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Sample Lab Report for Physics 210 lab:
The Ball-Drop Experiment
Wayne Hacker and William Flack
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Copyright Wayne
Hacker 2009. All rights reserved.
May 1, 2009
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Hackernotes: Sample Lab Report, Wayne
Hacker 2009. All rights reserved.
1
Contents
1 Introduction
4
2 Theory
4
3 Lab Procedure
5
4 Results and analysis
6
5 Conclusion
12
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Hacker 2009. All rights reserved.
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A note to the student on the sample report
This sample report is intended to give you, the student, a road map to start your journey
to learning how to write a technical report. Throughout this mock report, you will find
notes inserted that explain why we did things the way we did them. These notes should
not be regarded as part of the report, and there should be nothing analogous to them in
the reports you write for this course.
To distinguish these notes from the text of the report itself, we’ll preface them with the
phrase “Off the record” in square brackets; and we’ll use a smaller font for them, like
this:
[Off the Record:] In this part we did not show the y-intercept because . . .
The sample report will begin on the next page. Your reports should look very much like
it, but without the [Off the Record:] comments.
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[Off the Record:] (sample title page)
Instructor: Tony “The Tiger” Pitucco
[Off the Record:] (Don’t use nicknames!)
Lab report for the ball-drop experiment
William Flack
Wayne Hacker
[Off the Record:] You should list the names of the experimenters in alphabetical order. This
makes it easier for the instructor to record the grades, and makes it less likely that someone’s
grade will accidentally not be recorded. Points will be deducted for names not in alphabetical
order.
September 21, 2010
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4
Introduction
A steel ball was dropped from various heights y and its fall times t measured. Data were
fitted to an equation of the form: y = ktn , yielding experimental values of k = 4.98
m/s2 and n = 2.03. From the value of k, the experimental value for the acceleration of
gravity gexp = 2k = 9.97 m/s2 was computed. These compared to expected values of
g = 9.79 m/s2 and n = 2. Deviation from expected values was apparently systematic,
and was greatest for small values of y. The experimenters hypothesize that nonrandom
deviations were caused by a fixed time delay, possibly in the ball-release mechanism;
and by a variable time delay in the timer-stopping apparatus at the bottom of the ball’s
fall. Repeating the calculation without the three data points of lowest y value yielded
gexp = 2k = 9.78 m/s2 and n = 2.01. The relative error of these is of the same order of
magnitude as the random error in the time measurements.
Notational convention: Values of y and t obtained by direct measurement, or as an
average of direct measurements, will be given the subscript “best”. Experimental values
obtained by more involved computations with measured data will be given the subscript
“exp”. Values predicted from theory will be given the subscript “th”.
2
Theory
The position of a body moving vertically under the influence of gravity is
1
(1)
y(t) = y0 + v0 t + ay t2 ,
2
where y0 is the initial height, v0 is the initial velocity, and ay is the acceleration due to
gravity. If the initial height is taken as y0 = 0; if the body is initially at rest so that
v0 = 0; and if the positive y-direction is chosen to be downward, so that y increases as
the body falls, and ay = g > 0; then the equation becomes
1
y(t) = gt2 .
2
(2)
If this relationship were unknown, it might seem reasonable to hypothesize that y and t
followed a power-law relationship, with the form
y = ktn
(3)
where k and n are real constants. Such a hypothesis could be tested by collecting data
consisting of values of y and the associated values of t, then plotting ln y vs. ln t. Taking
the natural logarithms of both sides of equation (3) yields
ln y = ln k + n ln t
(4)
Thus if equation (3) is correct, the points (ln t, ln y) should lie on a straight line with slope
n and intercept ln k. From (2), it should be found that n = 2 and that k = (1/2)g ⇒
g = 2k = 2eln k .
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5
Lab Procedure
This experiment was conducted on April 15, 2009, 11:15 a.m.-12:45 p.m.
A steel ball was dropped from various heights, and its fall times recorded. The ball was
dropped over ten distances ranging from y1 = 10 cm to y10 = 100 cm in 10-cm increments.
For each value of yi = 10i cm (i = 1, . . . , 10), the ball was dropped ten times.
The apparatus consisted of a meter stick held vertically by clamps. A Pasco free-fall
adapter was clamped at various heights on the stick and used to drop the ball, which fell
onto a drop pad at the base of the stick. The free-fall adapter and the drop pad were
connected electrically to a timer.
The free-fall adapter consists of two parallel vertical jaws, connected by two parallel
horizontal rods. One of the jaws can slide back and forth along the two rods, toward
and away from the other jaw. A spring between the two jaws tends to push the sliding
jaw horizontally away from the fixed one. The fixed jaw has a conical protrusion with
a shallow circular recess in it, into which the ball can be fitted; the sliding jaw is then
pushed against the spring to hold the ball in place. A latch is then engaged to keep the
sliding jaw from moving outward. When the ball is to be released, a knob is turned to
move the latch, allowing the spring to push the sliding jaw away from the fixed one.
When the jaws are closed on the ball, an electrical current passes from the fixed jaw
through the ball to the sliding jaw. When the ball is released, the circuit is broken; this
starts a timer operating in gate mode.
When the ball lands on the drop pad, it strikes a horizontal metal plate. The force of
the ball compresses a spring and brings the plate into contact with an electrode beneath
it, completing an electrical circuit that stops the timer.
In early tests of the apparatus, it was found that the free-fall adapter could generate bad
data points. It appeared that the sliding jaw did not move away from the fixed jaw and
the ball quickly enough, which sometimes caused the circuit to be broken very briefly
and then re-connected. Re-connecting the circuit after it had been broken stopped the
timer, producing spurious fall times on the order of milliseconds. Such spurious times
could immediately be recognized, as theoretical calculations indicated that the fall times
for the lowest values of y should be greater than 100 ms. When this occurred, the
trial was immediately discarded and its result not recorded. There were no ambiguous
times: either they were greater than 100 ms, as predicted by theory; or less than 10 ms,
indicative of a false time.
To reduce the incidence of such false times, a washer was installed between the fixed jaw
and the spring. This increased the compression of the spring when the jaws were closed,
and thus increased the acceleration of the sliding jaw away from the fixed jaw and the
ball when the latch was released. The experimenters also discovered that it was necessary
to turn the latch-release knob quickly through a full half-circle; turning it more slowly
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apparently created vibrations that could break and immediately re-connect the electrical
circuit through the jaws and the ball.
The meter stick was clamped with its zero mark at the level of the drop plate. A plastic
right triangle was used to measure the height of the bottom of the ball. One side of the
triangle was held vertical against the meter stick; the triangle was then moved upward
until the horizontal upper side of the triangle touched the bottom of the ball. To keep the
apparatus from swaying horizontally as the release knob was operated, it was necessary
to stabilize the top of the meter stick with one hand while the knob was twisted with the
other.
Although the parameters of the ball are not a part of the theoretical calculations, it
was later realized that they might be relevant to the discussion of error. One of the
experimenters (Hacker) returned to the lab on April 17, 2009, and measured them. The
mass of the ball was 27.75 g ± 0.01 g; its diameter was 1.9 cm ± 0.1 cm.
4
Results and analysis
For each value of y, the mean and standard deviation of the ten measurements of t were
calculated. These appear in Table 1. The measured values of y and t are given in the
standard measured number form: ybest ± δy and tbest ± δt (best estimate + uncertainty).
y (meters)
0.100 ± 0.001
0.200 ± 0.001
0.300 ± 0.001
0.400 ± 0.001
0.500 ± 0.001
0.600 ± 0.001
0.700 ± 0.001
0.800 ± 0.001
0.900 ± 0.001
1.000 ± 0.001
t (seconds)
0.1459 ± 0.0005
0.2047 ± 0.0003
0.2495 ± 0.0005
0.2875 ± 0.0006
0.3211 ± 0.0004
0.3519 ± 0.0005
0.3802 ± 0.0003
0.4058 ± 0.0005
0.4308 ± 0.0013
0.4534 ± 0.0009
Table 1: Experimental values of y and t
The error in y is based on the fact that it was measured using a ruler with markings
of 1 mm. The digital timer registered time to the nearest 0.0001 s; since the standard
deviations were several times this, they were taken for the error values of t.
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From the experimental values of y and t in Table 1, a plot was made of ln y versus ln t.
This plot appears as Figure 1. The data appear to fit closely to a regression line as given
in equation (4); using Excel’s slope and intercept functions, the parameters of the
regression line were calculated as n = 2.028 and ln k = 1.606.
Figure 1: A plot of ln y versus ln t, with the values of y and t obtained from the experiment. The regression line has the form ln y = n ln t + ln k, with n = 2.028 and
ln k = 1.606. The latter yields g = 2k = 9.969 m/s2 .
[Off the record:] Notice that we did not show where the line actually intersects the ln y-axis.
The reason for this is to focus the reader’s attention to the data points and where they lie with
respect to the line.
The regression line yields an experimental value for g of gexp = 2k = 9.969 m/s2 . The
NOAA website http://www.ngs.noaa.gov/cgi-bin/grav pdx.prl was given the parameters found in http://en.wikipedia.org/wiki/Tucson: latitude = 32.2217◦ ; longitude =
110.9264◦ ; elevation = 2400 ft. It yielded the value: g = 9.7924 m/s2 , which was taken
to be the true value. However, to account for the variation of the location of the West
campus open physics lab, where the measurements were taken, to the above coordinates,
only two-place accuracy (three significant figures) was used. Under this assumption,
gtrue ≡ 9.79 m/s2 . Thus gexp had a relative error of 1.80%. The regression line yielded an
experimental value of n = 2.028; compared to a true value of n = 2, this has a relative
error of 1.40%.
These relative errors were significantly larger than the estimated relative error for y,
which ranged from 0.1% to 1.0%; and significantly larger than the relative error for t,
which ranged from 0.07% to 0.30%. Accordingly, evidence of systematic error was sought.
[Off the record:] This is where you show your instructor that you deserve a good grade for
your report. You’ve got results that aren’t as good as you’d hoped for; now you try to find
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explanations for that, and to justify them using your data. A two-pronged approach is called
for. On the one hand, you should manipulate your data in various ways and look for patterns
in the error. On the other, you should think about the experimental apparatus and technique,
what could have gone wrong with it, and what patterns that would have produced in the data.
Equation (2) was solved for t. The measured values ybest of the drop distance and the
“true value” gtrue = 9.79 m/s2 were used in this equation to calculate the theoretical time
values tth :
r
2ybest
(5)
tth =
gtrue
Table 2 shows, for each measured value ybest , the measured fall time tbest ; the theoretical
fall time tth ; the difference ∆t = tbest − tth ; and the relative difference ∆t/tth .
ybest (m) tbest (s)
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
0.1459
0.2047
0.2495
0.2875
0.3211
0.3519
0.3802
0.4058
0.4308
0.4534
tth (s)
0.1429
0.2021
0.2476
0.2859
0.3196
0.3501
0.3782
0.4043
0.4288
0.4520
∆t = tbest − tth (s) ∆t/tth
0.0029
0.0026
0.0019
0.0016
0.0015
0.0018
0.0020
0.0015
0.0020
0.0015
0.0206
0.0128
0.0078
0.0057
0.0046
0.0051
0.0054
0.0037
0.0046
0.0032
Table 2: Comparisons of experimental and theoretical values of t. Theoretical values
tth were derived from experimental fall distances values ybest using equation (5) and
gtrue = 9.79 m/s2 . Note that the error ∆t is always positive; and that it decreases with
drop distance.
This analysis strongly supports the hypothesis that significant nonrandom error occurred
in the experiment. For every value of ybest , it is the case that tbest > tth . It is highly
unlikely that such a distribution could come about through random error alone.
[Off the record:] We’ve started by looking at the differences between the measured and the
theoretical values of t for each measured value of y. We’ve immediately seen an interesting
pattern. Now we need to think about possible causes for that pattern. Often, it’s a good idea
to generate various graphs and see what they show about the error.
Figure 2 on page 9 shows ∆t = tbest − tth graphed versus ybest . It does not indicate
a constant time delay, but one that starts relatively large and then decreases with increasing ybest . This tends to rule out one possible explanation for the difference between
experimental and theoretical values: air resistance. Since air resistance increases with a
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body’s speed, its effect would be greater as the body fell farther. Then tbest − tth would
increase with ybest rather than decreasing.
[Off the record:] Don’t just give the causes that you think might account for your error; give
the causes that you’ve ruled out, and explain why. This is especially true with obvious possible
problems like friction, air resistance, etc. You want the reader to know that you’ve considered
these possibilities.
Figure 2: A plot of the time error tbest − tth versus measured drop distance ybest . The
theoretical predicted time tth was calculated from the measured drop distance ybest using
equation (5) and gtrue = 9.79 m/s2 .
[Off the record:] It’s often a good idea to look at the graph of relative error. A constant
relative error can be a strong indication of a problem—for example, if ∆t/tth were constant, it
would strongly suggest that your timer was fast or slow. We didn’t look at relative error in this
case, because figure 2 shows that ∆t decreases with increasing ybest . Since tth increases with
ybest , it’s obvious that the relative error will decrease.
The pattern of error in figure 2 might result from a systematic error in measuring the
drop heights ybest . The experimenters measured the distance from the bottom of the
ball’s initial position to the top of the target pad. It was presumably necessary for the
top of the target pad to be pushed down some small additional distance in order to stop
the timer. It is also possible that the experimenters did not correctly align the bottom
of the meter-stick with the top of the drop pad, and that the ball had to fall some small
additional distance on each trial. Either or both of these could have led to the ball’s
falling a total distance of ybest + yadd , where ybest is the measured distance and yadd is a
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constant error.
[Off the record:] Don’t assume that you couldn’t have made an error in the experiment. Your
instructor will be more impressed by your catching your own mistake than by an assertion that
you couldn’t possibly have made one.
[Off the record:] Now that you’ve thought of a possible cause for the discrepancy, you need to
think of ways to test that hypothesis using the data. We hypothesized that the height might’ve
been measured incorrectly in a consistent way; now we have to think of how we could detect
such an incorrect measurement in the data.
If this were the case, then equation (2) would become
1
(6)
yth = ybest + yadd = gt2best
2
where ybest and tbest are the measured values of y and t, and yth is the theoretical value
of y obtained from the measured value tbest in equation (2).
Figure 3 is a graph of yth − ybest versus tbest . According to equation (6), if the assumption
of a constant distance error yadd were true, then the points in the graph would lie on a
horizontal line. This does not appear to be the case: there is a general upward trend in
the points.
Figure 3: A plot of yth − ybest versus tbest . The theoretical predicted fall distance yth was
calculated from equation (2) using the measured fall time tbest and g = 9.79 m/s2 .
[Off the record:] That idea didn’t work either. Nevertheless, include it in the report: it shows
that you thought of it, and that you ruled it out.
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There does not appear to be a single cause for the discrepancy between theory and
experiment. A closer look at figure 2 suggests that the discrepancy between theoretical
and measured time is high for small fall distances, and decreases with increasing fall
distance to level off at a constant value between 1.5 and 2.0 ms.
The experimenters hypothesize that the discrepancies are produced by a time delay with
two components. One of these is constant; the other is variable, and is greatest when the
fall distance is small.
[Off the record:] We couldn’t find a single easily testable cause for our problem results. Now
we’re forced to make hypotheses that we can’t test using the data. Notice that we use words
like “hypothetical” and “possible” a lot more often in this last part. It should be very clear that
we’re now speculating, and that we can’t prove our speculations with the data available to us.
Of course, our hypotheses should be consistent with the data and with a good understanding
of the apparatus.
If there is a constant component of delay, one possible source is the drop mechanism.
When the ball is held in place before its release, a small portion of it extends into the
circular recess in the fixed jaw. Thus when the ball is released, it does not fall straight
down; it must move a short distance horizontally to get clear of the recess, and until this
has occurred the edge of the recess exerts a force with an upward component on the ball.
Since this delay occurs at the beginning of the ball’s fall, it is not affected by the fall
distance.
To explain the higher values of tbest − tth seen in the leftmost points of Figure 2, the
experimenters hypothesize a variable time delay. A possible cause for this is the design
of the target pad. When the falling ball strikes the upper surface of the pad, it must
overcome some spring resistance to bring the upper surface into contact with the electrode
and close the circuit that stops the timer. This resistance is presumably low, and the
distance through which the spring must be deformed is presumably small; but they must
be large enough to keep the mechanism from being triggered by air currents or dust
particles.
One can imagine a situation in which a very small ball was dropped from a very low
height, and struck the target pad with insufficient energy or weight to push down the
impact surface and close the circuit. If the ball were slightly larger or the drop distance
slightly greater, then the circuit would be closed; but the spring resistance would slow the
ball in its fall, and thus increase the measured time. Increasing the mass of the ball or the
drop distance would reduce this effect: a more massive ball would experience a smaller
upward acceleration from the force of the spring; a greater fall distance would result in
the ball’s moving faster when it reached the drop plate, lessening the time necessary to
deform the spring.
[Off the record:] We’ve come up with some hypotheses to explain our problem results. We
can’t test those hypotheses with the data we’ve got. How could we modify the experiment to
test them, and possibly to produce better results next time? What new problems could the
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proposed modifications cause? This is an opportunity to impress your instructor with how
clearly and carefully you’ve thought about the experiment.
It would be instructive to repeat the experiment with a ball of significantly greater mass.
The hypothetical mechanism for variable delay predicts that the effect would be reduced
by increasing the ball mass. If such an experiment showed reduced values of tbest − tth
for the smaller values of ybest , it would tend to corroborate the hypothesis.
An alternative approach would be to replace the drop pad with a photogate to stop the
timer. However, it might be difficult to ensure that the ball’s lowest point passed through
the light beam. It would probably be desirable to replace the ball with a flat-bottomed
weight; and that would have unknown effects on the air resistance.
It would be difficult to devise an experiment to test the hypothesis of constant delay. It
might be useful to repeat the experiment with a different release mechanism: for instance,
releasing the ball by shutting off the current to an electromagnet. However, it is possible
that the magnetic field would take some nontrivial time to decay, during which the ball
would experience an upward magnetic force even while it was supposedly in free fall.
This experiment could probably be improved by eliminating the smallest values of yexp
and adding larger ones. This would increase the ball’s downward speed when it struck
the target pad, thus reducing the hypothesized variable delay; and since the ball would
have a greater fall time, the fixed delay would represent a smaller percentage error.
[Off the record:] If you think that certain data points are a problem, you can redo your
calculations with them left out. However, you should always make it clear that you’re doing
so, and you should always do the analysis with all the data points first. It’s never acceptable
to throw out a point or points without explanation.
Although it was not possible to add new data points for greater fall times, it was possible
to eliminate the ones for small ybest . The experimenters calculated a new regression
line on ln ybest versus ln tbest , again using Excel’s slope and intercept functions, but
with the points corresponding to ybest = 0.100, 0.200, and 0.300 m omitted. The new
regression line had parameters n = 2.008 and ln k = 1.587. The latter yielded gbest =
2k = 9.776 m/s2 . The relative error in n was 0.39%; the error in gbest relative to the
value g = gtrue = 9.79 m/s2 was −0.14%.
It is possible that further improvement could be produced by eliminating additional lower
values of ybest and adding higher ones: for instance, by varying vbest in 10-cm increments
from 0.5 m to 2 m. This process of improvement could not be continued indefinitely: as
ybest continued to increase, air resistance would eventually become significant.
5
Conclusion
Fitting the experimental data points to a curve of the form in equation (3) yielded a value
for the gravitational acceleration of g = 2k = 9.969 m/s2 , and a value of the exponent of
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n = 2.028. These compare to theoretical values of g = 9.79 m/s2 and n = 2.
The experimental values were within 2% of the theoretical values. This significantly
exceeded the relative random measurement error. Patterns within the data suggested
that systematic error was present.
To explain the systematic error, the experimenters hypothesize two time delays. The first
of these is a fixed delay, believed to result from the design of the drop mechanism: when
the ball is released, there is a brief period when it is subject to an upward and horizontal
force; during this brief period, the downward acceleration of the ball is less than the full
acceleration of gravity, increasing the fall time.
A second hypothetical delay, this one variable, is produced by the fact that the falling
ball must compress a spring in the drop pad to stop the clock. This delay can be reduced
by increasing the mass of the ball and/or the drop distance, and thus the momentum of
the ball when it strikes the drop pad.
Eliminating the three data points associated with the three smallest drop distances
yielded values of g = 9.776 m/s2 and n = 2.008. The relative errors for these values
were of the same order of magnitude as the standard deviation of the time measurements.
If the experiment were to be repeated, it would be desirable to use a ball with greater
mass, to eliminate the shortest drop distances, and to add additional drops at greater
distances still. Increasing the mass of the ball and the drop distance would reduce the
hypothesized variable delay; increasing the drop distance would also increase the fall
time, thus reducing the percentage error represented by the hypothesized fixed delay.
The experimenters speculate that using drop distances ranging from 50 to 200 cm would
produce closer agreement between theoretical and experimental results. However, caution
should be exercised in increasing the drop distance, since at some point air resistance
will begin to produce a significant delaying effect.