PHYSICAL QUANTITIES, UNITS & MEASUREMENT

Scalars and vectors, Measurement techniques, Units and symbols

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Goal

Define physical quantity Differentiate base quantities and derive

quantities State the base unit Understand prefixes

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Physical Quantities

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A physical quantity is one that can be measured and that consists of numerical magnitude and unit.

Physical Quantity

Base Quantity

Derived Quantity

Base Quantities & Units

There are 7 base quantities

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length

mass

timetemperature

luminous intensity

amount of substances

electric current

Derive Quantities & Units

All other quantities are derived from this base quantities

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Derived quantity

Relation with base and derived quantities

Symbol for unit

Special name

volume length × width × height

m3

density mass ÷ volume kg m3

Speed distance ÷ time m s-1

acceleration

change in velocity ÷ time

m s-2

force mass × acceleration kg m s-2 (N) newton (N)

pressure force ÷ area kg m-1 s-2 (N m-2)

pascal (Pa)

work force × distance kg m2 s-2 (N m)

joule (J)

power work ÷ time kg m2 s-3 (J s-1) watt (W)

Prefixes

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When we measure quantities which are very big or very small, it is difficult to write down the measured values.

The chances of making mistakes may be quite high.

We uses prefixes to simplify the writing of such numbers.

Some prefixes of SI units

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Prefix Abbreviation

Power

pico p 10-12

nano n 10-9

micro 10-6

milli m 10-3

centi c 10-2

deci d 10-1

kilo k 103

mega M 106

giga G 109

tera T 1012

Example

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1. Rewrite the following quantities using suitable prefixes

1. 5 000 000 J2. 48 000 g3. 0.0009 s4. 0.000 007 m

2. Rewrite the following measurements in the units suggested

1. 760 mm in m2. 3.2 × 103 m in km3. 4.5 s in s4. 2.5 ms in s5. 8000000 µm into km

Goal

To measure length using A rule A vernier Caliper

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From 1889 to 1960, the meter was defined to be the distance between two scratches in a platinum-iridium bar.

Platinum-Iridium bar10

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The metre is defined such that the speed of light in free space is exactly 299,792,458 metres per second (m/s)

orange-red line of krypton-86 propagating in a vacuum11

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The Metre

The metre or meter is a base unit of length in the metric system used around the world for general and scientific purposes.

Historically, the metre was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent 1⁄10,000,000 of the distance from the equator to the north pole through Paris.

In 1983, it was redefined by the International Bureau of Weights and Measures as the distance travelled by light in free space in 1⁄299,792,458 of a second.

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Measurement of Length

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Metre rule is used to measure length of object.

Precaution to be taken when using a ruler: Avoid parallax error – the position of eye must

be in line with the reading to be taken. Avoid zeros errors and end errors – if the ends

of the ruler are worn-out, it is advisable that measurements should start from the 1 cm mark of the scale

Any instrument that are out of adjustment or with some minor fault is still accurate as long as the zero error is added or subtracted form the reading shown on the scale.

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Example

1. A girl uses a rule to measure the length of a metal rod. Because the end of the rule is damaged, she places one end of the rod at the 1 cm mark as shown.

How long is the metal rod?

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2. A piece of cotton is measured between two points on a ruler.

When the length of cotton is wound closely around a pen, it goes round six times

What is the distance once round the pen?

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3. The diagram shows one method of measuring the diameter of a beaker.

What is the diameter of the beaker?

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4. The diagram shows a thick-walled tube. The thickness of the wall is 3 mm.

What is the internal diameter d of the tube?

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5. A floor is covered with square tiles. The diagram shows a ruler on the tiles.

How long is one tile?

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6. A ruler is used to measure the length of a nail.

What is the length of the nail?

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Measure thickness or diameter of object correct to 2 decimal places of decimals of a centimetre

Vernier Caliper21

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Reading Vernier Scale

The vernier scale consists of a 9 mm long scale divided into 10 divisions.

The scale shows the object being measure is between 2.4 cm and 2.5 cm long.

To find the second decimal number, look for a making on the vernier scale which coincides with a marking on the main scale.

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The inside jaws which can be used to measure the internal diameters of tube and containers.

The depth bar at the end is used to measure the depth of a container.

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24

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Example

Write down the reading shown by the following

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(a) (c)

(b) (d)

0 5 10

7 8

0 5 10

4 5 A B

Q P

0 5 10

6 7

0 5 10

0 1

Revision 1

The diagram shows a vernier V placed against a scale S.

What is the vernier reading?

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Revision 2

The diagram shows part of a vernier scale.

What is the correct reading?

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Revision 3

The diagram shows a vernier scale.

What is the reading on the vernier scale?

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Goal

Using micrometer screwgauge to measure length

Accuracy of each instrument to measure length

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Measure diameter of wire or thin rod correct to 3 decimal places of centimetre

Micrometer screwgauge36

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barrel

thimblecircular scale

main scale

spindleanvil

rachet head

Micrometer screwgauge is used to measured the diameter of fine wires, the thickness of paper and similar small lengths.

It has two scales: the main scale on the sleeve and the circular scale on the thimble which have 50 divisions. One complete turn of the thimble moves the spindle by 0.50 mm.

Hence each divisions represents a distance of

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mm 01.050

mm 50.0

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39

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There are number of precautions one should take when using a micrometer: The thimble should never be tightened too

much. Clean the ends of the anvil and spindle

before making a measurement. Check for systematic error by closing the

micrometer when there is nothing between its anvil and spindle.

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Example

1. Write down the reading shown by the micrometer screw gauge.

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(a) (c)

(b) (d)

25

300 5

40

5 10 1545

5

100

30

350 5

2. Determine the reading of the following micrometer screw gauge

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(a)

(b)

0 5

15

20

Range & Precision

Instrument Range of measure

ment

Precision

Measuring tape 0 – 5 m 0.1 cm

Metre rule 0 – 1 m 0.1 cm

Vernier calipers 0 – 15 cm 0.01 cm

Micrometer screw gauge

0 – 2.5 cm 0.001 cm

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Time

Time is measured in years, months, days, hours, minutes & seconds.

The second is the SI unit of time. All timing devices make use of some

regular process such as regularly repeating motions called oscillations.

One regular oscillations is referred to as the period of the oscillation.

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This clock depends on the oscillation of caesium-133 atom.

The accuracy is to 1 second loss or gain in every 20 millions year.

Caesium Atomic Clock50

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Other Clocks

Pendulum clock is used to measured long intervals of time. The gravitational potential energy from the descending mass is used to keep the pendulum swinging.

Watches also used to measure long intervals of time. It depend on the vibration of quartz crystals to keep accurate time. The energy keeping the crystal vibrate comes from small battery, coiled springs or by kinetic energy.

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Stopwatch Stopwatches are used to measure short

intervals of time. There are two types; the digital stopwatch and

analog watch. The digital stopwatch is more precise as it can

measure time intervals of 0.01 seconds while the analogue stopwatch measures in intervals of 0.1 seconds.

One common error in using stopwatches is the reaction time in starting and stopping the watch which is few hundredths of a second (typically 0.3 s)

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Precision

Measuring Instrumen

t

Smallest Division

Precision

Examples

Analogue stopwatch

0.1 s 0.1 s25.1 s, 25.2 s, 25.3 s

Digital stopwatch

0.1 s0.1 s

25.1 s, 25.2 s, 25.3 s

0.01 s0.01 s

25.12 s, 26.13 s, 26.14 s

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Example 1

The diagram shows a stopwatch, originally set at 00:00. When a car was first seen, the stop-start button was pressed. When the car passed the observer, the stopwatch showed 01:06.

How long did the car take to reach the observer in second?

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Example 2

A stopwatch is used to time a race. The diagrams show the watch at the start and at the end of the race.

How long did the race take?

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Example 3

The diagrams show the times on a stopclock at the beginning and at the end of an experiment.

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Example 4

A pendulum is set in motion and 20 complete swings are timed. The time measured is 30 s.

What is the time for one complete swing of the pendulum?

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Physical Description

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PHYSICAL QUANTITY

VECTOR QUANTITY

SCALAR QUANTITY

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SCALAR QUANTITY

Magnitude NO Direction

Example of Scalar

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distance

speed

time

mass

pressure

energy

volume

density

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VECTOR QUANTITY

Magnitude Direction

Example of Vector

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velocitydisplacement

weight

acceleration

force

Adding Scalar

The addition of scalar quantities is very simple just by adding or subtracting it arithmetically.

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Adding Vector

In adding two vectors we need to consider the direction of the vector quantities.

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(a)

(b)

(c)

(d)

(e)

30 N 50 N

4 m 8 m

500 N 800 N

73 m 26 m

6 m/s 6 m/s

1. Which of the following groups of physical quantities consists only of scalars?

A. acceleration, force, velocityB. acceleration, mass, speedC. force, time, velocityD. mass, speed, time

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2. Which is the correct statement about force and velocity?

A. Force and velocity are both scalars.B. Force and velocity are both vectors.C. Force is a scalar, velocity is a vector.D. Force is a vector, velocity is a scalar.

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3. When two forces are combined, the size of the resultant depends on the angle between the two forces.

Which of the following can not be the magnitude of the resultant when forces of magnitude 3N and 4N are combined?

A. 1N B. 3N C. 7N D. 8N D

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4. A student studies some equations. power = work / time force = mass × acceleration velocity = displacement / time1. How many vector quantities are

contained in the equations?

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A 1 B 2 C 3 D 4

D

5. Which of the following correctly lists one scalar and one vector quantity?

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B

Vector Diagram

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Triangle Method

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Parallelogram Method

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Example

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1. You walk 7 m south and then 3 m west. What is your displacement from your starting point?

2. A toy car is moving 12 m eastwards. A child then pushes it 2.6 m northward. What is the resulting displacement of the car.

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3. An airplane is headed directly east at 340 m/s when the wind is from the south at 45 m/s. What is it velocity with respect to the ground?

4. A motorboat moved across a stream that flows at 3.5 m/s. In still water the boat can do 4.6 m/s. Find (a) the angle stream at which the boat must be pointed (b) the resulting speed of the boat in the cross-stream direction.

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5. By using a geometrical instrument, find the resulting vector for each of the following;

A. A displacement of 5 m and 7 m acting at 45° to one another.

B. A velocity of 6 m/s and 8 m/s acting at 60° to one another.

C. A force of 5 N and 4 N acting at 90° to one another

D. Two forces of 4 N and 6 N acting on a body with an angle of 50° between them.

Use scale of 1 cm : 1 unit for your drawing

Further Example

1. Two forces act at right angles at a point O as shown.

1. What is the resultant of the forces?

A

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2. Which diagram correctly shows the addition of a 4N and a 3N force?

A

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3. Forces X and Y act on a block in the directions shown on the scale diagram.

1. In which direction is the resultant force acting?

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4. Forces of 3 N and 4 N are acting as shown in the diagram.

1. Which diagram may be used to find the resultant R of these two forces?

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A

PHYSICAL QUANTITIES, UNITS & MEASUREMENT

LEARNING OUTCOME

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Scalars and vectors

Define the terms scalar and vector. Determine the resultant of two vectors

by a graphical method. List the vectors and scalars from

distance, displacement, length, speed, velocity, time, acceleration, mass and force.

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Measurement techniques

Describe how to measure a variety of lengths with appropriate accuracy using tapes, rules, micrometers, and calipers using a vernier as necessary.

Describe how to measure a variety of time intervals using clocks and stopwatches.

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Units and symbols

Recognise and use the conventions and symbols contained in ‘Signs, Symbols and Systematics’, Association for Science Education, 1995.

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