PHYS 1712 Physics Laboratory I Purpose of this course :

PHYS 1712 Physics Laboratory I
Purpose of this course :
Why you (as a physics major) need to learn to do experiment?
Because physics is an experimental science!
That is the only way we can find out how Nature behaves
In this course you will :
1. Learn how to use basic equipment such as conventional
and digital oscilloscopes, power supplies, signal
generators, digital multimeters, etc. to do experiment and
make measurements.
2. Learn how to collect, analyze, handle data and estimate
their errors by using Excel.
3. Learn how to describe your results to other people by
writing a short report.
A brief outline of this course
• You’ll need to do 6 experiments in this term.
• You are required to write a short report on each
experiment.
You should have received hardcopies of the Lab manual and
Short notes on error analysis
Other useful materials relating to this course is available in
the course webpage :
http://www.phy.cuhk.edu.hk/course/2013-2014/2/phys1712
• When you come to each lab session, do not come
unprepared!
• Study the lab manual thoroughly before you come
• Write the first page of the lab report before you come
• Here is an e.g. of the first page.
• In the first page of the report you
should state the objective of the
experiment, and describe how the
experiment will be performed.
• Use your own words, don’t merely
copy from the manual!
• Lab opening hours : 2:15 pm – 6:30 pm
• You may come earlier and familiarize yourself with the
equipment, but do not start the experiment before 2:30 pm
• Marks will be deducted if you come after 2:30 pm
• After you arrived in the lab :
• Sign the attendance sheet
• Hand in the first page of the lab report of the present
experiment to the TA. The TA will give it back to you
after checking
• Hand in the lab report of the previous experiment
• Marks will be deducted for late submission
• The graded report will be returned to you in the next lab
session
Marking scheme for the lab
report :
(Also available from the course
webpage)
• Before you come to the lab, download the data sheet from
the course webpage into a memory stick
• You should be able to finish the measurements of each
experiment within 2 hours. Try to finish the calculation and
data analysis in the lab, so that in case you have questions
you may ask the TA. Save you data in your memory stick.
• The TA will stay until 6:30 pm, after which the lab will be
closed
• If you wish, you may stay in the lab until 6:30 pm to write
the lab report, or you can complete the lab report at home.
You are not expected to remain in the lab for any other
business than that related to the experiment
• Before you leave the lab, remember to upload your data,
clean up the table and the computer
The lab report :
• Clear, concise, informative and no-nonsense. About 4
pages excluding the appendices should be adequate
• Hand-written is okay, typing is not necessary. The hand
writing must be clear
• Draw your own diagram (hand-drawn is okay), do not
copy and paste the diagram from the manual or other
sources from the web
• Try to use your own words and avoid copying the
writings from the manual. Use past tense and passive
voice.
Lab regulations :
• You may discuss the experiment with your classmates,
but please speak softly
• No eating or drinking
• The lab is strictly for the purpose of doing experiment,
not for social gathering. Please do not hang around in
the lab after you finished your measurement if you are
not writing your report
Course assessment
• Preparation
Performance in lab
Lab reports
15 %
15 %
70 %
• Different Sessions of the class are graded independently
• You’ll fail the course if you
• Absent without good reasons in 3 labs, or
• Failed to hand in 3 lab reports
• If you absent in an experiment with compelling reasons,
a make-up experiment can be arranged
Any questions?
Taking measurements and analyzing data
• We all make mistakes, especially when we are exploring
the unknown in which no one knows the ‘correct’ answer
• To avoid making careless mistake, use your physicist
intuition from time to time to judge if your data or result
makes any sense (e.g. use an order-of-magnitude
estimation)
• Provided all the careless mistakes are avoided, there
could still be differences between your measurements
and the ‘true’ values – experimental error
• By error we don’t mean a mistake, it means how much we
can trust about a measurement – that is, the uncertainty of
the measurement
• We always present a measurement or result together with
its uncertainty. The magnitude of the uncertainty is
dependent on the method and nature of the measurement.
e.g. If I am given a stick, I can judge by my eyes that the
stick has a length somewhat between 0.8 and 0.9 m. I’ll
present my ‘measurement’ as 0.85 ± 0.05 m
If instead I used a measuring tape, with a marking
down to the millimeter scale, and find the length to be
between the 0.832 and 0.833 m marks. I could present my
measurement as 0.8325 ± 0.0005 m
• The error should have one or at most two non-zero digits.
The no. of significant digits of the data should be
consistent with the size of the error
The following are incorrect
presentations of data :
21.23556 ± 0.2
11.23 ± 0.337543
2.03 ± 0.00032
The correct format
should be :
21.2 ± 0.2
11.23 ± 0.34
2.03000 ± 0.00032
e.g. the elementary charge e has been very accurately
determined to be
e = (1.602176487 ± 0.000000040) × 10-19 C
or, in concise form, e = 1.602176487(40) × 10-19 C
• However, in most of our experiment, we usually have
accuracy up to 1 or 2 digits only .
• Note: a common mistake is to copy all the digits displayed
in a calculator or when using a spreadsheet (e.g. Excel)
Digital equipments are generally more preferable than
analog equipments(e.g. a measuring tape), but can we
trust all the digits as displayed by the equipment?
The manufacturer of the equipment usually states the
accuracy of the equipment in its specification
e.g. According to the specification of the model GDM-8135
digital multimeter, the error of the DC voltage reading is
“(0.1% of reading + 1 digit)”.
Suppose the displayed is 18.5 mV. The uncertainty is
18.5 × 0.1% + 0.1 = 0.12 ≈ 0.1, so the data is (18.5 ± 0.1) mV.
Suppose the displayed is 118.5 mV. The uncertainty is
118.5 × 0.1% + 0.1 ≈ 0.2, so the data is (118.5 ± 0.2) mV.
• In many situations, the error appears in a random fashion.
e.g. Random fluctuation of the equipment due to
temperature, humidity, power supply instability, etc.
Randomness is also an intrinsic nature of some physical
processes, e.g. emissive decay of radioisotopes.
• Random errors can be reduced by repeated measurements.
However, repeated measurements could not help to reduce
instrumental or systematic errors, e.g. parallax error in
reading a scale, mis-calibrated equipment, etc.
• If the error is random, repeated measurements gives us the
• Best estimation of the ‘true’ value – the mean
• Best estimation of the uncertainty – the standard error
Suppose we repeat a measurement n times and obtain the
data: x1, x2, x3 … xn. We can count the occurrence, or
frequency of the measured value that lies within selected
ranges. If we display the data in the form of a histogram
we’ll get sometime like :
40
30
20
10
What can we say about our data? First of all, we can
calculate the mean x of our data by
n
x  (  xi ) / n
i 1
Usually, the mean is the best estimate of the ‘true’ value
from our measurement
We can also calculate the standard deviation  , which is
the averaged deviation from the ‘true’ value. It represents
the spread of our data :
1

n 1

n
i 1
( xi  x )2
The distribution of our measured values can be assumed to
obey the normal distribution if the error is truly random :
0.4
0.3
f ( x)
0.2
0.1
0.0
-3
-2
-1
0
1
2
3
(x  x ) /
1
 ( x  x ) 2 / 2 2
f ( x) 
e
 2
, where



f ( x )dx  1
The area under the curve in region x    x  x   is 0.68.
This means if you take a measurement one more time, you
have a 68% chance to get a measurement x falling within the
interval x  
We estimated the ‘true’ value by the mean x . How reliable is
this estimation? In other word, if we take another set of n
measurements and calculate the mean x , how would it
differ from the previous x ?
It turns out that the various x obtained also follow a normal
distribution, with the standard deviation  x given by

n
1
2
x 

(
x

x
)

i
n(n  1) i 1
n
This  x is a good estimate of the probable error of the
measurement. It is called the standard error.
e.g. Suppose we did an experiment to measure the free-fall
time t of an object over a distance. We obtained the values
5.43 , 5.14 , 5.41 , 5.82 , 5.58 , 5.05
We calculate the mean =
Standard error =
second
1
5.43 + 5.14 + 5.41 + 5.82 + 5.58 + 5.05
6
1
0.025
6(5)
2
+ 0.265
2
+ 0.005
2
+ 0.415
2
+ 0.175
2
= 5.405
+ 0.355
= 0.115
Therefore, we experimentally obtained t = 5.4 ± 0.1 s
This means that there is about 68 % chance that the
‘true’ t value lies between 5.3 and 5.5 s
2
Propagation of error
Suppose we want to measure a quantity p, which is the sum
of two independently measured quantities q and s. If we
measured q = 2.8 ± 0.2 and s = 4.1 ± 0.3, what is p?
Of course we’d have p = q + s = 6.9, but what is its error?
One answer is 0.2 + 0.3 = 0.5, but it may overestimate the
error because it assumed the maximum error occurred at
the same time for both q and s.
It turns out that a better formula to use is
dp =
(0.2)2  (0.3)2 = 0.36  0.4
So that p = 6.9 ± 0.4
There is a general rule to calculate the propagation of error,
which is briefly outlined in the notes. At this stage, we shall
simply employ the formulas for special cases:
1) If k 
m
c p
i 1
then d k 
i
i
, where ci are constant coefficients,
m
c d p 
i
i 1
2) If k 
2
i
m
( pi)
i
(with constant
i 1
then
dk
 d pi 
 ( i ) 

k
i 1
 pi 
m
2
2
i
),
e.g. In a free-fall experiment we wish to find g by
measuring the distance d and the time t and use the
equation d = gt2/2. Suppose d = 1.095 ± 0.001 m and
t = 0.472 ± 0.002 s, what is g from this experiment?
2(1.095)
2d
 9.83
We have g  2 , so that g 
2
(0.472)
t
We also have
Hence
𝛿𝑔
=
𝑔
δg = 9.83 ∙
(1)2
0.001
1.095
We obtain g = 9.83 ± 0.08
2
𝛿𝑑
𝑑
+ (−2)2
2
𝛿𝑡
𝑡
0.002
+4
0.472
2
2
= 0.084
Least square fit to a linear graph
If x and y are related by a linear equation :
y = ax + b
and we wish to find a and b by measuring x and y. The best
way is to vary x and correspondingly measure y to arrive
with a set of data points (xi , yi)
14
12
Distance (cm)
10
8
6
4
2
0
0
2
4
6
8
Time (sec)
10
12
14
16
14
12
Distance (cm)
10
8
6
4
2
0
0
2
4
6
8
Time (sec)
10
12
14
16
Given the set of n data points: (xi,yi), i = 1, 2,…n.
a and b can be obtained by minimizing the mean-square
deviation S, with
n
1
S   ( yi  axi  b)2
n i 1
where yi  axi  b is the deviation of the point (xi, yi)
from the straight line
Examples of using Excel for data analysis …
• Calculate the mean and standard error …
• Draw a histogram of the data …
• Calculate the slope and y-intercept by linear least square fit
• Plot the x-y data …
• Insert error bars