Download Measurement

Measurement
Level: Secondary 4 / 6
Project aims:
The main objectives of this project are to let students:
1. Understand the importance of measurement in our daily life.
2.
3.
4.
Explore different methods for measurement.
Understand that measurements can never be exact.
Compare the accuracy of different measuring methods.
Learning objectives:
1.
2.
3.
4.
Understand the importance of measurement in science.
Gain familiarity with a variety of measuring instruments and methods.
Understand that measurements are never exact through error analysis.
Recognize the limitation of measuring instruments used.
Tasks:
In the following project tasks, students are asked to design their own methods for
measuring particular object(s), building(s) and distance(s). They have to explain how
they make the measurements, identify the possible sources of error of each method,
and compare and estimate the accuracies of their measurements.
Task 1: Measuring length
Measure the length of an object (shorter than one foot long recommended), say a
pencil, using a one feet ruler without finer scale. Students may use other tools in their
measurement when necessary. However, the tools should not contain any scale or
mark, which can directly be used to measure the length of the object.
Task 2: Estimating number
Estimate the number of small objects inside a container, say, the number of beans in a
jar without counting the number directly, or to estimate the number of tiles in a floor
which the laboratory is located by measuring the number of tiles per square meter.
Task 3: Measuring height
Measure the height of a given tall building or structure, such as the Lion Rock or the
International Finance Centre in Hong Kong.
Task 4: Measuring distance
Measure the distance between any two particular places, without the use of any
electronic equipment or professional devices
Apparatus/Materials:
Any appropriate apparatus and materials proposed by students are welcomed. Typical
examples include meter ruler (scale covered with paper), identical coins (many),
protractor, beaker, balance, measuring tape, tennis ball (as a falling object), stop watch,
string / rope with a weight, meter wheel and so on.
Pre-knowledge:
Measurement is of central importance in our daily life. It is one of the concrete ways
we deal with our world. Measurement is particularly important in science. Science is
concerned with the description and understanding of nature and measurement is one
of its most fundamental tools. One may say that science cannot exist without
measurement and we start learning physics by learning how to measure a few physical
quantities. Physics attempts to describe nature in an objective way through
measurement.
However, in every measurement, there is uncertainty associated with it – there is no
such thing as a perfectly accurate measurement. There are uncertainties arise from the
measuring instrument, the people doing the measurement, the method of making the
measurement and the characteristics of the systems being measured. We need to learn
to deal with these uncertainties in measurement through error analysis.
Questions:
1.
2.
Which method of measurement is more accurate? Can you account for this?
How could you deal with the difficulty in measurement involving slope and
gradient, like staircase?
Extension:
1.
2.
Measure the thickness of a thin sheet of paper that is too thin to be measured
using normal measuring instrument.
In room temperature and pressure, one mole of air molecules occupies a volume
3.
of 24.5 liter and has a mass of 14.4g. Estimate the mass of air inside a room.
Which is the “heaviest”? The people in a room, the chairs and tables, or the air in
that room?
Let the “cleaning area” of an object be the total area of that object that you wish
to clean. (For example, the top surface of a desk and the inner surface of a
cabinet but not those outside your visibility.) Clean a small area and record the
time needed. Try estimating the time needed to clean the whole cleaning area, the
error in estimating the cleaning area and the cleaning time. Raise out the factors
we haven’t took into account if this value is outside your expectation.
Appendix:
Propagation of error in different forms
(This appendix is adopted from the draft Error Estimation - 誤差估算 released in
October 2005 by Science Education Section of Curriculum Development Institute,
EMB.)
In many cases, a particular result will be obtained by combining a number of
measurements of different quantities in some ways.
For instance, when two independent measurements ( A  A ) and ( B  B ) are
combined to give a result X, the associated error terms X will be given by:
Combinations of A and B
Error of X
X  A B
X  A B
X  A  B
X  AB
X
A B


X
A
B
A
X 
B
X  kA ( k is a constant)
X
A

or X  k A
X
A
X  An
X
A
n
X
A
X  AB n
X
A
B

n
X
A
B
The combinations involve addition and multiplication with more variables or more
terms can be resolved into combinations listed above.
Example 1: Combinations of error involving addition and multiplication
A student finds the constant speed of a moving trolley with a stopwatch. The equation
used is v  s t . The distance s is measured with a meter ruler, the time t is measured
with a stopwatch. Their values are s = (4.01 ± 0.01) m; t = (3.2 ± 0.2) s.
The speed v
= 4.01 / 3.2 ms-1
= 1.253 ms-1 (calculator display)
= 1.3 ms-1
This result is corrected to 2 significant figures since the time measured is
only known to 2 significant figures.
The estimated error in speed, v
 s t 
= v


t 
 s
 0.01 0.2 
= 1.253

 ms-1
4
.
01
3
.
2


-1
= 0.081437 ms (calculator display)
= 0.1 ms-1
Since v has 1 decimal place, the error term ∆v should also has 1 decimal
place, i.e. v = 0.1 ms-1.
Therefore the speed of the trolley is 1.3 ± 0.1 ms-1.
Handling the trigonometric functions of an angle
In the tasks, students may measure an angle and apply trigonometric functions of the
angle for their analysis. Here shows two examples to deal with the tangent of an
angle.
Example 2: Handling the trigonometric functions with multiple measurements
Students used a protractor to measure the angle of elevation to a building and apply
the formula h  d tan  to calculate the height of the building. If
d  (8.23  0.005) m and θ = 27º, 29º, 28º, 31º
Then the average value of height is
h
 27  29  28  31 
 8.23 tan 

4


 4.515m
Since the tangent of an angle is an increasing function, the extreme values
of the height are
hmax  8.235 tan 31 = 4.948m
hmin  8.225 tan 27 = 4.190m
hmax  h = 4.948 – 4.515 = 0.433m
h  hmin = 4.515 – 4.190 = 0.325m
So, maximum error = 0.433m
After rounding off to have the same decimal
h  (4.52  0.43) m
place
with
d,
Example 3: Handling the sine of the angle with single measurement
If a student has only one measurement for the angle, and have result
d  (8.23  0.005) m,   (29  0.5)
Then the mean and extreme values are:
h
 8.23 tan 29 = 4.561m
hmax  8.235 tan 29.5 = 4.659m
hmin  8.225 tan 28.5 = 4.466m
hmax  h = 4.659 – 4.561 = 0.098m
h  hmin = 4.561 – 4.466 = 0.095m
So, maximum error = 0.098m
After rounding off to have the same significant figure with d,
h  (4.56  0.10) m
Further Example
(This method is provided for students with strong mathematical background.)
For any function f(x), we can find the error by
d

 f  x    f x  x
 dx

f x  x 
 f x 
f x 
 f x 
f x  x
x x
x  x x x  x
we have
 f x  1 f x  x   f x  x 

x
2
x
for an infinitesimally small error, that is, x  0 ,
or, for a small value of error,
lim
 f  x  df  x 

x
dx
d

 f  x    f x  x
 dx

Example 4: Handling error by calculus
A student measures the angle of elevation of a building. He finds the angle is
  34  0.5 . Find the value of sin  and its error.
34  34 

180
0.5  0.5 
rad  0.593

180
rad  0.00872
sin   sin 0.593  0.559
 d

sin    
sin     cos 0.593  0.00872  0.00723
 d

sin   sin    0.559  0.007
* Notice that we should use radian scale because we are using calculus.