Measurements : Appendix I
Measurements : Appendix IThe VERNIER Scale A Vernier scale is a
small, moveable scale placed next to the main scale of a measuring
instrument. It is named after its inventor, Pierre Vernier (1580 -
1637). It allows us to make measurements to a precision of a small
fraction of the smallest division on the main scale of the
instrument. (In the first example below the "small fraction" is one
tenth.) Vernier scales are found on many instruments, for example,
spectroscopes, supports for astronomical telescopes etc. One
specific example, the Vernier calliper, is considered below.
Using a Vernier ScaleFigure 1 shows a Vernier scale reading
zero. Notice that 10 divisions of the Vernier scale have the same
length as 9 divisions of the main scale.
figure 1
In the following examples we will assume that the smallest
division on the main scale is 1mm so the divisions on the Vernier
scale are 09mm each. The position of the zero of the Vernier scale
tells us the number of cm and mm in our measurement. For example,
in figure 2, the reading is a little over 12cm.
figure 2
To find a more precise reading, consider figure 3 (which is a
magnified view of part of figure 2).
figure 3
We are, in effect, trying to find the distance, x.
To find x, find the mark on the Vernier scale which most nearly
coincides with a mark on the main scale. In figure 3 it is
obviously the third mark.
Now, it is clear that ............x = d - d
Remembering that each division on the main scale is 1mm and that
each division on the Vernier scale is 09mm, we have:
x = 3mm - 3(09)mm = 3(01)mm
Therefore, the reading in the example is: 123cm
Similarly, if it had been, for example, the seventh mark on the
Vernier scale which had been exactly opposite a mark on the main
scale, the reading would be: 127cm
Hence, the level of precision of an instrument which has a
Vernier scale depends on the difference between the size of the
smallest division on the main scale and the size of the smallest
division on the Vernier scale.
In the example above, this difference is 01mm so measurements
made using this instrument should be stated as: reading 01mm.
Another instrument might have a scale like the one shown in
figure 4.
figure 4
Therefore, the precision is: 1mm - (49/50)mm = (1/50)mm =
002mm.
Results of measurements made using this instrument should
therefore be stated as: reading 002mm.
This principle is used in the Vernier calliper shown below.
The diagrams below illustrate how to use a Vernier calliper to
measure:
A. the internal diameter of a hollow cylinderB. the external
dimensions of an objectC. the depth of a hole in a piece of
metal.
A.B.
C.
Links to other pages about measurements:
Introduction
How many Decimal Places?
How does an uncertainty in a measurement affect the FINAL
result?
Using the vernier calipers and micrometer screw gaugeThe
precision of length measurements may be increased by using a device
that uses a sliding vernier scale. Two such instruments that are
based on a vernier scale which you will use in the laboratory to
measure lengths of objects are the vernier callipers and the
micrometer screw gauge. These instruments have a main scale (in
millimetres) and a sliding or rotating vernier scale. In figure 1
below, the vernier scale (below) is divided into 10 equal divisions
and thus the least count of the instrument is 0.1 mm. Both the main
scale and the vernier scale readings are taken into account while
making a measurement. The main scale reading is the first reading
on the main scale immediately to the left of the zero of the
vernier scale (3 mm), while the vernier scale reading is the mark
on the vernier scale which exactly coincides with a mark on the
main scale (0.7 mm). The reading is therefore 3.7 mm.
Figure 1 : The reading here is 3.7 mm.
Figure 1 : The reading here is 15.8 mm.
This Java applet will help you to understand how to read a
vernier scale.
The vernier calipers
The vernier calipers found in the laboratory incorporates a main
scale and a sliding vernier scale which allows readings to the
nearest 0.02 mm. This instrument may be used to measure outer
dimensions of objects (using the main jaws), inside dimensions
(using the smaller jaws at the top), and depths (using the
stem).
Figure 3: The vernier calipers
To measure outer dimensions of an object, the object is placed
between the jaws, which are then moved together until they secure
the object. The screw clamp may then be tightened to ensure that
the reading does not change while the scale is being read.Watch
this short movie to see how to do this.
Here is a nice vernier calipers applet.
The first significant figures are read immediately to the left
of the zero of the vernier scale and the remaining digits are taken
as the vernier scale division that lines up with any main scale
division.
Some examples: Note that the important region of the vernier
scale is enlarged in the upper right hand corner of each
figure.
Figure 4: The reading is 37.46 mm.
In figure 4 above, the first significant figures are taken as
the main scale reading to the left of the vernier zero, i.e. 37 mm.
The remaining two digits are taken from the vernier scale reading
that lines up with any main scale reading, i.e. 46 on the vernier
scale. Thus the reading is 37.46 mm.
Figure 5: The reading is 34.60 mm.
In figure 5 above, the first significant figures are taken as
the main scale reading to the left of the vernier zero, i.e. 34 mm.
The remaining two digits are taken from the vernier scale reading
that lines up with any main scale reading, i.e. 60 on the vernier
scale. Note that the zero must be included because the scale can
differentiate between fiftieths of a millimetre. Therefore the
reading is 34.60 mm.
Figure 6: The reading is 40.00 mm.
In figure 6 the zero and the ten on the vernier scale both line
up with main scale readings, therefore the reading is 40.00 cm.
Try the following for yourself.
Figure 7: Click here for the answer.
Figure 8: Click here for the answer.
Figure 9: Click here for the answer. The micrometer screw
gauge
The micrometer screw gauge is used to measure even smaller
dimensions than the vernier callipers. The micrometer screw gauge
also uses an auxiliary scale (measuring hundredths of a millimetre)
which is marked on a rotary thimble. Basically it is a screw with
an accurately constant pitch (the amount by which the thimble moves
forward or backward for one complete revolution). The micrometers
in our laboratory have a pitch of 0.50 mm (two full turns are
required to close the jaws by 1.00 mm). The rotating thimble is
subdivided into 50 equal divisions. The thimble passes through a
frame that carries a millimetre scale graduated to 0.5 mm. The jaws
can be adjusted by rotating the thimble using the small ratchet
knob. This includes a friction ?clutch? which prevents too much
tension being applied. The thimble must be rotated through two
revolutions to open the jaws by 1 mm.
Here is a useful applet to learn how to use the micrometer
screwgauge.
Figure 10: The micrometer screw gauge
In order to measure an object, the object is placed between the
jaws and the thimble is rotated using the ratchet until the object
is secured. Note that the ratchet knob must be used to secure the
object firmly between the jaws, otherwise the instrument could be
damaged or give an inconsistent reading. The manufacturer
recommends 3 clicks of the ratchet before taking the reading. The
lock may be used to ensure that the thimble does not rotate while
you take the reading. Watch this short movie to see how to do
this.
The first significant figure is taken from the last graduation
showing on the sleeve directly to the left of the revolving
thimble. Note that an additional half scale division (0.5 mm) must
be included if the mark below the main scale is visible between the
thimble and the main scale division on the sleeve. The remaining
two significant figures (hundredths of a millimetre) are taken
directly from the thimble opposite the main scale.
Figure 11: The reading is 7.38 mm.
In figure 11 the last graduation visible to the left of the
thimble is 7 mm and the thimble lines up with the main scale at 38
hundredths of a millimetre (0.38 mm); therefore the reading is 7.38
mm.
Figure 12: The reading is 7.72 mm.
In figure 12 the last graduation visible to the left of the
thimble is 7.5 mm; therefore the reading is 7.5 mm plus the thimble
reading of 0.22 mm, giving 7.72 mm.
Figure 13: The reading is 3.46 mm.
In figure 13 the main scale reading is 3 mm while the reading on
the drum is 0.46 mm; therefore, the reading is 3.46 mm.
Figure 14: The reading is 3.56 mm.
In figure 14 the 0.5 mm division is visible below the main
scale; therefore the reading is 3.5 mm + 0.06 mm = 3.56 mm.
Try the following for yourself.
Figure 15: Click here for the answer.
Figure 16: Click here for the answer.
Figure 17: Click here for the answer.
Taking a zero reading
Whenever you use a vernier calipers or a micrometer screw gauge
you must always take a ?zero reading? i.e. a reading with the
instrument closed. This is because when you close your calipers,
you will see that very often (not always) it does not read zero.
Only then open the jaws and place the object to be measured firmly
between the jaws and take the ?open? reading. Your actual
measurement will then be the difference between your ?open? reading
and your ?zero? reading.
Recording the result of your vernier measurement
Let us say you take a reading with an object between the jaws of
a vernier calipers and you see the following:
Say that you decide that the best estimate of the reading l 1 is
37.46 mm.
What about the standard uncertainty u(l1) in this reading?
Using a triangular probability density function, you might
decide that you are 100% sure that the reading is not 37.42 mm and
100% sure that the reading is not 37.50 mm.
Then mm = 0.0163 mm
When you remove the object and read the vernier calipers with
the jaws closed, you might decide that the best estimate of the
"closed" reading l0 = 0.04 mm with standard uncertianty u(l 0) =
0.0204 mm
What should you then record as the best estmate of the length of
the object you are measuring?
The best estimate of the length l = l 1 - l0 = 37.46 - 0.04 =
37.42 mm
with a standard uncertainty = = 0.0261 mm
Therefore l = 37.420 0.026 mm (65% level of confidence).
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