Measurement of the acceleration due to gravity.
Physics 216

In this lab we will experimentally determine the acceleration of gravity g near the surface of the Earth, by
measuring the time of flight for balls dropped from a known height. We will also verify that the acceleration
due to gravity does not depend on the mass of the ball.

Pre-lab quiz:

1. Read these lab instructions carefully.

2. In preparation for your quiz, You should derive a formula for the acceleration, g, of a ball falling from
rest in terms of the time it takes for the balls to fall t, and the height from which it falls, y (ignore air

Introduction: Near the Earth’s surface, the acceleration due to gravity g, is nearly constant. For a ball
starting at rest and undergoing constant acceleration the position as a function of time can be expressed as:

h =


where the initial position (i.e., the position at t = 0) has been set to h and the position where the ball hits
is set to zero.

Figure 1: Apparatus.

Lab equipment: The laboratory equipment consists of two steel balls of different diameters, electromagnet,
Pasco free fall apparatus, and a National Instruments Data Acquisition Device.

Lab instructions:

Measuring Distance: The vertical distance that the ball falls can be adjusted with the perpendicular
clamp. First set one of the balls into the adapter by closing the switch to the power supply thus
turning on the electromagnet and place the ball over the electromagnet and the metal plate. Set the
distance (h) so that the distance from the bottom of the ball to the top of the receptor pad equals is
about 50 cm, then measure this distance as accurately as possible.

Measuring Time: Turn on the Computer and open the Free Fall program. When you are ready to record
the time it takes for the ball to fall to the receptor pad click on the run arrow at the top of the screen.
The light on the screen will turn green and the run arrow will turn black indicating the apparatus is
ready to take measurements. Release the ball by opening the switch. Once the ball is released the
stopwatch is automatically activated and stops when the ball hits the receptor pad. Record this data,
thit, and repeat this procedure 10 times for the first ball.

Second ball Change the height by some amount (a few centimeters) and repeat the measurements above
with the second ball.

Error analysis

Your lab report should include a determination of the final uncertainty in your measured values of g which is
based on the individual uncertainties of the quantities measured using the rules in Errors and the Treatment
of Data.

δthit Use the average time for each ball to determine g (i.e., you should determine two separate values for g,
one for each ball). This formula for average is taken directly from Errors and the Treatment of Data:

t̄hit =



thiti ,

Use the equations for standard deviation and error in the mean to determine the uncertainty in the
time measurements. Note that the general name for the error in the mean is σµ, in our case it is the
uncertainty in the time measurement so we will call it δt̄hit. These formula for standard deviation and
error in the mean are taken directly from Errors and the Treatment of Data:

σ =


(N − 1)


(t̄hit − thiti)

) 1

; σµ = σ/


δh Using a good meter stick, distance measurements normally have an uncertainty of ±0.0005 m (half a
millimeter). Note that this assumes that you make very careful measurements. This is certainly true
for the position of the top of the ball before it drops, but the position of the bottom of the ball when
the timer turns off is less certain. Think carefully about how far the ball actually falls before the timer
stops. Examine the pad carefully to determine how and when it turns the timer off. When you have
determined a reasonable estimate for the uncertainty in this measurement, you must combine the two
numbers using the rule for addition (in this case subtraction) of values. Call the combined error δh.
In equation form,

δh =

δh2top + δh2bot.

Now you must use the rules for propagation of errors to determine the uncertainty in g (i.e., δg) using the
values of δt̄hit and δh above. As a challenge, and to prepare you for future labs, you should make an
effort to determine this equation yourself using the rules laid out in Errors and the Treatment of Data.
You can rely on your lab TAs or your professor for help if needed.

Lab report In the results section of your report, state the results of your experiment in the form g =
x.xx± x.xx m/s2. You should also address the following questions:

1. Do your results for the two balls agree within their uncertainties? Do your results indicate that
g is independent of mass?

2. Do your results match the actual value of g (9.80 m/s2) within the their uncertainties?

3. The uncertainty that you have calculated is only random (or statistical); if your results don’t agree
it may indicate the presence of systematic error(s) in the experiment. If you don’t understand the
difference, then re-read that section in Errors and the Treatment of Data. If your results indicate
that systematic error(s) may be present, try to determine some possible sources of systematic
error in the experiment. For each suggestion, determine if it would affect the result in the same
direction as the discrepancy in your result.

4. When you receive your graded report for this laboratory, you must keep it for a reference of your
results to be used later in the semester.


Laboratory Report Guide
Physics 216-218

Included below are some general guidelines for writing laboratory reports. Some of the sections may not
be applicable to a given lab, but this general format should always be followed. Try to write your report
in such a way that a person familiar with physics but not familiar with this particular experiment would
be able to follow what you did and why you did it.

Title Title of the experiment, date performed, your name, name of all your lab partners.

Abstract This should provide a concise summary of your work.

Introduction State the theoretical basis for the experiment (i.e., the relationship between the measured
quantities and the desired quantity) and why you are investigating this phenomena. Include a
derivation of the formula being used, if applicable.

Data and Analysis Present your data clearly and neatly in columns or tables with clear headings and
units included. Be sure to pay attention to significant figures. It is often appropriate to include
the results of any computations and error analysis in this same table. Also show any formulae you
used for calculations. Be sure to include any equations that you used for error analysis inserting the
appropriate variables for your particular lab and/or briefly explain any statistical methods used for

Figures Graphs and figures should be large enough to allow the viewer to visualize the data easily. As a
general rule, make all graphs fill at least half the page. Be sure to clearly mark the data points, their
errors where appropriate, show any quantitative values that result (i.e. your slope), label your axes
with units and title your graph.

Conclusions State your results (qualitative and quantitative) clearly and concisely. If the result is the
measured value of a physical constant use the standard format for uncertainty. (For example always
say “Our result was g = 9.67 ± .11 m/s2.” Do this even if your result is written elsewhere in the
report.) Your conclusion should address how well your results match the theory presented in your
theory section of the lab. This is the appropriate place to include possible “sources of error”, but
be careful that you can back-up your claims with numerical estimates (example if g is too small and
does not fall within the uncertainty, what measurement needs to be higher to make g larger? Would
making this change be reasonable?) . Stating, for example, that “the equipment was bad” is too
general and not an acceptable answer. Remember, the point of the experiment is to do the best you
can with the equipment available, and to make a reasonable assessment of the uncertainties in your

Summarizing When looking over your lab report for completeness you should be able to answer the
following questions;

1. What was being measured?

(a) The introduction section should present the physical basis and detailed numerical derivation,
if applicable, to answer this question.

2. To what are you comparing your final results?

(a) Does your analysis section address what you are measuring and comparing results with their

3. What are your uncertainties?

(a) Are your numbers in your data section reasonable considering the precision of the equipment
used in the laboratory?

4. Do your final values/results compare within their uncertainties? Consider the range each result
covers if you add and subtract the uncertainty, your goal is to have an overlap of the ranges.

(a) If results do not overlap within uncertainties, begin by looking at the fractional uncertainty
of your measured values. The quantity with the highest fractional uncertainty is a good
place to start tracking down errors by running quantitative scenarios (i.e. change values
within your final equations to see how they affect your final result.) Explain in detail what
you have done to track down your errors in your results section to receive full credit.

Grading Policy: Each lab report will be worth 100 points. Reports will be graded primarily on the
clarity of presentation, and the completeness of the analysis. Although the labs will not be graded on the
actual numerical result as long as the uncertainty can be justified, obvious blunders and/or carelessness
will be penalized. Late lab reports will also be penalized. Pay careful attention to areas which will result
in an automatic deduction of points: (a) Graphs are too small. (b) Results are not restated in the results
section. (c) Uncertainty not calculated where appropriate. and (d) Missing sections.


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