c)
d)
e)
f)
triple-beam balance
digital balance
vernier calipers
micrometer
2. Use the following general error propagation equation to analyze the errors involved
in making calculations involving measurements with their own uncertainty.
Lab Skills: Using
Calipers, and
∂Rulers,
f
∂f Vernier
∂f
σ =
σ +
σ +
σ
∂x
∂y
∂z ∗
Micrometers
General
Error Propagation Equation
2
2
f
2
x
2
2
y
2
z
THEORY
Refer to lab handout on Error Propagation.
1
Using the
Metric Ruler
Using the Metric Ruler
Consider the following
standard
metric
ruler.
Consider
the following
standard
metric ruler.
The ruler is incremented
in units
of centimeters
(cm).TheThe
smallest
scaleis division
is a
The ruler is incremented
in units
of centimeters (cm).
smallest
scale division
a tenth
of a centimeter
or 1 Therefore,
mm. Therefore,
theuncertainty
uncertainty ∆x = smallest increment/2 = 1mm/2 =
tenth of a centimeter
or 1 mm.
the
0.5mm = 0.05cm. Note that a measurement made with this ruler must be stated to a tenth
∆x = (smallest increment) / 2 = 1mm/2 = 0.5mm = 0.05cm.
1
Note that a measurement made with this ruler must be stated to a tenth of a centimeter
since the uncertainty is stated to a tenth of a centimeter. In the example above, the length
of the object is clearly longer than 2.7 cm and less than 2.8 cm. It looks closer to 2.8 cm, so
the value would be stated as x = 2.77 cm ± 0.05 cm.
(You might think it looks more like x = 2.78 cm ± 0.05 cm, that would also be fine, it
is still inside the uncertainty range. Make the best estimate you can.)
2
Using the Vernier Calipers
The Vernier caliper is an instrument that allows you measure lengths much more accurately
than the metric ruler. The smallest increment in the vernier caliper you will be using is
(1/50) mm = 0.02 mm = 0.002 cm. Thus, the uncertainty is
∆x = 12 0.002 cm = 0.001 cm.
The vernier scale consists of a fixed metric scale and a sliding vernier scale. The fixed
scale is divided into centimeters and millimeters, while the vernier scale is divided so that 50
divisions on it cover the same interval as 49 divisions on the main scale (at least this is the
way the ones De Anza has are constructed). Thus, the length of each scale vernier division
∗
These notes are largely drawn from the first lab explanation of Prof Eduardo Luna.
1
The Vernier caliper is an instrument that allows you measure lengths much more accurate
than the metric ruler. The smallest increment in the vernier caliper you will be using is
(1/50)mm = 0.02mm = 0.002cm. Thus, the uncertainty is ∆x = (1/2)0.002 cm = 0.001 cm.
5
1. THE VERNIER SCALE
Equipment List:
C
3 X 5 card
C
one vernier caliper
C
one ruler incremented in millimeters
What you will learn:
This lab teaches how a vernier scale works and how to use it.
The vernier scale consists of a fixed metric scale and a sliding vernier scale. The fixed scale
I. Introduction:
is divided into centimeters and millimeters, while the vernier scale is divided so that 50
A vernier
scale
(Pierre
Vernier,
ca.
can
bemain
used
onThus,
anythemeasuring
device with a
divisions
on itof
cover
the same
interval
as 491600)
divisions
on the
scale.
length
is 49/50 the
length
a main
scale
division.
Close
the jaws
completely
and note that the
of each
scaleoften
vernier division
is 49/50
the length
of a main
division.
Close the jaws
graduated
scale.
Most
athe
vernier
scale
is found
onscale
length
devices such as
first line
atcompletely
the
far and
leftnote
onthat
scale
the
zeroscale
ormeasuring
indextheline)
thevernier
first line at
the far(called
left on the
vernier
(called
“zero”coincides with the
vernier
or micrometers.
A vernier
increases
the
measuring
precision
“index”
line) scale.
coincidesCarefully
with
the zerocompare
lineinstrument
on the main
Carefully
and see division
zerocalipers
line onor the
main
andscale.
see
that
thecompare
first
vernier
is 0.02
that
the first normally
vernier division
is with
0.02 mm
short
of the firstmeasuring
main scale division,
thelike
second
beyond
what
it
would
be
an
ordinary
scale
a
ruler
or
meter
stick.
mm short of the first main scale division, the second vernier division is 0.04 mm away from
vernier division is 0.04 mm away from the second main scale division, and so on. If the
the secondjaws
main
scale opened
division,
andto so
on. fraction
If theofjaws
arescale
slightly
are slightly
it is easy
tell what
the main
divisionopened
the vernierit is easy to tell
indexof
has
moved
byworks:
noting
vernierthe
division
best coincides
with amoved
main scale
what afraction
the
main
scalewhich
division
vernier
index has
by noting which vernier
II. How
vernier
system
division.
division
best coincides
withacross
a maina scale
A vernier
scale slides
fixeddivision.
main scale. The vernier scale shown below in
A measurement
by closing
the jaws on the object
to be divisions on the main
figure 1 is subdivided
sois made
thatwith
tena vernier
of itscaliper
divisions
correspond
to nine
measured and then reading the position where the zero line on the vernier falls on the main
1 main scale divisions
scale.
ten
vernier
divisions
arehow
compressed
intofraction
the
space
ofscale
nine
2.1When
More
details
about
Scale
works
scale.
The
measurement
is incomplete
untilaanVernier
additional
of a main
division
is determined. This
is obtained
by noting
which
line on the vernier
scale
(0,2,4,6,8)
we say the vernier-scale
ratio
is 10:9.
So
the
divisions
on the
vernier
scale are not of a
coincides
best with
a lineaonfixed
the main
scale.scale. The vernier scale shown below in figure 1 is
A vernier scale
slides
across
main
standard
lengthso(i.e.,
but thetodivisions
on theonmain
scale scale.
are always
subdivided
thatinches
ten of or
its centimeters),
divisions correspond
nine divisions
the main
(10/9
As an example, let’s consider measuring the length of the aluminum block below.
some
standard
length
millimeters
orvernier
decimal
inches.are
A compressed
vernier scale
enables
an of nine
is easier
to see
thanlike
49/50.)
When ten
divisions
into
the space
2
unambiguous
between
the smallestratio
divisions
thethe
main
scale.
main scale interpolation
divisions we say
the vernier-scale
is 10:9.onSo
divisions
on the vernier
scale are not of a standard length (i.e., inches or centimeters), but the divisions on the main
scale are always some standard length like millimeters or decimal inches. A vernier scale
enables an unambiguous interpolation between the smallest divisions on the main scale.
Figure 1: A 10/9 Vernier scale.
Figure 1: Ten vernier divisions in the space of nine
main scale divisions, a scale ratio of 10:9.
1
This section is drawn from the first lab explanation of Prof David Newton.
Since the vernier scale pictured above is2 constructed to have ten divisions in the space
of nine on the main scale, any single division on the vernier scale is 0.1 divisions less than a
division on the main scale. Naturally, this 0.1 difference can add up over many divisions. For
example, after six divisions have been spanned by both scales, the difference in length
between the vernier and main scale would be 6 X 0.1 = 0.6 divisions.
Since the vernier scale pictured above is constructed to have ten divisions in the space
of
on 2
thebelow,
main scale,
division
on thescale
vernier
scale
is 0.1 at
divisions
lessAfter
than a
In nine
figure
both any
the single
vernier
and main
start
evenly
the left.
division on the main scale. Naturally, this 0.1 difference can add up over many divisions.
distancea of
six increments they differ in length by 0.6 increments as is indicated by the two
For example, after six divisions have been spanned by both scales, the difference in length
dotted lines.
between the vernier and main scale would be 6 × 0.1 = 0.6 divisions. In figure 2 below, both
the vernier and main scale start evenly at the left. After a distance of six increments they
differ in length by 0.6 increments as is indicated by the two dotted lines.
Figure 2: In the span of six divisions, the difference
between the vernier and main scale is 0.6 divisions.
In practice, the left sides of the two scales are not matched up as above. Instead, the
Figure 2: In the span
of six
divisions,
theofdifference
between
vernier and main scale is
Figure
2:
In
theby
span
six divisions,
thethe
difference
two left sides
of each scale
are
offset
an
amount
corresponding
to the length measured. In
between the vernier and main scale is 0.6 divisions.
0.6 divisions.
figure 3, the two scales are still off by 0.6 divisions as in figure 2 above, however in figure 3
the scales match up along the dotted vertical line on the right side instead of matching up on
When using the Vernier calipers for a measurement, the object in the jaws will cause an
Ininpractice,
left
sides
the
two
scales
areIntwo
not
matched
upscales
as above.
Instead,
the side
left side asoffset
figure
2.theIn
figure
we
would
say
the
dotted
vertical
lines
on
in the two
scales
on
the3,of
left
edge
of the
scales.
figure
3, the two
are
still
off the
by left
twofigure
left0.6
sides
of
each
offset
by an amount
to
the
length
measured.
In in
divisions
as inscale
figure
2 above,
however
figurecorresponding
3 in
thethe
scales
match
up along
dotted
of the
are
separated
byare
0.6
divisions
ofin length
same
sense
thatthe
the
two lines
linescales
on by
the are
right
sideoff
instead
of divisions
matching
up
side
in figure
2.3Intofigure
3,
figure
3,vertical
the
two
still
by 0.6
ason
inleft
figure
2asabove,
however
in figure
3 how
figure
2 are
separated
0.6
divisions.
Study
and compare
figures
2 and
understand
we
would
say
the
two
dotted
vertical
lines
on
the
left
side
of
the
figure
are
separated
by
the scales
match up along the dotted vertical line on the right side instead of matching up on
a vernier
system
0.6 divisions of length in the same sense that the two lines in figure 2 are separated by 0.6
left side as in figure 2. In figure 3, we would say the two dotted vertical lines on the left side
works. divisions. Study and compare figures 2 and 3 to understand how a vernier system works.
of the figure are separated by 0.6 divisions of length in the same sense that the two lines in
figure 2 are separated by 0.6 divisions. Study and compare figures 2 and 3 to understand how
a vernier system
works.
Figure 3: The offset on the left is 0.6 divisions, so now the 6th line aligns exactly with the
Figure 3: A length at the left has a 0.6 division
main scale.
difference also.
Figure 3: A length at the 3
left has a 0.6 division
difference also.
2.2
Making a measurement
A measurement is made with a vernier caliper by closing the jaws on the object to be
measured and then reading the position where the zero line on the vernier falls on the main
scale. The measurement is incomplete until an additional fraction of a main scale division is
determined. This is obtained by noting which line on the vernier scale (0,2,4,6,8) coincides
best with a line on the main scale. As an example, lets consider measuring the length of the
aluminum block below.
Note that the zero line on the vernier scale falls between the 4.4 cm and 4.5 cm mark on the
main scale. Thus, the first significant digits are 4.4 cm. The remaining two digits are
obtained
notingon
which
on the vernier
bestcm
withand
a line4.5
on cm mark on
Note that the
zerobyline
theline
vernier
scale scale
falls(0,2,4,6,8)
betweencoincides
the 4.4
the Thus,
main scale.
closely at the picture
thatThe
the 46remaining
line lines up the
the main scale.
theLooking
first significant
digitsbelow
are indicates
4.4 cm.
two digits are
closest. Therefore, the reading is 4.446 cm. Or in standard form 4.446 cm ± 0.001 cm.
obtained by
noting
thescale
vernier
scale the
(0,2,4,6,8)
coincides
best
Note
that the which
zero line line
on theon
vernier
falls between
4.4 cm and 4.5
cm mark on
the with a line on
main scale.
Thus, the
first significant
are 4.4below
cm. Theindicates
remaining two
digits
are4.6 line lines up the
the main scale.
Looking
closely
at thedigits
picture
that
the
obtained by noting which line on the vernier scale (0,2,4,6,8) coincides best with a line on
closest with
a
mark
on
the
main
scale.
Therefore,
the
reading
is
4.446
cm. Or in standard
the main scale. Looking closely at the picture below indicates that the 46 line lines up the
form 4.446closest.
cm ±Therefore,
0.001 cm.
the reading is 4.446 cm. Or in standard form 4.446 cm ± 0.001 cm.
3
3
4
complete revolution of the circular scale moves the thimble 0.5 mm along the linear scale so that t
distance between the jaws is also changed to 0.5 mm. Since the circular scale has 50 divisions, rotati
it through one circular-scale division, will cause the rod to move through a distance equal to 1/50 of 0
or 0.01 mm. Thus the numbers on the circular scale represent hundredths of a millimeter.
3 Using mm,
The
Micrometer Caliper
micrometer caliper can be used to measure lengths directly to 0.01 mm, and by estimating tenths o
circular scale division, it can be used to estimate lengths to 0.001 mm. Measurements made with
The micrometermicrometer
caliper hascaliper
a linear
engraved
on its sleeveofand
a circularThe
scale
canscale
be estimated
to thousandths
a millimeter.
totalengraved
micrometer caliper readi
on what is properly
called
thereadings
thimble.on The
linear
is divided
is the sum
of the
the main
andscale
circular
scales. into divisions of 1 mm
and is 25 mm long. Half-millimeter marks are provided below the linear scale. The circular
scale has 50 divisions. One complete revolution of the circular scale moves the thimble 0.5
In the figure to the left, notice the main scal
mm along the linear scale so that the distance between the jaws iswhich
also changed
to with
0.5 mm.
is marked
a 0 and a 5. Thes
Since the circular scale has 50 divisions, rotating it through one circular-scale
division,
will the main scal
indicate millimeters. Below
are or
marks
the Thus
half way
cause the rod to move through a distance equal to 1/50 of 0.5 mm,
0.01 atmm.
the points betwee
marks. These
l/2 mm
numbers on the circular scale represent hundredths of a millimeter.mm
A micrometer
caliper
canmarks are presen
of the thimble move
be used to measure lengths directly to 0.01 mm, and by estimatingbecause
tenthsone
of arevolution
circular scale
the thimble only 1/2 mm down the barrel.
division, it can be used to estimate lengths to 0.001-0.002 mm. Measurements made with a
micrometer caliper can be estimated to thousandths of a millimeter. The total micrometer
caliper reading is the sum of the readings on the main and circular scales. In the figure to
Possible settings and readings of the micrometer caliper are shown here. In the first example, t
the left, notice the
main scale which is marked with a 0 and a 5. These indicate millimeters.
sixth main scale mark is visible just to the left of the circular scale. This means we have a readi
Below the main scale
are marks
at the
waymm.
points
mm
marks.
l/2themm
somewhere
between
6.0 half
and 6.5
Thebetween
line on the
main
scale These
points to
barrel about halfw
marks are present
because
one 20
revolution
the thimble
moves thecalliper
thimble
onlyis1/2
between
19 and
divisions. of
Therefore,
the micrometer
reading
6.0 mm + 0.19 mm + 0.0
= 6.195settings
mm = 0.006195
meters. of the micrometer caliper are shown here.
down the barrel. mm
Possible
and readings
In the second example, the half-millimeter mark to the right of the sixth main scale mark is visible. S
In the example,
the half-millimeter
mark 6.5
to the
right
of The
the line
sixth
scale
mark
is to a point of th
the reading
is somewhere between
and 7.0
mm.
on main
the main
scale
points
just is
slightly
past thebetween
41 mark.6.5
So and
we can
the last
of the
reading
visible. So thebarrel
reading
somewhere
7.0estimate
mm. The
lineplace
on the
main
scaleto be 0.002 mm
The of
reading
is thenjust
6.5slightly
mm + 0.41
0.002
mm So
= 6.912
mmestimate
= 0.0069the
12 last
meters. This show
points to a point
the barrel
pastmm
the+41
mark.
we can
that
we
can
estimate
micrometer
readings
to
one
thousandth
of
a
millimeter.
place of the reading to be 0.002 mm. The reading is then 6.5 mm + 0.41 mm + 0.002 mm
= 6.912 mm = 0.0069 12 meters. This shows that we can estimate micrometer readings to
one thousandth of a millimeter. The final value is (6.912 ± 0.005) × 10−3 m.
4
5