Free Fall- Acceleration due to gravity

Free Fall: Acceleration Due to Gravity Introduction A freely falling body – dropped or thrown straight upward or downward – experiences uniformly accelerated motion in a straight line. The acceleration is due to the force of gravity. The purpose of this lab is to study the motion of a falling body and to determine g , the value of the acceleration due to the force of gravity. You will be graphing your data. Be sure to follow the instructions for graphing found in the beginning of your lab manual Theory The one dimensional motion of a freely falling body is given by the kinematic equations, v = v 0 + gt [1] and y = v 0 t + gt 2 [2] where we have taken the y axis as downward, and assumed that the starting position is y = 0. 𝒗 𝟎 is the velocity at time t = 0 and g is the (constant) acceleration due to gravity. For this Laboratory, we will also use the following theorem: For constant acceleration, the instantaneous velocity at the mid-point in time of a time interval is equal to the average velocity over the whole time interval. Average velocity is defined by v avg = y = y 2 − y 1 [3]  t t 2 − t 1 So, from our theorem, the instantaneous velocity at (for example) t = 5/60 th s is given by the the average velocity over the time interval from 4/60 th s to 6/60 th s. This allows us to determine instantaneous velocity from measurements of time and position. Apparatus y 1 t 1 t n y n y = 0 Spark-tape y brass weight spark tape Spark apparatus The spots on the Spark -tape, are a measure of the position of the falling weight at the time of each spark

Procedure Tape a 200 gram weight to the end of your spark tape. Make sure there is enough “loose” tape to pull through the apparatus as the weight falls to the floor. Turn on the sparker and release the weight. Be careful not to impart any upward or downward motion to the weight. The spark-tape gives a recording of the position of the falling body as a function of time. The apparatus sparks 60 times per second, therefore the time between spots on the tape is 1/60 th of a second. Note: handle your tape carefully, as it is easy to make extraneous marks that might be confused with the spark-marks. To help isolate the spark marks from other extraneous marks on the tape, lightly circle the trail of spark-marks in pencil. (The marks in this trail should be increasingly farther apart as you go.) Beginning at the large smudgy spot, measure the position of each successive spark. Record the position and times for your sparks in a table of the form shown here. 13/60 (.2166) 39.5 284.85 14/60 (.2333) 44.3 301.21 15/60 (.25) 49.44 315.15 16/60 (.2666) 54.70 329.10 17/60 (.2833) 60.30 358.48 18/60 (.3) 66.20 212.84 t(s) y(cm) V(cm/s) 0 0 0 1/60 (.0167) 1 42.42 2/60 (.0333) 2.4 90.91 3/60 (.05) 4 106.06 4/60 (.0667) 5.9 120.12 5/60 (0.0833) 8 135.13 6/60(.1) 10.4 152.69 7/60 (.1167) 13.1 171.17 8/60 (.1333) 16.1 295.76 9/60 (.15) 22.85 316.81 10/60 (.1667) 26.65 232.73 11/60 (.1833) 30.60 250 12/60 (.2) 34.9 269.70 t(s) y(cm) V(cm/s) Y 0 0 0 0 1/60 (.0167) 1 42.42 .84 2/60 (.0333) 2.4 90.91 1.94 3/60 (.05) 4 106.06 6.49 4/60 (.0667) 5.9 120.12 10.12 5/60 (0.0833) 8 135.13 14.55 6/60(.1) 10.4 152.69 20.02 7/60 (.1167) 13.1 171.17 26.45

Note that 𝑣 is not defined for the top-most and bottom-most rows of the table. When you have completed your table, plot (on graph paper) 𝑣 as a function of t . Draw a best-fit straight line for this data and calculate the slope. This is your experimental value for the acceleration due to gravity, g . Finding the slope: 96.1-81.1/2-1= 15/1=15, the slope for this graph is 15, which means g=15 On a new graph plot as dots with a circle around them your measured (experimental) values of y as a function of t . What is the functional form for this data? eg. straight line, hyperbola, etc Is this what you expected? The functional from of this data is a curved line, that keeps increasing, the quicker the metal falls the higher the gravity it is. Now use your experimental value of g to construct a new (third) table. The first column of your new table should contain the times from your experimental data. The second column should contain positions calculated using equation 1 . [Think about how the data for this experiment was obtained. What should be the value of 𝒗 𝟎 in this calculation?]

Plot the data points from this table on the first graph, using little x marks for your points. How well does this curve match the original data? Conclusion : Your conclusion should list the theoretical and experimental values determined in this lab. It should also give the % error for the experiment. What are the significant sources of error in this lab? Does the calculated position vs time plot reasonably agree with the observed data? FZ