Brown University
Physics Department
Physics 0030
Lab 1
LAB 1: MEASUREMENT AND PROPAGATION OF ERRORS
In this introductory lab session you will learn the fundamentals of measurement and
propagation of error and other skills important to any practicing scientist.
It is important that you master error analysis during this session because you will be
required to have an error analysis section in every report you write for the entire year.
Lab results presented without some numerical estimation of error will not be taken
seriously by the scientific community. To quote Kelvin “I often say that when you can
measure what you are speaking about, and express it in numbers, you know something
about it. But when you cannot express it in numbers, your knowledge is of a meager and
unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your
thoughts advanced to the state of science, whatever the matter may be.”
Experiment 1: HEART RATE MEASUREMENT
Count the number of your heart pulses that occur in 20 seconds. Your partner counts the
seconds without looking at a watch while you hold one wrist with your other hand, to feel
your pulse. Then exchange roles. Note: the goal is to generate measurements that are
NOT accurate. Take and record at least 8 repeated measurements.
CALCULATIONS, RESULTS, DISCUSSION
What is the uncertainty you associate with the 20 second count of heart pulses? From the
repeated measurements, calculate the average count and the uncertainty in this average.
Then, give your result for your heart rate (beats per minute) with the appropriate
uncertainty. How do you find this uncertainty? [See the following Exp. and the separate
handout on the Analysis of Experimental Uncertainties.] Discuss your result and its
uncertainty. What sources of systematic uncertainty may be present, in addition to the
random measurement error, and how would their presence affect the result? How might
you test for the presence of systematic errors and modify the procedure to reduce them?
Note you will measure your heart rate with an oscilloscope/EKG in a later physics 40 lab.
Please keep your data so that you can compare the accuracy/precision of the two
methods.
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Brown University
Physics Department
Physics 0030
Lab 1
Experiment 2: FREE FALL VELOCITIES
This experiment studies uniform acceleration in one dimension by systematic
measurements of a falling body's position and instantaneous velocity. You will have
equipment to measure the time when a freely falling body passes a given vertical
position. A description of the timing equipment is in the Guide to Laboratory
Measurements.
REFERENCES
Young and Freedman, University Physics (12th Ed), Chapter 2.
BASIS OF THE EXPERIMENT
It is shown in many texts (page 49 in Young and Freedman) that if an object moves with
constant acceleration ax at some time t its position x is given by
(1)
x = x0 + vot +
1 2
ax t
2
where x0 , v0 are its position and velocity at t = 0.
Use the apparatus to make measurements of the times when a freely falling body passes
two positions. From the combined measurements of x,t determine the gravitational
acceleration. The accepted value of the acceleration due to gravity is about 9.81 m / s 2 .
PLAN OF THE EXPERIMENT
We use photobridges across the path of the falling body to measure the time intervals we
need. The apparatus consists of a rigid vertical rod adjacent to the body's trajectory, on
which the photobridges, marked U, M, and L in Figure 1, are mounted. The body, latched
mechanically at Z until released, defines an exact zero point in time, distance and
velocity. Two electronic timers marked TUM and TUL in the figure, are set to operate in
pulse mode. Not intended to be a wiring diagram (these are present in the laboratory) the
figure indicates the logic flow of signals from the photocells to the timers. The pulse
from the U cell as the body first cuts its beam is passed to both the UM and UL timers,
starting both counters.
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Physics Department
When the body first cuts the M beam,
its photocell sends a second pulse to the
TUM timer, which causes the timer to
stop, giving the time of fall from U to
M. The TUL timer continues until the
beam is cut to photocell L, at which
time its pulse stops the timer with the
time of fall from U to L.
All the bridges are movable on the rod.
Suppose we start with the U bridge
high on the rod and the L bridge
mounted about a meter below it. Now
let the M bridge be placed midway in
space between the other two. When a
drop is made, the UL timer will contain
the total fall time through the bridges,
and the UM timer will show the fall
time from U to the space midpoint.
The latter, because of acceleration, will
be larger than one-half the UL reading.
Physics 0030
Lab 1
z
U
START
z1
M
TUM
(pulse)
STOP
z2
L
TUL (pulse)
STOP
Figure 1. Diagram of apparatus
We can refer to the distance between UM as z1, and the time interval t1, and the distance
between UL as z2, with the time interval t2. The following relationships will then hold:
1
z1 = v0 t1 + gt12
2
1
z2 = v0 t2 + gt22
2
Eliminating v0 the unknown velocity at the start gives
! z2 $ ! z1 $
# &−# &
" t % " t1 %
g=2 2
(t2 − t1 )
Distance measurements are critical. Note that measurements at the rest position always
refer to the lower edge of the body, because that is the edge that activates the photobeam
“switches”. The distance from the rest level to the upper photobeam can be made a onetime problem by choosing a good location for the upper bridge (one that allows easy
access for placing the mass at the rest position) and locking it there for the entire
experiment. All measurements to or between photobeams are best made by using the
well-defined metal frame of the photobridge itself. If we assume the photobeams
themselves are always at the same distance offset from the top surfaces of the bridges,
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Brown University
Physics Department
Physics 0030
Lab 1
then the distance between photobeams, for example, is exactly the distance between
corresponding upper edges of their photobridge frames. You should verify for yourself if
such an assumption is reasonable.
PROCEDURE AND DATA
KEEP A RECORD OF YOUR PROCEDURE THROUGHOUT THE EXPERIMENT.
Align the apparatus so that the beams are cut reliably over the entire drop length. Small
shifts of the mounting board on the floor, and small rotations of the bridges, may be
needed. Be sure that there is a box at the base to catch the body.
Set the top bridge position high, but allow ready access to the launch position. Make
several drops to check for good alignment, for repeatability at fixed bridge positions, and
to decide on a good range of positions for the lowest bridge. Note that the highest
position of the lowest bridge should not be such as to give small (two-digit or very low
three digit) time readings, since any digital reading can inherently be in error by one in
the lowest digit.
Measure carefully the constant distances discussed above and record them. In your
notebook set up a Table in which to enter your data in a clear, understandable way.
Always record the numbers as you measure them - leave calculations, even simple ones,
for later. Include units for all numbers.
Take measurements of pairs of positions and times, for the lower bridge set at least 5
different positions. Remember to record an uncertainty with each data point.
CALCULATIONS
For each setting of the lowest bridge, calculate the acceleration, applying the equations of
motion in free fall.
RESULTS
The best value obtained from a series of N measurements of a quantity is the mean value,
simply the arithmetic average of the individual measurements.
Using all N independent acceleration determinations a i , (where N is at least 5), calculate
a “best value” for your experiment as the mean, or average of the individual values a i ,
a=
1
N
N
∑a ,
i
i =1
No experimental result is complete or meaningful without an estimate of the experimental
uncertainty. A good measure of the uncertainty in the mean is the standard deviation of
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Brown University
Physics Department
Physics 0030
Lab 1
the mean, S.D., which is obtained from the root mean square deviation (RMS) of your
measured values from their mean:
S.D. =
1 N
(a − ai )2 ,
∑
N −1 i=1
and the Standard Error,
S.E. =
RMS
,
N
where the ai are your individual determinations of a.
A final best value with its uncertainty is then a ± S.E.
DISCUSSION AND CONCLUSIONS
Compare your measured value to the accepted value of the acceleration of gravity, and
discuss the result, taking into account your experimental uncertainty and the
reproducibility of measurements with the apparatus. Include a discussion of sources of
experimental uncertainties. Discuss possible systematic errors, ways to test for their
presence, and steps to take to reduce them.
Note: In your report you are not expected to repeat the plan of the experiment as given in
the handout, but to say briefly what you actually did and mention any problems you
encountered.
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