Measurement

In this presentation you will: investigate different instruments for measuring length explore the meaning of the terms accuracy, reliability

and error

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Measurement

In this presentation, you will explore how different measuring instruments measure with different degrees of accuracy.

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You will also see how all measurements are subject to errors and that these errors can be estimated.

Measurement

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Choice of Measuring Instrument

It would not be sensible to try to measure the thickness of a human hair with a ruler marked in millimeters.

Neither would it be sensible to measure the distance around the planet with a ruler.

For the purpose of this presentation we will be concentrating primarily on the measurement of length.

All measurements need to be made with an appropriate measuring instrument.

Measurement

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Measuring Length

Length could also be distance, height, width, depth, thickness, breadth, perimeter, radius, circumference, diameter and several others.

If we concentrate on lengths less than 1 m, our first useful piece of equipment is a meter rule.

If a measurement is made with a ruler, it should be stated to the nearest millimeter.

Rulers are usually marked in centimeters with a smaller mark for every millimeter.

Alternatively, for lengths less than 30 cm, a 30 cm ruler can be used instead.

Measurement

1

Which of the following would not be measured in meters?

Question

A) Perimeter

B) Radius

C) Height

D) Area

Measurement

10 cm

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Measuring Length

Suppose a block of wood is exactly 10 cm long when measured with a ruler. How do we write it down?

Here are some possibilities:

Which of the above is (or are) correct?

• 10 cm• 10.0 cm• 100 mm• 0.1 m• 0.10 m

Two answers are correct, and three are not correct because they are not accurate enough.

Measurement

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Measuring Length

If we can measure to the nearest millimeter then we must show that in our answer.

10 cm shows a measurement to the nearest centimeter, 10.0 cm shows it to the nearest millimeter.

You may have been taught that you can leave off zeros after a decimal point. That’s true for math – but not for science.

The number of zeros after a decimal point tells us about the accuracy of the measurement.

Measurement

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Equivalent Units

We have eliminated one answer (10 cm) and approved one answer (10.0 cm). What about the other answers?

All the other answers use units other than centimeters. You need to know the relationships between units to work out the other correct answer.

There are 10 mm in 1 cm and 100 cm in 1m.

Therefore 10.0 cm is equivalent to 100 mm and 0.100 m.

The other correct answer is therefore 100mm.

Measurement

2

A meter rule has a thousand graduations on it. What does each graduation represent?

Question

A) 0.5 mm

B) 1 mm

C) 5 mm

D) 1 cm

Measurement

3

You have measured an exact length of 5 cm with a ruler graduated in millimeters. Which of the following shows how it would be written to show the accuracy of the measurement?

Question

A) 5 cm

B) 5.0 cm

C) 5.00 cm

D) 5.000 cm

Measurement

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Preferred Units

In the international metric measurement system, known as the SI system, the standard unit of length is the meter.

Other units that are allowed to be used are shown in the table.

You will notice that these preferred units are always 1000 times larger or smaller than each other.

That means that the centimeter is neither a standard unit nor a preferred unit. For this reason it is only occasionally used in science.

Name Symbol Factor Power of 10

nanometer nm 0.000000001 1 × 10-9

micrometer

µm 0.000001 1 × 10-6

millimeter mm 0.001 1 × 10-3

meter m 1 1

kilometer km 1,000 1 × 103

Measurement

4

Which of the following is equivalent to a length of 10 mm?

Question

A) 0.00010 m

B) 0.0010 m

C) 0.010 m

D) 0.10 m

Measurement

5

Which of the following is NOT a preferred unit of length?

Question

A) mm

B) cm

C) nm

D) km

Measurement

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Standard Form

Standard form is a way of expressing very large or very small numbers more conveniently, using powers of ten.

Let’s consider how we would show a measurement such as 54 mm in standard form.

For example, we could use it to express, in meters, a small length that was measured in millimeters.

5.73 × 10-2

6.25 × 103

1.09 × 10-8

9 × 104

2.94 × 103

Measurement

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Standard Form

Large numbers can also be written in standard form.

Standard form requires that the number before the multiplication sign is less than ten. So, although we can write 54 mm as 54 × 10-3 m, it becomes 5.4 × 10-2 m when put into standard form.

For example, 9500 m can be written as 9.5 × 103 m.

If you remember that there are 1000 mm in 1 m, then 1 mm is 0.001 m. This can be written in standard form as 1 × 10-3 m.

1 mm = 1 × 10-3 m

54 mm = 5. 4 × 10-2 m

9500 m = 9.5 × 103 m

Measurement

6

Which of the following shows the number 0.000542 in standard form?

Question

A) 54.2 × 10-4

B) 5.42 × 10-5

C) 542.0 × 10-6

D) 5.42 × 10-4

Measurement

7

Which of the following shows 64920 expressed in standard form?

Question

A) 6.492 x104

B) 64.92 x103

C) 0.6492 x105

D) 64920 x10-4

Measurement

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Errors

We can use a ruler to measure the length of an object to an accuracy of 1 mm.

That gives us a total possible error of 1 mm, the size of the smallest graduation.

If we were stating the error, we could say that the measurement is

54 mm ± 0.5 mm.

54 mm

If the object’s length is measured as 54 mm, this means that the actual length of the object may be anywhere between 53.5 mm and 54.5 mm.

Measurement

8

A dimension in a drawing is given as 35 mm ± 1.5 mm. What is the lowest acceptable value of this dimension?

Question

A) 33.5 mm

B) 35.0 mm

C) 36.5 mm

D) 38.0 mm

Measurement

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Percentage Errors

Knowing an absolute error allows us to calculate the error as a percentage, which is useful when comparing errors.

If we have a total possible error of 1 mm in a reading of 54 mm, then the percentage error is calculated as follows:

You should be able to see that the percentage error will increase as the length being measured with a ruler gets smaller.

54 mm

1 mm of error

% Error = x 100%

= 1.85%

1

54

Measurement

12 mm

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Percentage Errors

If our reading is 12 mm then the percentage error is:

This is becoming too large and we need to look at other ways of measuring small distances.

Note that this has nothing to do with any inaccuracy on the part of the observer. This is an intrinsic error due to the use of a ruler graduated in millimeters.

% Error = x 100%

= 8.33%

1

12

Measurement

9

What is the percentage error of a 1 mm error in a reading of 20 mm?

Question

A) 1%

B) 5%

C) 10%

D) 20%

Measurement

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Calipers

The different types of calipers with scales can be summarized as:

These get increasingly accurate – and increasingly expensive.

These calipers can measure:• length• internal diameter• external diameter• depth

• Vernier• Dial• Electronic

The next instrument we should consider for the measurement of small distances is the caliper.

Measurement

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Vernier Calipers

A vernier caliper uses a vernier scale to get extra precision.

It has ten graduations which are slightly displaced from the scale on the fixed body.

Only one of these marks lines up exactly with a mark on the fixed scale.

A vernier scale is an extra scale that sits on a moveable slider.

The position of this mark gives an extra figure of precision.

Main scale reading, opposite the vernier’s zero, is 4.6

Vernier reading, where lines match up, is 7

Final reading is 4.67 cm

Measurement

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Dial Calipers

A dial caliper uses an extra scale on a dial to get extra precision.

In either case it gives two extra figures to the precision.

The needle on the dial will rotate either one full revolution per millimeter or possibly one half revolution per millimeter.

Measurement

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Electronic Calipers

An electronic caliper replaces the dial with electronic sensing to gain extra precision.

The reading is taken directly from the scale.

As with all calipers, it is important to check that the reading is zero when the jaws are touching each other.

Measurement

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The Micrometer

The final measuring instrument to be examined is the micrometer.

The thimble is turned to open the jaws. The object to be measured is then placed between the jaws, and the jaws are closed. When nearly closed, it is advisable to use the ratchet to prevent over-tightening of the jaws.

The micrometer is usedto measure very small lengths – down to fractions of a millimeter.

The reading is either taken from the barrel and rotating scale or from the electronic display, depending upon which type of micrometer is being used.

Ratchet

ThimbleJaws

Measurement

10

Which of the following is a picture of a dial caliper?

Question

A)  

B)  

C)  

Measurement

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Showing Errors

In some experiments it can be helpful to show the degree of error involved in a measurement.

When drawing a graph it is possible to show the error by using a bar instead of a point. The length of the bar shows the degree of error.

Length in mm

Error

125.0 ±0.5 mm

198.0 ±0.5 mm

254.0 ±0.5 mm

An extra column or row could be added into a table to accommodate this.

Measurement

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Showing Errors in Calculations

If you have calculated errors then it is useful to use these errors within calculations.

The general rule is that errors are added.

Suppose you have measured three sides of a cube as:

The percentage error is:

1.5 × 10-3 m ± 0.05 × 10-3

x 100%

= 6.66%

0.1 x 10-3

1.5 x 10-3

Measurement

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Showing Errors in Calculations

To find the volume we multiply the lengths together.

However, to find the percentage error, the individual errors are added together.

The percentage error of the volume is:

The percentage error is therefore almost 20%.

(1.5 ×10-3 m) x (1.5 ×10-3 m) x (1.5 ×10-3 m)

= 3.375 × 10-9 m3

6.66 + 6.66 + 6.66 %

= 19.98%

Measurement

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Showing Errors in Calculations

It follows from this that it is important to reduce all errors in measurements, particularly if the figures are then to be used in calculations.

This can be achieved by always using the most suitable measuring instruments.

Measurement

Summary

After completing this presentation you should be able to:

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show knowledge and understanding of a range of instruments used to measure length

show knowledge and understanding of errors and percentage errors

show knowledge and understanding of the way errors can increase when measurements are used in calculations