Physics E-1a
Expt 2a: Galileo & Naturally Accelerated Motion
Fall 2006
Introduction
Preparation: Before coming to lab, read this lab handout and the reading in Giancoli
(through Chapter 4, p. 72-97). Answer the bold numbered questions that appear
throughout this lab handout. Seven of these pre-lab answers are to be handed in at the
beginning of your lab — they appear at the end of this document. See the Lab
Companion for further information and hints on preparing for an experiment. Be sure to
bring to lab: writing paper, graph paper, a ruler, a calculator and your copy of the Lab
Companion.
Post-Lab Questions: At the beginning of each lab section, you will be given an
additional handout with a series of questions to be answered and handed in at the end of
the experiment. Try to answer these questions with one or two concise sentences. For
experiment 2a you will also hand in the graph described below under Analysis. To
satisfactorily answer the post lab questions and prepare proper graphs you should pay
special attention to all text that appears in italics in this handout.
Historical Background: For nearly two thousand years until the seventeenth century,
the teachings of Aristotle (4th century BC) were the basis of scientific knowledge.
According to his beliefs, all motion on earth is linear (bodies travel in straight lines)
while all motion is space is curved, heavy bodies fall faster than light bodies, and a
continuous force is necessary for motion to occur. Now, Aristotle based his teachings on
observation. After all, heavy objects DO fall faster than light ones (compare the fall of a
stone with that of a feather). Motion in space IS curved (a baseball, either thrown or hit,
does not travel in a straight line). And cars come to a stop when the ignition is turned off
(but Aristotle didn't know that).
Galileo was as aware of these facts as was Aristotle. However, he also recognized the
importance of friction. By carrying out experiments in which friction was reduced, he
found that, in the limit of zero friction, Aristotle's laws did not hold. Heavy and light
objects fall at the same rate in vacuo. In the absence of gravity and friction, motion in
space is rectilinear. And, in the absence of friction and all other forces, a body moving
with velocity v will continue to move with that same velocity and not come to rest. This
was the first time that idealization was introduced into scientific thinking. In this
experiment we shall replicate an experiment of Galileo in which he "diluted" gravity and
was able to find the relation between distance traveled and elapsed time for a constant
acceleration.
Inventory:
• 1 inclined plane set-up
• 2 lab jacks
• 1 2-meter stick
• 1 ball bearing
• 1 stop watch
• 1 water clock
Experimental Setup: The apparatus shown in the following figure consists of an
inclined plane, a ball, and a clock. You will measure the time required for the ball to roll
several different distances down the inclined plane.
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Objective: You will attempt to verify Galileo’s hypothesis that the distance traveled by
an object undergoing uniform acceleration is proportional to the square of the time it is in
motion. You will also calculate a value for g, the acceleration due to gravity, and
compare it to the accepted value.
Theory
1. → Draw a free-body diagram for the ball rolling down the incline. Clearly label
all the forces. Ignore friction.
2. → Each of the forces in your diagram can be described mathematically. Using these
forces, write out an expression for the acceleration of the ball, a, in terms of the
acceleration of gravity, g, and the angle of the incline, θ. You will need this equation
in class, so write it on this lab sheet as well as on your pre-lab.
3. → Consult your textbook to find a relationship between distance, d, and time for
an object undergoing uniform acceleration. You will be measuring the amount of time
it takes a ball to travel a given distance, so you will need this equation — write it down.
4. → Using the equations in steps 2 and 3, combine and manipulate them to solve for g
in terms of t, d, and θ.
Procedure
1.Set your incline to a shallow angle (<10o ). Adjust the heights and positions of the two
supporting lab jacks until the incline is completely straight. Once you are satisfied with
the angle and straightness of your inclined PVC half, measure its length, L, and the
height, h, of its highest end. Use these measurements and trigonometry to calculate the
angle, θ, of the incline.
2. Galileo didn’t have a stopwatch for measuring the amount of time that it took
things to travel down his incline. Instead, he used a water clock that had a roughly
constant stream of water over short periods of time. Rather than starting and stopping
a stopwatch as a ball started and stopped traveling down the incline, he started and
stopped the water stream coming from the water clock. The amount of water that
came from the water clock was proportional to the length of time measured, with
double the time having double the volume of water. Like Galileo, you will use a
water clock, however, while Galileo was satisfied to find the acceleration of gravity,
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g, in terms of ratios between distances and volumes of water2 , you want to find g in
units of meters per second2 . To calculate the conversion factor between milliliters of
water and seconds of time, use your stopwatch to determine how long it takes for
your water clock to stream some easily measured volume of water (e.g. 15ml or 35ml
of water). Make multiple measurements and average them. Use the average of your
measurements to calculate how many milliliters of water flow from your clock each
second. Estimate your error. The amount of water you measure should be similar to
the amount that will be displaced while the ball is rolling the length of the incline.
Remember to put the displaced water back into your water clock at the end of each
measurement!
3. Release the ball from a 5 different positions on the incline. Using the provided
white board marker, mark off 5 or more (more is better) different distances on the
incline. Record the distances from each mark to the lower end of your incline in the
table you prepared before coming to lab. Leave space to record 4 or more trials for
each distance. Hint: The time it takes for the ball to roll short distances will be
especially important.
4.Using the water clock, measure the amount of time that it takes the ball to roll down
the incline from each of your starting points. Start with your largest distance!
Perform 4 or more trials for each of your 5 or more distances. Record how many
milliliters of water are displaced during each trial and convert from ml of water to
seconds of time. Estimate your error for the largest, shortest, and middle distances
and use this as a representative error. After completing all your trials (for each
distance), determine the average time for each distance and plot that distance against
time and time2 on the plots you prepared in the pre-lab. The measurements for your
greatest length will define the longest distance and time on your plot. After plotting
this first point, label your axes.
The appendix to this handout is an excerpt from Galileo’s Dialogues Concerning Two
New Sciences. Read the description of the experiment conducted by Galileo carefully
before coming to lab.
5. What are the significant differences between your apparatus and procedure and
those described by Galileo?
6. To what extent do you believe these differences will affect your results, rendering
them either more or less accurate than Galileo’s?
Analysis
7 → Your data table should include the following entries: distance (d) measured in
meters, time (t) measured in milliliters. Use the individual measurements of time at each
distance to calculate the average time at that distance as well as the uncertainty;
include the uncertainties in your table. Also, calculate the average value and
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uncertainty for (t) 2 in (ml) . Hint: Refer to the Lab Companion for help in determining
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errors. How will you calculate the error in (t) ?
8 → Plot a graph of distance versus time and a similar graph of distance versus
(time)2. Using a single piece of graph paper oriented horizontally, plot distance on the
vertical axis then plot time and (time)2 on the horizontal axis (see figure below). Make
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sure to include error bars on all graphs.
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5
4
4
3
3
2
2
1
1
Distance
(meters)
0
0
0
20
40
60
80
100
120
0
2000
4000
6000
2
1 104
8000
1.2 104
2
Time (millilmeters of water)
Time (milliliters of water)
For each plot, draw a smooth curve that passes through all the error bars. Never just
connect the dots. Remember that the most you know about any measurement is the range
in which it falls, not an exact number.
9→ Compare the plots of distance versus time and distance versus (time) 2 . Do they have
different shapes? Is the plot d vs. t2 a straight line?
10→ From your plot of d vs. (t) 2 , calculate the acceleration, a, of the rolling ball.
Hint: The general equation for a straight line is y = cx + d, where c is a constant and
equal to the slope of the line.
Be sure to include a value for the uncertainty in the acceleration, Δa.
Hint: Construct maximum-slope and minimum-slope lines on your plot of d vs. (t)2 .
11→ Using your water clock calibration, convert acceleration into SI units. Referring
to your solution to bold question #2, calculate the acceleration for an object in freefall,
g, including uncertainty, Δg.
12→ Does the accepted value of g fall within your uncertainty, i.e., [g-Δg, g+Δg]? If
not, what is the absolute error? What is the relative (percent) error? Are these errors
systematic, random, or a combination of both? Think about the sources of the errors and
uncertainties.
Pre-Lab Checklist
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1. Answer the first 7 questions in the handout. Answer them carefully but concisely.
The answers will be collected at the beginning of the hour. If you feel that they will
be helpful during the lab, make a Xerox for yourself.
2. Prepare a data table for your measurements, following the suggestions of question 7.
3. Prepare the blank graphs that you will use to plot your lab results. These must be on
graph paper.
Post-Lab Checklist
1. Hand in the answers to the post-lab questionnaire, answered carefully but concisely.
The questions will reflect questions 8 – 12 in the handout.
2. Hand in the graphs of question 8
3. Hand in all measurements (data table) and calculations. Be sure that everything is
clearly labeled and that all units are included. Make sure that your final results follow
the significant figure rules.
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Appendix: Excerpt from Two New Sciences
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Pre-Lab Questionnaire
Galileo’s Inclined Plane
1.
Draw a free-body diagram for the ball on the illustration of your laboratory
apparatus shown below. Include and clearly label all forces.
2.
Write out an equation for the acceleration of the ball, a, in terms of the acceleration
of gravity, g, and the angle of the incline, θ.
3.
Write down the equation relating an object’s constant acceleration, a, distance
traveled, d, and travel time, t. Reference your answer using the name of the book you are
using, the page number and equation number.
4.
Combine the equations from question 2 and question 3 and solve for g in terms of t,
d, and θ.
5. What are significant differences between your apparatus and procedure and those
described by Galileo?
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6. To what extent do you believe these differences will affect your results, rendering them
either more or less accurate than Galileo’s?
7.
Show how to calculate the total error in your calculated value for g based on the
error in your measured values of t, d, and θ. Note there is a factor of t2 in this equation.
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