Allan Hancock College Physics 161 Lab

Allan Hancock College Physics 161 Lab

Free-Fall Acceleration

Purpose: To test kinematic theory as it relates to freefall, and to measure the acceleration due to gravity.

Theory: If the only force acting on an object is gravity, then the object is said to be in freefall. Since the force of gravity near the surface of the Earth is constant, the object’s acceleration is also constant. Using the assumption that the object’s acceleration is constant and equal to g, and the definition of acceleration, a = dv/dt, it can be shown with calculus that the speed of the object as a function of time is given by

v(t) = vo – gt

and its position as a function of time is given by

y(t) = yo + vot – ½ gt2

Apparatus: Tape marker operating at 40 Hz, tape, clamps, rods, meter stick, computer with Excel (for data analysis).

Procedure: Place tape in the marker apparatus so that it passes under the carbon paper:

Figure 1: Marker apparatus. Note the position of the paper tape.

Fix a mass to the end sufficient to overcome the drag of friction and air resistance on the tape (up to 500 g). Also, detach your strip from the spool and take precautions to minimize the effects of drag and friction. Set the apparatus at 40 Hz and release the tape.

Figure 2: Markings on paper tape. Note that all measurements are taken from the first data point.

The tape marker marks the position of the tape (and also the falling mass) every 1/40th of a second. Fix the tape securely to a horizontal surface and identify the position marks. Choose the first clear mark and label it 0, and each successive mark 1, 2, 3, etc. These data points correspond to the tapes position at t = 0s, t = 1/40s, t= 2/40s, etc.

Data: Using Excel, make a table of the data:

Figure 3: Data table made using Microsoft Excel.

Recall that v = Dy/Dt and a = Dv/Dt, so on your table v1 = (y2 – yo) / (t2 – to) and a1 = (v3 – v1) / (t3 – t1).

Instructions for entering data using Excel

1) Record your points in col. A.

2) Click on the cell in col. B that corresponds to the 1st entry in Col. A (point = 0). Type “=Ai/40” and hit enter. Here, “i” refers to the row number of the data point in col. A.

3) Highlight all of col. B down to the last entry in col. A. Include the 1st cell containing to. Pull down the edit menu, go to fill---down. Col. B should now contain all your time data.

4) Enter your position data in col. C.

5) To record your velocities, click on the cell in col. D corresponding to you 2nd data point (y1). Type “= (Cj-Ci)/(Bj-Bi)”, where j refers to the row number below the cell you are in, and i refers to the row number above the cell you are in, i.e. j = 3 and i = 1.

6) Repeat step 3 for col. D, but stop one cell above your last data point.

7) To record your instantaneous accelerations, click on the cell in col. E that corresponds to your 3rd data point (y2), and now type “= (Dj-Di)/(Bj-Bi)”, where j refers to the row number below the cell you are in, and i refers to the row number above the cell you are in, i.e. j = 4 and i = 2.

8) Repeat step 3 for col. E, but stop 2 cells above your last data point.

9) Sum col. E to get you average acceleration and compare it to the accepted value for g, g = 981 cm/s2.

Note: you can convert your data into meters after you have entered your position data in centimeters by clicking on the 1st cell in the next column and typing “=Bi/100”, where i is the 1st row in col. B. Then repeat step 3, and now use this data column to calculate the velocity.

Analysis: You can analyze the data by graphing y vs. t and v vs. t. The position equation,

y = yo + vot – ½ gt2,

is a quadratic equation of the form,

y = c + bx + ½ ax2,

where x is equivalent to t, and a is the acceleration. A graph of the y vs. t should produce half of a parabola and a regression analysis will produce the coefficient a.

Likewise the velocity equation,

v = vo – gt,

is a linear equation of the form,

y = b + mx,

where v(t) is the independent variable y, vo is the y intercept b, x is again time, and the slope of the line m is the acceleration. Here a graph of v vs. t will produce a line, the slope of which is the acceleration.

Graphing instructions

Before printing anything, read all the instructions!

1) Highlight the 2 columns you want to graph; there are various methods of doing this, so if you are not comfortable with the computer, get some help.

2) From the toolbar, choose chart, and then select scatter plot from the chart window. Go to next.

3) You now will see a graph in the chart window. Check that the time data is on the horizontal axis. There are tabs at the top of the window, which represent various options for formatting the chart, check all of them! Select the gridlines tab, and deselect major gridlines to remove the gridlines from the graph. Go to next.

4) You now have the option to have your graph on a separate sheet or embedded on the same page as your data. When you are satisfied with the graph, click on finish.

5) Click on the gray background of the graph. From the color palette window that pops up, choose the white square from the area portion. This removes the gray background from the graph. Do this step before you print!

6) From the chart menu select add trendline. For the position graph choose polynomial fit, and for the velocity graph choose linear fit. Before you finish, go to the options. Select display equation to show the equation on the graph. This is your result!

7) Be sure that the names of your lab group are on all the documents before you print them.

Results: If the tape and mass were in freefall, the coefficients m and a from your graphs and the average acceleration should equal 9.81 m/s2. Do they? What is the discrepancy between the theoretical acceleration 9.81 m/s2 and your experimental acceleration? Why do you think there is a discrepancy? What theoretical assumptions were made that were not met by the experiment?