033 0.05 0.067 0.083 1.8 3 4.5 6.3 1.8 3 4.5 6.3 54.545 70.588 88.235 112.500 6 7 0.1 0.1167 8.4 10.6 8.4 10.6 123.529 131.737 8 9 0.133 0.15 13.1 16 13.1 16 153.374 170.588 10 11 0.167 0.18 19.1 22.5 19.1 22.5 182.353 261.538 12 0.2 26.2 26.2 185.000 13 14 15 0.2167 0.233 0.25 30.1 34.3 38.8 30.1 34.3 38.8 233.533 257.669 264.706 16 17 0.267 0.283 43.5 48.4 43.5 48.4 276.471 306.250 18 19 20 21 0.3 0.316 0.333 0.35 53.7 59.2 65.1 71.1 53.7 59.2 65.1 71.1 311.765 343.750 347.059 352.941 22 0.367 77.6 77.6 382.353 23 24 25 26 27 0.383 0.4 0.416 0.43 0.45 84.2 91.1 98.4 105.6 113.4 84.2 91.1 98.4 105.6 113.4 412.500 405.882 456.250 514.286 390.000 28 29 30 0.467 0.483 0.5 121.5 129.7 138.2 121.5 129.7 138.2 476.471 512.500 500.000 LINEST function of velocity vs. time gave us: 986.1587 31.77874 - 22.02963 9.392481 Slope: 986.2 cm/s^2; Slope Error: 31.78 cm/s^2 o Convert to m/s^2: 986.2cm/s^2 * 1m/100cm = 9.862m/s^2 31.78cm/s^2 * 1m/100cm = 0.3178m/s^2 So, the slope of the velocity vs. time is 9.862 m/s^2 - Y-intercept: 22.03 cm/s; Y-int error: 9.392 cm/s o Convert to m/s: 22.03cm/s * 1m/100cm = 0.2203m/s 9.392cm/s * 1m/100cm = 0.09392m/s So, the y-intercept of the velocity vs. time is 0.2203 m/s  Graphs:  Questions: Describe the motion of the bob as it is falling. When the bob is released, it goes straight down to the bottom of the floor. How well does the best fit line match your data points? It matches the data points very closely. There are not many data points that skew the line of best fit. As the graph shows, our R^2 values (correlation coefficient) is very close to 1 for both graphs (0.951 for position vs. time graph & 0.972 for the velocity vs. time graph). The line of best fit very closely models the data points. From the v vs. t graph, determine the acceleration of the falling bob? We have done this calculation in the data section of the lab. Slope: 986.2 cm/s^2. Convert to m/s^2: 986.2cm/s^2 * 1m/100cm = 9.862m/s^2. So, the acceleration is 9.862 m/s^2. Using the values obtained with the LINEST function, determine the acceleration with the most probable error of the falling bob. We have done this calculation in the data section of the lab. Slope: 986.2 cm/s^2; Slope Error: 31.78 cm/s^2. Convert to m/s^2: 986.2cm/s^2 * 1m/100cm = 9.862m/s^2 31.78cm/s^2 * 1m/100cm = 0.3178m/s^2 So, the acceleration is 9.862 m/s^2 with a most probable error 0.3178m/s^2. (9.862 m/s^2) +/- (0.3178 m/s^2) = (9.5442 m/s^2, 10.1798 m/s^2) Does your acceleration with probable error include the accepted value of 9.81m/s2? If not perform a percent error calculation of the two values. Yes, it does. 9.81m/s^2 is included in the interval from 9.5442 m/s^2 to 10.1798 m/s^2.  Results: We obtained a slope of 9.862 m/s^2 (calculation: 986.2cm/s^2 * 1m/100cm = 9.862m/s^2) and a slope error of 0.3178m/s^2 (calculation: 31.78cm/s^2 * 1m/100cm = 0.3178m/s^2), which gave us an interval between (9.862 m/s^2) +/- (0.3178 m/s^2) = (9.5442 m/s^2, 10.1798 m/s^2). We also obtained a Y-intercept of 0.2203 m/s (calculation: 22.03cm/s * 1m/100cm = 0.2203m/s) and a Y-intercept error of 0.09392 m/s (calculation: 9.392cm/s * 1m/100cm = 0.09392m/s), which gave us an interval between (0.2203 m/s) +/- (0.09392 m/s) = (0.1264 m/s, 0.3142 m/s).  Conclusion: The experimental value of g we obtained by conducting this experiment is 9.862 m/s^2, which is very close to the accepted value of gravity (known as 9.81 m/s^2). We didn’t have to calculate a percent error value since the accepted value falls in the interval that we obtained from the graphs [(9.862 m/s^2) +/- (0.3178 m/s^2) = (9.5442 m/s^2, 10.1798 m/s^2)]. We can conclude that this experiment was successful in calculating an accurate value of g by calculating the distance (cm) and time (sec) of small segments. Our R^2 values (correlation coefficient) is very close to 1 for both graphs (0.951 for position vs. time graph & 0.972 for the velocity vs. time graph). The line of best fit very closely models the data points, and the lab was successful in attaining the originally states purpose/goal due to these reasons.  Error Analysis: There are possible sources of error in this lab that we could have made that made the results les accurate and precise than they could have been. Systematic error could have occurred in measuring and recording the distance of the dot markings on the spark tape. We could have reduced experimental error by repeating this experiment multiple times to create more trials and a more accurate result. Those are examples of some of the errors that could have occurred.