Experiment 2
Uniform Acceleration in One Dimension
Preparation
Prepare for this week's experiment by reading about velocity and acceleration. Review definite
integrals in your calculus book.
Principles
Objects dropped near the surface of the earth will accelerate downwards. While the value for the
acceleration due to gravity, g, varies from place to place on the planet, all objects dropped at one
place will accelerate downward at the same rate. In today's experiment you will take data that
will enable you to calculate a value for g.
Suppose that you could drop an object and make a record of its position at specific times. How
could you use that data to find the value for g?
Consider the motion of a particle traveling in one dimension that is under going constant
acceleration. We can choose an instant in time and call this time t = 0. Its position is xo, its
velocity is vo and its acceleration is a. xo and vo are fixed values which may or may not be equal
to zero. At some later time, t, the particle has position, x, and velocity, v, which are both
functions of time.
From your class work and readings you know that the position of this particle is given by
x = x0 + v0 t + 1 2 at 2 .
If you plot the position as a function of time the graph will be a parabola. It will open upward if
the acceleration is positive.
At any point on the line the tangent or instantaneous slope will equal
dx
=v.
dt
If you differentiate the equation for position you can see that the velocity equals
v = v0 + at .
If you plot the velocity as a function of time you will get a straight line. The slope of that line
will equal
dv
=a.
dt
Suppose you dropped an object and marked its position at a number of times so that the time
intervals are all equal and fairly small. If you selected a starting point and measured the position
1
of the falling object relative to that point you could plot the positions as a function of time. The
graph you get would not be perfectly smooth, but it would come close to the ideal curve.
You wouldn't be able to find the instantaneous slope at any point on this curve, but you could
find the average slope between two adjacent points. Designate each position as xi, as i goes from
1 to n, where n is the total number of points. Do the same for the times at which the marks were
made; call them ti. The average slope equals the average velocity of the object over that time
interval such that
vi =
xi+1 - xi
.
ti+1 - ti
The slope between two
close points gives you the
average velocity over the
time interval.
Position
Since the object is accelerating at a
(cm)
constant rate, some of the time it will
be going faster than the average
velocity and some of the time it will be
going slower. At the midpoint of the time
interval, 1 2 ( ti +ti+1 ) , the average velocity and the
instantaneous velocity will be equal. This value
1
2 ( ti +ti+1 ) is called the mid mark time.
The average velocity
equals the
instantaneous velocity
at the midmark time.
Mark
time #1
Midmark
time
Mark
time #2
Time (1/60 sec)
You would now know how fast the object was moving at
specific times. If you plot these velocities as a function of time the slope of the line will give a
value for the acceleration.
The inverse of the time difference is the frequency, f, of the sparker
f=
so
Equipment
1
1
1
,
ti+1 - ti
vi = f (xi+1 –xi)
length of heat-sensitive tape
2-meter stick
masking tape
Procedure
The free fall apparatus has an electromagnet to hold the object until a record of its fall is to be
made. When the switch to the coils is opened the object will fall freely into a cup at the bottom
of the apparatus. The object, which is insulated except for a metal ring around its center of mass,
will fall between two stretched wires which are connected to a high voltage spark generator
which produces pulses 60 times per second. A strip of heat sensitive paper is stretched over one
of the wires. As the object falls a spark will jump from one wire, through the conducting ring,
2
through the paper to the other wire. This produces a record of the position of the object at 1/60
second intervals.
The lab instructor will demonstrate the apparatus if it is available and give each group of students
a prepared tape. Because the apparatus is potentially dangerous, you will not be allowed to make
your own tape.
Use centimeters for your distances. Measure positions to the nearest millimeter and estimate
hundredths of centimeters.
Find one or two people to work with.
2.
Get a prepared tape from your instructor. Stretch it out on the table so that the
distances between the spots are increasing from left to right and secure the ends
with masking tape. Find the part of the tape where the marks are about 2 cm
apart. Go down the tape circling every mark except for the last one. Check the
tape to be sure that no marks are missing.
.. . .
3
Take data in ink and write it directly into the notebook. Write down the names of
you lab partner or partners. Write down your spark frequency. Start a column
labeled "Position". Put the correct unit in parentheses. There is no need to
number the positions or to write anything else on your data page.
.
4.
Lay a two meter stick on its side and place it on the tape so the first circled mark
is at the 10.00 cm mark on the meter stick. This corresponds with position xo at
an arbitrary t=0. Without moving the meter stick, read the position of each
subsequent mark to the nearest hundredth of a centimeter and record it directly
into your notebook. This means that you will have to estimate the values between
the mm marks. If your measurements all seem to end in 0 or 5, you are not
measuring correctly; start again and measure more carefully.
5.
Do not destroy your tape until you've completed the analysis. You may need to refer to it
again.
.
1.
.
.
.
Data
The data for this experiment consists of the spark frequency (60 Hz) and the positions of the
marks. Be sure you have your instructor check and sign your data before you leave for the
day.
Analysis
You must take data and do most of the analysis correctly to get credit for participating in the lab
for this week. Do as much of the analysis as you can before you leave for the day. You should
at least finish the spreadsheet before you go. Have your instructor check it when he or she
initials your data.
3
The mark times for this experiment are 0, 1/60 s, 2/60 s, etc. This is very cumbersome for
calculations and graphing. A way around this is to make the unit of time 1/60 s. This is
perfectly legal as long as you keep track of the units throughout the analysis.
1.
Use Excel to analyze your data. Label one column 'mark times' and make the unit 1/60 s.
Enter the mark times (0, 1, 2, 3, etc.). Make a column for the positions, including the units
and enter the positions on the spreadsheet.
2.
Calculate the mid mark times in term of 1/60 s. They should be 0.5, 1.5, 2.5 (1/60 s).
Calculate the interval velocities using the equation
vi = (xi+1 - xi )f .
Calculate the average acceleration over each interval using
ai = (vi+1 - vi )f .
3.
Look at the interval accelerations. There will be some variation, but all of them should be
close to g. Are any of them way off? These may be the result of a typo or of a faulty
measurement. If this happens check your spreadsheet for typos. If there are no typos,
measure the relevant parts of the tape again.
4.
Find the slope and intercept of the interval velocities vs. mid mark time. The slope will be
incorrect because of the change in the time unit. The slope is dv/dt and you in effect
multiplied each value of dt by 60. Find the correct value for the slope. Find the percent
error for your acceleration value from the accepted value of 979.5 cm/s2.
5.
Graph mark position vs. time using the Cartesian graph paper. Sketch in the best fitting
curve.
6.
Graph velocity vs. mid mark time. Use the slope and intercept to calculate two points, plot
them and draw the best fitting line.
7.
When you are satisfied with your analysis, show your results to your instructor and have
him or her initial the data. You will not get credit for the lab without initialed data. Throw
the tape away and leave your lab area clean.
Questions
1.
About how long did it take the object to fall? How did you determine this?
2.
What was the initial velocity of the object at the time you chose as t=0? What was the final
velocity at the point where you stopped measuring? Do you think the values you got are
reasonable? Why or why not? Hint: Look at the answer to question 1.
4
Consider some of the sources of error in this experiment. The wooden meter stick can
shrink or expand by as much as 0.1% as the humidity in the room changes. Suppose you
did the experiment perfectly but the meter stick expanded by 0.1%. Would g be too big or
too small? By how much would it be wrong?
4.
v or a
Many of Albert Einstein’s famous ‘thought experiments were
conducted on moving trains or rocket ships. Do some thought
of the train
experiments of your own and consider two cases:
What would the results of the free fall experiment be if you set
the apparatus up and took data on a train moving at constant velocity?
What if the train were accelerating?
.
What would the results of this experiment be if you did this experiment
repeatedly on a rocket ship traveling from earth to the moon at constant
velocity? Assume the tape is parallel to a line connecting the centers of the
earth and moon.
.
5.
.
Assume the tape is oriented sideways relative to the direction of travel and
that the object is falling down the length of the tape. Draw the dot pattern you
would see and give the value for g that you would calculate in each case.
.. . .
3.
.
.
Direction the object
falls
If this applies to you, write at the end of your report "I have not cheated on this lab report" and
sign your name. The report will not be graded if you cannot do this.
Grading
You may choose how much work you do on the lab report. This means you get to choose how
high your maximum grade can be. You must come to the lab, take data, complete the analysis,
and turn the notebook in to get credit for lab participation. You must have credit for at least ten
experiments to pass the course. Correct data and analysis are worth four points. Any of the
questions you answer correctly will add to your score for the week.
Questions must be answered clearly and in detail. Write in complete sentences. Most answers
should consist of at least a paragraph.
Even the best students occasionally find themselves overwhelmed with work. If that happens to
you, turn in the analysis and move on.
4 pts
Data and Analysis: This is the compulsory part. The data should be clear, reasonable,
and signed. The analysis includes the spreadsheet printout, sample calculations, the
5
calculations for g, the percent error, and the graphs. Everything must be neat, well
organized, and correct. Everything must be clearly labeled; units and significant digits
must be correct for the work to get full credit.
3 pts
each for questions 1, 2 and 3
4 pts
for question 4
3 pts
for question 5
Your lab instructor will tell you when your labs are due. Turn the notebook in by sliding it under
the door of room 214 NSC. There is a five point per week penalty for late work beyond the
analysis. Only the data and analysis part of your work will get credit if you turn it in more than
two weeks late.
If your analysis is not substantially correct it will be returned to you ungraded and you will have
one week to correct your work. After that it will turn into a zero. Work without sample
calculations is not acceptable.
You will lose points if your work is missing units or labels, if the significant digits are not
correct, or if it is illegible or poorly organized.
Your report will not be graded if there are no instructor initials on the data.
6