Unit 3 – Essential Mathematics 40S

Measuring Devices
 In this learning experience you will explore the use of various
measuring tools such as a ruler, Vernier caliper and micrometer.
 Even though you may have used some or all of these tools in a
previous course, it is important that you review the features of
these tools before you consider their appropriate applications
and possible limitations.
 These measurement instruments have either metric or imperial
units or both. You should already be familiar with measurements
using a ruler. A ruler's level of accuracy is usually to the nearest
1/10 or 1/16 of an inch or 1/10 of a centimetre (millimetre).
However, when constructing intricate items such as an airplane
engine, measurements must often be made to the nearest
1/1000" or even 1/10 000". Without such exact measurements,
these engines would not run efficiently or last long, and some
may not function at all. Technicians or engineers who build these
engines may use, among other instruments, Vernier calipers and
micrometers to make precise measurements.

Vernier Caliper
 The vernier caliper is an instrument used in
precision measurement.

Vernier Caliper
 Examine the Vernier Caliper's various scales on the
diagram above.
 The fixed scale does not move and looks like a ruler.
The moving (or Vernier) scale moves.
 There are one or two systems of measurement:
imperial and/or metric. The imperial scale can be in
decimal or fraction form and the metric scale is decimal.
 This instrument measures in three ways:
 outside jaws — to measure outer dimensions of objects, e.g.,
outer diameter of a pipe
 inside jaws — to measure inner dimensions of objects, e.g.,
inner diameter of a pipe
 stem or depth gauge — to measure depth of objects, e.g, depth
of a small container

Vernier Caliper
 Examine the Vernier Caliper's various scales on the
diagram above.
 The fixed scale does not move and looks like a ruler.
The moving (or Vernier) scale moves.
 There are one or two systems of measurement:
imperial and/or metric. The imperial scale can be in
decimal or fraction form and the metric scale is decimal.
 This instrument measures in three ways:
 outside jaws — to measure outer dimensions of objects, e.g.,
outer diameter of a pipe
 inside jaws — to measure inner dimensions of objects, e.g.,
inner diameter of a pipe
 stem or depth gauge — to measure depth of objects, e.g, depth
of a small container

Metric Scale
 The fixed scale is divided into millimetres. Each
millimetre can be further subdivided into smaller units
(the same amount as indicated on the vernier). The
vernier is broken into 10 units. Each represents 1/10 of
1 mm or 0.1 mm.

Vernier Caliper Simulation

Notebook Assignment
Page 227 - 229
Q. 1 - 4

Micrometers
 A micrometer measures small lengths, such as
the diameters of pipes, rods, wires, bolts, and
washers. The following picture depicts the key
parts of a micrometer.

How to Use A Micrometer

How to Use A Micrometer
 Note the following parts of the instrument:
 jaws: anvil and spindle (to measure small
lengths
 two measurement scales
 — a scale on the barrel (fixed scale)
 — a scale on the thimble (moving scale

How to Use A Micrometer
 The micrometer will either measure in metric or
imperial units
 In the micrometer examples used, one
complete rotation of the thimble advances the
thimble 0.5 mm on the fixed scale.
 Since the thimble is marked with 50 divisions,
each division represents

Micrometer Simulation

Notebook Assignment
Page 235 - 237
Q. 1 - 5

Advantages and Limitations of
Measuring Devices
 Vernier Caliper:
The most distinct advantage of the Vernier caliper is
that it can provide very accurate measurements over a
large range. In general, pocket models can measure
from 0 to 3 inches, however, sizes are available all the
way to 4 feet. It can be used for both internal and
external surfaces. The most obvious limitation to using
a Vernier caliper is that they are not always easy to
read and operate. In order to achieve a highly reliable
measurement, the caliper must be properly aligned and
used correctly.

Advantages and Limitations of
Measuring Devices
 Micrometer:
This instrument is often used by mechanics and
mechanical engineers for sensitive measurements.
They are very accurate, can be reset for zero, most
have friction lock, and over tightening protection. Like
the caliper they are not always easy to read and
operate. Every instrument is manufactured for special
needs so they may function in a slightly different way.

Practice
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 PRECISION MEASUREMENT
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Worksheet #1

Practice
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Accuracy vs Precision
 You can examine a measurement in terms of
either its accuracy or its precision. In everyday
language the terms accuracy and precision are
often used interchangeably. However, in
mathematics and science a distinction is made
between the two terms. In this lesson, you will
discover the difference between accuracy and
precision.

Accuracy vs Precision

Accuracy vs Precision
 The term accuracy can be defined as how
close a measurement comes to its true value.
Precision is how close the measured values
are to each other.
 Consider the following example of Accuracy vs
Precision :

Accuracy vs Precision
 On the first target, all the shots are close to the centre
so the average accuracy is good. Since all five shots
are quite spread out the precision is poor.
 On the second target, all the shots are far away from
the centre so the average accuracy is poor. Since all
five shots are quite close together, the precision is
good.
 On the third target, all the shots are close to the centre
so the average accuracy is good. Since all five shots
are quite close together the precision is also good.
 So, if you are playing a sport like soccer and you
always hit the left goal post instead of scoring, then you
are not accurate, but you are very precise!

Accuracy and Precision With
Measuring Devices
 The term accuracy is defined as how close a
measurement comes to its true value. So, the more
closely a measuring device comes to an actual
measure, the more accurate its measurements can be.
 The term precision can also be defined as the degree
of agreement among several measurements made with
the same instrument in the same way. The precision of
a measuring instrument is usually stated as the
smallest scale division of measurement. The smallest
scale division on a measuring device is taken to be the
uncertainty of the device. The greater the number of
divisions of a measuring device, the more precise it is.

Accuracy and Precision With
Measuring Devices
 A common metric ruler that is divided into
millimetres is said to have a precision of 1
millimetre. A common imperial tape measure
that is divided into 1/16th of an inch is said to
have precision of 1/16th of an inch. The metric
Vernier caliper has a precision of one-tenth of a
millimetre. The metric micrometer is even more
precise with a precision of 1/100th of a
millimetre.

Accuracy and Precision With
Measuring Devices
 Is it possible to have a device that is precise but
not accurate? Consider a tape measure that is
used to measure a certain length. This tape
measure is precise to 1/8th of an inch. If you lay
this tape measure down to measure an item but
it is not laid down flat, the measure would not
be accurate because it would not measure
close to the item's true value.

Precision Measurement of
Objects
Example 1:
 Measure the black line below the ruler in
imperial measure. How precise is the
measurement?
Sample solution:
 The tape measure is divided into eighths of an
inch. This line measures 3 and 5/8th of an inch.
The measurement is precise to the nearest
eighth of an inch.

Precision Measurement of
Objects
Example 2:
 The black lines on the ruler below are the same
length. Measure each line in metric and
imperial units. How precise are the
measurements?

Precision Measurement of
Objects
Sample solution:
 The top part of the ruler is in imperial measure using inches. Each
inch is divided up into 1/16th of an inch. Reading the top line it is 2
and 11/16th of an inch. The measurement is precise to the nearest
sixteenth of an inch.
 The bottom part of the ruler is metric measure using centimetres.
Each centimetre is broken up into 1/10th of a centimetre. Reading
the bottom line it is 6 cm and 8/10th of a cm. The measurement is
precise to the nearest tenth of a cm.
 In some cases you may be interested in a level of precision that is
actually less than what is indicated by your measuring device.

Precision Measurement of
Objects
Example 3:
 Measure the length of each line precise to the
nearest indicated unit, using the imperial ruler
given.
a) ½ inch b) ¼ inch c) 1/8 inch

Precision Measurement of Objects

Precision Measurement of
Objects
Example 4:
 A homeowner is planning to build a fence
around her property. The dimensions of the
property, precise to the nearest metre are 65 m
by 32 m. How precise are the dimensions?
What is the perimeter of her property?

Precision Measurement of
Objects
Solution:
 The dimensions are precise to the nearest metre. The perimeter of
the distance around the property is 65 + 32 + 65 + 32 = 194 m.
Note that in this example, the length of 65 m is precise to the
nearest metre which means that the length could be as small as 64
m or as large as 66 m. The width could be as small as 31 m or as
large as 33 m.
Taking the lower values gives a perimeter of 190 m and the larger
values gives a perimeter of 198 m. This means the actual
perimeter could be anywhere between 190m and 198m. Because
calculations involving measurements may introduce more
uncertainty, the answer is sometimes less precise than the original
measurements.

Precision Measurement of
Objects
Example 5:
 A photo has the dimensions 7.8 cm by 9.4 cm.
 Calculate the area using the given
dimensions.
 State the precision of each dimension.
 Calculate the least possible area.
 Calculate the greatest possible area.
 Would the area found in part (a) be precise to
the nearest hundreth of a cm?

Precision Measurement of
Objects
Solution:
 Using the given dimensions, the area is 7.8 cm x 9.4 cm = 73.32
cm2
 Each dimension is precise to the nearest tenth of a centimetre.
 The shortest length could be 7.7 cm and the shortest width could
be 9.3 cm. The least possible area could be 7.7 x 9.3 = 71.61 cm2.
 The longest length could be 7.9 cm and the longest width could be
9.5 cm. The greatest possible area is 7.9 cm x 9.5 cm = 75.05 cm2
 No, the area in part (a) would not be precise to the nearest
hundreth of a cm. This example illustrates how calculations can
introduce less precision. Although the original dimensions are
precise to the nearest tenth of a centimetre, the area could be as
small as 71.61 cm2 or as large as 75.05 cm2. This tells us that the
area is less precise than the nearest tenth of a cm, so it is definitely
less precise than the nearest hundreth of a centimetre.

Practice
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PRECISON vs elit. Vivamus et
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Worksheet #2
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Extra Worksheets for Practice

Practice
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Tolerance Levels
 In this lesson you will learn why the concept of
tolerance is important and how to calculate the
tolerance level in different situations. When
objects are manufactured, none are exactly the
same size. This is because of errors in
manufacturing that cannot be totally overcome.
As a result, designers allow for error, but only to
a set limit. This is called tolerance level.

Definition of Tolerance
Levels
 Tolerance levels are the limited permissible
variations in the physical dimensions in a certain
situation.
 When carpenters construct a house, the tolerance
level for the measurements will likely be higher
than when manufacturing equipment for use in the
medical industry. Medical devices, such as a blood
pressure meter, must have a low tolerance level for
variations in accuracy. Even though measurement
accuracy is important for a house, the very fact that
a measurement may be off by a small amount will
not affect the function of the house.

Definition of Tolerance
Levels
 Some measurements may have tolerance
levels of ±¼ inch or as much as ±5 inches,
whereas others may allow for more or less
tolerance. The aerospace industry will likely call
for the small level (± 0.001 mm) because most
of the equipment requires a high degree of
precision to operate under high stress for long
periods of time.

Finding Tolerance Levels
 Tolerance limits are usually given as the maximum limit
and minimum limit. Other terms, such as upper and
lower limit may also be used. These limits provide the
values that the physical dimensions should not go
beyond.
Example 1:
 A measurement calls for 8.231 ± 0.002. What are the
measurement's upper and lower limits?
Solution:
The upper limit is 8.231 + 0.002 = 8.233 and lower limit is 8.231 −
0.002 = 8.229.

Finding Tolerance Levels
Example 2:
 What are the maximum and minimum limits of a
measurement of 92" ± ¼"?
Solution:
 The maximum limit is 92 + ¼ = 92 ¼" and the minimum
limit is 91 − ¼ = 91 ¾".
Example 3:
 Find the tolerance limits for the length of an item
measuring 30.0 ± 0.1 mm.
Solution
 Upper limit: 30.0 + 0.1 = 30.1 mm
Lower limit: 30.0 − 0.1 = 29.9 mm

Finding Tolerance Levels
 Tolerance is calculated by taking the maximum limit
minus the minimum limit. This gives the total
permissible variation in size for a given dimension.
Example 1:
 A measurement calls for 8.231 ± 0.002. What would the
tolerance be?
Sample Solutions:
 The maximum limit is 8.233 and minimum limit is 8.229. So, the
tolerance would be 8.233 − 8.229 = 0.004.
Did you notice that the tolerance is equal to twice the value that
is added or subtracted? In this example you found that the
tolerance for the measurement 8.231 ± 0.002 is 0.004. This can
also be found by doubling 0.002, 0.002 x 2 = 0.004.

Finding Tolerance Levels
Example 2:
 What is the tolerance for the measurements 92" ± ¼"?
Sample Solutions:
 The maximum limit is 92 ¼" and the minimum limit is 91
¾". So, the tolerance would be 92 ¼" - 91 ¾" = ½".
OR
2(1/4) = 2/4" = ½"

Finding Tolerance Levels
Example 3:
 Find the tolerance for the length of an item measuring
30.0 ± 0.1 mm.
Sample Solutions:
 Upper limit: 30.0 + 0.1 = 30.1 mm
Lower limit: 30.0 − 0.1 = 29.9 mm
Tolerance = 30.1 − 29.9 = 0.2 mm
OR
2(0.1mm) = 0.2mm

Effect of Tolerance on Area and
Volume
 Tolerance levels will affect calculations such as area,
surface area, and volume, which will result in their own
tolerance values. These values are important because
they affect the amount of material required to build the
objects. The calculations for the tolerance levels of
area, surface area, and volume are performed with the
maximum and minimum limit sizes, using the same
formulas as you would for any linear measurement.

Effect of Tolerance on Area and
Volume
 Example 1:
Find the tolerance level and the upper and lower limits
on the area of the given rectangle.

Effect of Tolerance on Area and
Volume
 Example 2:
The diagram below shows a mould for casting an
object. The casting is a rectangle (A) with a square hole
(B). The diagram appears to only be 2-dimensional but
notice the item has a depth labelled as well. Find the
upper and lower limits of the volume of material that will
be required and the tolerance.

Practice
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Extra Worksheets for Practice

Practice
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