Galileo and the Inclined Plane
Introduction
In the early seventeenth century, Galileo makes a claim (or
hypothesis) concerning the natural motion (“natural motion” is
Aristotelian language) of a freely falling body. Galileo claimed that a
naturally falling object will gain equal amounts of velocity in equal
amounts of time. If correct, this means that i) the object’s speed
increases as it falls and ii) the rate at which it picks up speed does
not change during the fall. The cartoon at the right illustrates the idea.
However, practical problems make it difficult for Galileo to
put such a claim to any experimental test. Freely falling objects
move too quickly to study or record the motion directly. Because
seventeenth century clocks could not record the short times
involved, Galileo tried to slow down the motion by replacing the
falling object with a ball rolling down a gently inclined plane.
"In order to make use of motions as slow as possible ... I also
thought of making moveables descend along an inclined plane
not much raised above the horizontal" (Galileo, Two New
Sciences, p. 87).
Galileo’s hypothesis: A falling object picks up the
same amount of speed in the second (and third and
fourth) second of travel as it did during the first second
until it hits the ground. Notice that the object travels
further in the 2nd second of motion than during the first.
Picture from Hewitt et al (1994), Conceptual Physical
Science (New York: HarperCollins).
Notice that Galileo guessed (or assumed) that objects descending an incline speed up in exactly the same way falling
objects do! Essentially, he reasoned as follows. A ball rolling down a steep incline will pick up speed faster than a ball
rolling down a gentle incline, but the way in which its speed increases will be the same. Freefall, he reasoned, is
simply equivalent to a vertical ramp. The character of the ball’s motion in freefall should be the same as the character
of the motion of a ball “falling” down the inclined plane.
Another technological problem arose in measuring the velocity. If Galileo’s hypothesis is right, the velocity of
the freely falling object changes continuously. Galileo cannot measure changing velocities directly, but he can
measure distances and times. Galileo, therefore, uses math to transform his claim about times and velocities into a
claim about times and distances. His mathematical argument is summarized below:
If an object gains speed at a steady rate and
if the object is released from rest,
then the total distance traveled by the object will be proportional to the square of the time needed for that travel.
For example, if an object released from rest travels, say for three seconds, it will travel 3*3 or 9 times as far as it
would if it traveled for only one second after being released from rest. (Can you figure out the logic behind Galileo’s
mathematical argument? How can he derive the final statement from the first two?)
In this lab, you will put yourself in Galileo’s shoes. Your job is to collect and present convincing evidence in
support of Galileo’s idea about the motion of falling bodies. You will face some of the experimental design questions
and try to replicate the results of his famous experiment using an experimental setup similar to the one Galileo
describes in his Two New Sciences of 1638.
You will also use a modern device to record the travel of a freely falling object. This will enable you to test
whether the character of freefall motion is the same as that for the ball rolling down the ramp.
How do you measure time with 17th century clocks?
Stopwatches as we know them did not exist in Galileo’s time. It was not until the
1700s that craftsmen began to make small reliable mechanical clocks. In his book Two New
Sciences, Galileo describes the clock he uses:
For the measurement of time, we employed a large vessel of water placed in an
elevated position; to the bottom of this vessel was soldered a pipe of small diameter
giving a thin jet of water, which we collected in a small glass during the time of each
descent... the water thus collected was weighed, after each descent, on a very
accurate balance; the difference and ratios of these weights gave us the differences
and ratios of the times...
The basic principle behind this clock seems sound, but what guarantee is there that this “clock” doesn’t gain or lose
time? Since you want to make a specific mathematical claim involving time measurements, it is critical that your time
measurements be accurate.
Prelab Questions:
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In this passage, Galileo seems to be assuming that there is a specific mathematical relationship between
the amount of water collected and the amount of elapsed time. What assumption does Galileo seem to be
making about the mathematical relationship between the amount of water and the amount of time?
Would you expect this clock to run steadily? Why or why not? (Hint: Suppose the tank were nearly full and
you let it run for 1 sec and collected the water; then you did the same thing, this time starting with the water
level much lower than before. Would you collect the same amount of water in both cases? What might you
expect?)
Optional Challenge: Explain a procedure that you can perform to test whether this clock keeps good time.
Remember that you have no 20th century clocks to compare against!
Lab Instructions
1. Plan for data taking. Getting good data for any experiment requires planning. Put yourself in Galileo’s
shoes. You know good time measurements are essential, but you anticipate two practical difficulties.
Difficulty 1: You suspect the rate at which the water clock runs might depend on the water level in the tank.
What precaution(s) should you take to minimize the impact of this problem on your data? Explain.
Optional challenge: How could you assess the impact of this problem on your data? Explain.
Difficulty 2: Time measurements made by humans are notoriously inconsistent.
What precaution(s) should you take to minimize the impact of this problem on your data? Explain.
How will you assess the impact of this problem on your data? Explain.
2. Practice with the water clock. Do a few practice runs. Once you have a procedure that gives reasonably
consistent times for a given distance, time how long it takes the ball to go a given distance several times
and record your data below.
Remember to take all the precautions you identified above!
Distance: _____________ (Don’t forget units)
“Times”: ________________________________________________________ (Don’t forget units)
In Two New Sciences, Galileo boasts about the quality of his time measurements:
… we noted … the time that it consumed in running all the way, repeating the same process many
times, in order to be quite sure as to the amount of time, in which we never found a difference of
even the tenth part of a pulse-beat.
How consistent are your measurements? How does the repeatability of your measurements compare
with the level of consistency Galileo claimed?
Check your results with an instructor before continuing.
3. Plan your experiment. Galileo predicted a specific relationship between the time it took something to travel
down an incline starting from rest and the distance it traveled. Decide how you will do an experiment to
answer the following two questions:
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Does the motion along an inclined plane follow the pattern Galileo predicted?
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Do motions along inclined planes with different angles follow the same pattern?
Some things to consider as you design your experiment:
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You will keep the angle of the ramp the same throughout the experiment. A good choice of
angle will give you better data. Should you choose a really small angle or a really big one or
somewhere in between? How can you decide?
How can you collaborate with other lab groups to answer collect data to answer the second
question? (It’s OK to share data! Simply acknowledge the group that took the data when you
write up the experiment). You may need to be a little careful when you make the comparison,
however, because their procedure may not match yours.
You will be measuring distances and times, but which ones? Should you use short distances,
long distances or a combination?
How much data will you take? How will you decide when you have enough data?
How will you analyze your data to test Galileo’s idea?
Record your procedure here. Your description should be detailed enough that another P1 lab group could
follow it.
Check your procedure with an instructor before continuing.
4. Collect data. Take data to figure out the relationship between distance and time. Record the data in the
space below. Some tips:
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Graph the data as you take it. This will help you spot patterns in the data, detect potential
mistakes and give you some guidance about which data to take. (If you know how to use
Microsoft Excel or a similar spreadsheet program, this is very easy).
Leave the equipment set up after you are finished, in case you have to go back to check your
data or add to your data.
Record your data in some organized way (such as a table). Write your data on paper even if
you are using a computer to analyze the results (in case the computer crashes).
Check your results with an instructor before continuing.
5. Analyze the data. What is the mathematical relationship between distance and time according to your
data? Is it what you expect? Once you have enough data to figure out the pattern, do a few more trials to
develop a convincing case that this really is the pattern. Some ideas for analyzing the data.
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What mathematical relationship did Galileo expect to find?
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Suppose Galileo’s hypothesis is correct. Would you expect a graph of distance versus time to be a
straight line? what about a graph of distance vs. time squared? distance squared versus time?
Explain. (If you’re not sure of the answer, make some fictitious data that fits Galileo’s pattern and
try the various graphs).
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Make the appropriate graph(s) of your data to test Galileo’s hypothesis. (If you know how to use a
spreadsheet program like Microsoft Excel, you will likely want to use it to create the graph).
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Make appropriate graph(s) of another group’s data. Compare their results to yours. Does changing
the angle change the nature of the motion?
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Optional challenge: Try to figure out the rate at which the object speeds up from your graph.
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Optional challenge #2: Use the result of the previous challenge to estimate the rate at which
freely falling objects speed up.
Check your analysis with an instructor. Be prepared to discuss the reasoning behind your
analysis and whether your analysis supports (or doesn’t support) Galileo’s ideas.
Measuring freefall motion using 20th century technology
You will be using a spark timer and long strips of spark paper (also called spark tape) to study the motion of a falling
body. The paper gets pulled past the two metal tips in the spark timer. When the spark timer is on, two spark tips zap
the paper every 1/10 of a second (or every 1/60 of a second, depending on the setting you choose), leaving, a pair of
black dots on the paper tape. The spark timer allows you to record fast motions (such as free fall) because the time
between sparks is so short.
AB
xC
C
D
E
F
Spark
Timer
The diagram above shows the spark timer and tape in action. When point A on the tape was underneath the spark
tips, the timer was turned on and the person began pulling the tape to the left. As the hand moves to the left, so does
the tape. One spark interval later (either 1/10 of a second or 1/60 of a second), point B on the tape was under the
spark tips. Another spark interval later, point C was under the tips and so on. In this way, the spark tape makes a
record of the motion of whatever the tape is attached to. Galileo marked regular distance intervals with a ruler and
measured time with the water clock. The spark timer instead marks regular time intervals and you will measure the
distances with a ruler. You will study free fall motion by attaching a weight to the tape and orienting tape and timer
vertically.
Lab Instructions (spark timer)
1. Ask your instructor to demonstrate the operation of the spark timer.
2. Plan your data taking for the spark timer. Are there reasons to suspect that the data this apparatus will
not be those for the motion of a freely falling object? Explain. What precautions can you take to make sure
that this apparatus delivers a record of the motion that most closely resembles that of a freely falling object?
(Hint: Should you use a light weight or a heavy one?)
3. Collect data. Run the experiment. You may need to run the experiment more than once to get a tape with
good data. Once you have a good tape, decide which distances you will measure (and what times are
associated with each distance). Record your plan here.
Check with an instructor before continuing. The instructor will inspect your tape to make sure
the spark timer worked properly. The instructor will also ask you about how your plan for getting
distance and time information from the tape.
4. Record the distance and time data from the tape in the space below.
5. Analyze the data. (Hint: Graph it first). You may find it helpful to use your analysis for the water clock data
as a guide. Some questions to answer with your analysis:
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What is the mathematical relationship between distance and time according to your data? Is it what you
expect?
Compare the spark timer results to the water clock results. How are they similar? How are they
different? (You may notice some differences! You might speculate about the reasons for the
differences. You may find it helpful to identify some of the assumptions that underlie your analysis to
explain some of the observed differences).
Is the nature of freefall motion the same as motion of the ball rolling down the plane?
Post-lab Assignment
Write an abstract (200 to 400 words) that synopsizes the experiment: what was done, how the data were collected
and analyzed and whether your results support or refute Galileo’s claim about falling objects. You may include
multiple graphs or other diagrams. (A picture may be worth a thousand words, but it does not affect the word count).
Appendix: Galileo’s “Lab Report”
Galileo reports the inclined plane experiment in his book “Two New Sciences.” The book is written as a series of
intellectual conversations among three characters (Simplicio, Sagredo and Salviati) concerning the work of “the
author” (Galileo). In this excerpt, Simplicio, Sagredo and Salviati discuss the inclined plane experiment. (Note: One
bracchio is about 0.6 meters. Bracchio is the Italian word for arm.)
“Simplicio: Really I have taken more pleasure from this simple and clear reasoning of Sagredo’s than from the (for
me) more obscure demonstration of the Author, so that I am better able to see why the matter must proceed in this
way, once the definition of uniformly accelerated motion has been postulated and accepted. But I am still doubtful
whether this is the acceleration employed by nature in the motion of her falling heavy bodies. Hence, for my
understanding and for that of other people like me, I think that it would be suitable at this place [for you] to adduce
some experiment from those (of which you have said that there are many) that agree in various cases with the
demonstrated conclusions.
Salviati: Like a true scientist, you make a very reasonable demand, for this is usual and necessary in those sciences
which apply mathematical demonstrations to physical conclusions, as may be seen among writers on optics,
astronomers, mechanics, musicians, and others who confirm their principles with sensory experience that are the
foundations of all the resulting structure. I do not want to have it appear a waste of time on our part, [as] if we had
reasoned at excessive length about this first and chief foundation upon which rests an immense framework of infinitely
many conclusions--of which we have only a tiny part put down in this book by the Author, who will have gone far to
open the entrance and portal that has until now been closed to speculative minds. Therefore as to the experiments:
the Author has not failed to make them, and in order to be assured that the acceleration of heavy bodies falling
naturally does follow the ratio expounded above, I have often made the test in the following manner, and in his
company.
In a wooden beam or rafter about twelve braccia long, half a braccio wide, and three inches thick, a channel was
rabbeted in along the narrowest dimension, a little over an inch wide and made very straight; so that this would be
clean and smooth, there was glued within it a piece of vellum, as much smoothed and cleaned as possible. In this
there was made to descend a very hard bronze ball, well rounded and polished, the beam having been tilted by
elevating one end of it above the horizontal plane from one to two braccia, at will. As I said, the ball was allowed to
descend along the said groove, and we noted (in the manner I shall presently tell you) the time that it consumed in
running all the way, repeating the same process many times, in order to be quite sure as to the amount of time, in
which we never found a difference of even the tenth part of a pulse-beat.
This operation being precisely established, we made the same ball descend only one-quarter the length of this
channel, and the time of its descent being measured, this was found to be precisely one-half the other. Next making
the experiment for other lengths, examining now the time for the whole length [in comparison] with the time of one-half,
or with that of two-thirds, or of three-quarters, and finally with any other division, by experiments repeated a full
hundred times, the spaces were always found to be to one another as the squares of the times. And this [held]
for all inclinations of the plane: that is, of the channel in which the ball was made to descend, where we observed also
that the times of descent for diverse inclinations maintained among themselves accurately that ratio that we shall find
later assigned and demonstrated by our Author.
As to the measure of time, we had a large pail filled with water and fastened from above, which had a slender tube
affixed to its bottom, through which a narrow thread of water ran; this was received in a little beaker during the entire
time that the ball descended along the channel or parts of it. The little amounts of water collected in this way were
weighed from time to time on a delicate balance, the differences and ratios of the weights giving us the differences and
ratios of the times, and with such precision that, as I have said, these operations repeated time and time again never
differed by any notable amount.
Simplicio: It would have given me great satisfaction to have been present at these experiments. But being certain of
your diligence in making them and your fidelity in relating them, I am content to assume them as most certain and
true.”
The excerpt is taken from Galileo, Two new sciences, tranl. Stillman Drake (Madison: University of Wisconsin Press,
1974), pp. 169-70.