TOPIC 1
PHYSICAL QUANTITIES AND SI UNITS
The measurement and recording of quantities are central to the whole of physics. The
skills of estimating a physical quantity and having a feeling for which quantities are
reasonable and which are unreasonable are very useful for any physicist.
This topic introduces the SI system of units, which provides a universal framework of
measurement that is common to all scientists internationally.
Students should be aware of the nature of a physical measurement, in terms of a
magnitude and a unit, and will have experience of making and recording measurements in
the laboratory.
UNITS AND DIMENSIONS
All measurements of physical quantities require both a numerical value and a unit in which
the measurement is made. For example, your height might be 1.73 metres. The number and
the unit in which it is measured need to be kept together because it is meaningless to write
“height = 1.73”.
The numerical value is called the magnitude of the quantity and the magnitude has
meaning only when the unit is attached. In this particular case, it would be correct to write
“height = 173 cm”, since there are 100 cm in a metre.

A quantity is a physical property that can be measured (or calculated).

A unit is a measure of the quantity.

Base units are used to derive all other units – called derived units. The main base
quantities and units are:

 Length
m
 Mass
kg
 Time
s
 Temperature
K
 Electric current
A
 Amount of substance
mol
Derived units are SI units that are dependent on one or more base units. They are
products or quotients of the base units. To find the expression of a derived unit in
terms of the base units, it is necessary to use the definition of the quantity.
For example, the newton (N), as a unit of force, is defined by the equation
𝒇𝒐𝒓𝒄𝒆 = 𝒎𝒂𝒔𝒔 × 𝒂𝒄𝒄𝒆𝒍𝒆𝒓𝒂𝒕𝒊𝒐𝒏
1
So,

𝟏 𝑵 = 𝟏 𝑲𝒈 × 𝟏 𝒎𝒔−𝟐 𝒐𝒓
𝟏 𝑵 = 𝟏 𝑲𝒈 𝒎𝒔−𝟐
Homogeneous Equations – an equation is homogeneous if the units are the same on
each side of the equal sign. All physics equations are homogeneous. If you ever find
that the units on both sides of an equation are not the same, then either the
equation is incorrect or you have made a mistake somewhere.

Prefixes - Now you have units, you often need to group these into larger or smaller
numbers to make them more manageable. For example, you don't say that you are
going to see someone who lives 100,000 m away from you; you say they live 100 km
away from you. Here is a quick list of the common quantities used:
Name Symbol Scaling factor
tera
T
Common example
1012, 1,000,000,000,000 Large computer hard drives can be
terabytes in size.
giga
G
109, 1,000,000,000
Computer memories are measured in
gigabytes.
mega
M
106, 1,000,000
A power station may have an output of 600
MW (megawatts).
kilo
k
103, 1,000
Mass is often measured in kilograms (i.e.,
1000 grams).
deci
d
10-1, 0.1
Fluids are sometimes measured in decilitres
(i.e., 0.1 litre).
centi
c
10-2, 0.01
Distances are measured in centimetres (i.e.,
100th of a metre).
milli
m
10-3, 0.001
Time is sometimes measured in milliseconds.
micro
μ
10-6, 0.000001
micrometres are often used to measure
wavelengths of electromagnetic waves.
nano
n
10-9, 0.000000001
nanometres are used to measure atomic
spacing.
pico
p
10-12, 0.000000000001
picometres used to measure atomic radii.
2

Estimating physical quantities
In making estimates of physical quantities, it is essential that you do not just guess
a value and write it down. It is important to include the method you use, not just the
numerical values. You need to remember some important values/figures, to one
significant figure, in SI units. The list below is by no means complete but is a
starting point.
Quantity
Mass of an adult
Mass of car
Height of a tall man
Height of a mountain
Speed of a car on a high – speed road
Speed of a plane
Speed of sound in air at sea level
Weight of an adult
Energy requirement for a person for one day
Power of a car
Power of a person running
Pressure of the atmosphere
Density of water
Distance from the Earth to the Moon
Distance from the Earth to the Sun
Radius of the Earth
Mass of the Earth
Once you have some basic data, you can use it to find
Estimate
70 kg
1000 kg
2m
5000 m
30 m/s
300 m/s
300 m/s
700 N
10,000,000 J
60 kW
200 W
100,000 Pa
1000 kg m-3
400,000 km
150,000,000 km
6000 km
6 × 1024 kg
the approximate value for
many quantities. As a general rule, always get your values into SI units.
EXAMPLES
1)
Convert the following:
a) 2.86 kg into g;
b) 0.0543 kg into g;
c) 48 g into kg;
d) 3.8 hrs into sec;
e) 6500000 sec into days.
2)
Convert the following:
a) 1.0 m2 into cm2;
b) 7.38 m3 into cm3;
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c) 6.58 cm3 into m3;
d) A density of 3.45 g/cm3 into kg/m3;
e) A speed of 110 km/hr into m/s.
3)
Derive the base units for:
a) The joule;
b) The pascal;
c) The watt.
4)
Use base units to show whether or not these equations balance in terms of units.
a) E = mc2
b) E = mgh
c) P = Fv
d) p = ρgh
[E is energy; m is mass; c is speed; g is acceleration; h is height; F is force; v is
velocity; ρ is density; P is power; p is pressure]
5)
You might see some car specifications in a magazine that give fuel efficiency as 7.6
km per kilogram of fuel. Given that a mile is 1.609 km, a gallon is 3.785 litres, and a
litre of gasoline has a mass of 0.729 kg, what is the car’s fuel efficiency in miles
per gallon?
6)
Express each quantity in the standard SI units requested.
a) An adult should have no more than 2500 mg of sodium per day. What is the
limit in kg?
b) A 240 mL cup of whole milk contains 35 mg of cholesterol. Express the
cholesterol concentration in the milk in kg/m3 and in mg/mL.
c) A typical human cell is about 10 μm in diameter, modelled as a sphere. Express
its volume in cubic meters.
d) A low-strength aspirin tablet (sometimes called a “baby aspirin”) contains 81
mg of the active ingredient. How many kg of the active ingredient does a 100
tablet bottle of baby aspirin contain?
e) The average flow rate of urine out of the kidneys is typically 1.2 L/min.
Express the rate in m3/s.
7)
The concentration of PSA (prostate-specific antigen) in the blood is sometimes
used as a screening test for possible prostate cancer in men. The value is normally
reported as the number of nanograms of PSA per millilitre of blood. A PSA of 1.7
(ng/mL) is considered low. Express that value in
a) g/L b) Kg/m3 c) mg/L.
8)
The acceleration g of a falling object near a planet is given by the following
equation g = GM/R2. If the planet’s mass M is expressed in kg and the distance of
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the object from the planet’s centre R is expressed in meters, determine the units
of the gravitational constant G. The acceleration g must have units of m/s2.
EXPERIMENTAL UNCERTAINTY
Experimental uncertainty used to be called experimental error. However, the change was
made because “error” seems to imply that a mistake has been made and that is not true. All
readings have uncertainties.

A ruler might measure to the nearest millimetre;

A clock to the nearest second;

A thermometer to the nearest degree.
So one person using a metre rule might record the length as 86.0 cm and another person
measuring the same length might record it as 86.1 cm. This type of variation is called a
random uncertainty. It might come about through the limitations of the scale on an
instrument or through the way the instrument is used. This can be reduced by repeating
and averaging results or by plotting a graph and drawing a line of best fit e.g. reaction
time, temperature fluctuation, pressure fluctuation.
If the instrument itself is faulty or if it is being used incorrectly, there will be systematic
uncertainty. It causes readings to be larger or smaller than the accepted value. This
cannot be eliminated by repeating and averaging e.g. zero error, calibration error, parallax.
Systematic uncertainties are often much more difficult to detect. One way to check for
this error, with electrical instruments, would be to use a different meter and if it gives
the same reading there is unlikely to be a serious systematic error.
PRECISION AND ACCURACY
Accuracy – a measure of how far away the mean is from the expected value. This can be
improved by reducing or eliminating systematic uncertainties.
Precision – a measure of how close repeated measurements are from each other. Precision
is affected by random uncertainties.
Hence, any reading taken o high precision have low random uncertainty and readings taken
to high accuracy have low systematic uncertainty. This is illustrated by the diagram below
where an archery target is marked with the position of arrows fired at it.
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
Figure (a) shows that the archer is very skilled, so there is little random uncertainty
but that his equipment has a systematic error in it.

Figure (b) shows that the archer is unskilled, so there is considerable random
uncertainty but that his equipment has no systematic error in it. The average
position of his arrows is in the centre of the target.

Figure (c) shows that the archer is unskilled, so there is considerable random
uncertainty, and that his equipment has a systematic error in it.

Figure (d) shows that the archer has high precision equipment and great accuracy,
so there is minimal random uncertainty, and no systematic error in it.
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7
ESTIMATING UNCERTAINTIES
Uncertainty is the interval within which the true value can be considered to lie with a given
level of confidence or probability – any measurement will have some uncertainty about the
result, this will come from variation in the data obtained and be subject to systematic or
random effects. This can be estimated by considering the instruments and the method and
will usually be expressed as a range such as 20 °C ± 2 °C. The confidence will be qualitative
and based on the goodness of fit of the line of best fit and the size of the percentage
uncertainty.
When you repeat a measurement you often get different results. There is an uncertainty
in the measurement that you have taken. It is important to be able to determine the
uncertainty in measurements so that its effect can be taken into consideration when
drawing conclusions about experimental results. The uncertainty might be the resolution of
the instrument or, if the readings were repeated, the uncertainty might be half the range
of the repeats. If the uncertainty is predictable, i.e. it is systematic, then the uncertainty
should be subtracted from each reading, for example if there is a zero error on an
instrument.
Recording data

Results should be recorded to the resolution of the measuring instrument which
means there will be a consistent number of decimal places for the readings for any
one variable. Examples include;

i.
𝑑 = (7.80 ± 0.08) × 103 𝑘𝑔𝑚−3
ii.
𝑑 = (7.8 ± 0.1) × 103 𝑘𝑔𝑚−3
If you measure a length as 249 mm with an uncertainty of 1 mm at the zero and
another 1 mm at the other end then the reading, together with its uncertainty is
(249 ± 2) mm. this gives the actual uncertainty as 2 mm, the fractional uncertainty
as 2/249 or 0.0080 and the percentage uncertainty as 0.8%.

The final result must be expressed in numerical rather than percentage
uncertainties, for example, as (4.73 ± 0.03) N for a force measurement, rather than
(4.73 ± 6%) N.

When data is processed, these values should be recorded to a consistent number of
significant figures which is usually 3 as this is what we can usually confidently plot on
a graph.
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Example: A student measures the diameter of a metal canister using a ruler graduated in
mm and records these results:
Diameter / mm
Reading 1 Reading 2 Reading 3 Mean
66
65
61
64
The uncertainty in the mean value (64 mm) can be calculated as follows:
a) Using the half range
The range of readings is 61 mm – 66 mm so half the range is used to determine the
uncertainty. Uncertainty in the mean diameter = (66 mm – 61 mm)/2 = 2.5 mm
Therefore, the diameter of the metal canister is 64 mm ± 2.5 mm.
Since a ruler graduated in mm could easily be read to ± 0.5 mm, it is acceptable to
quote the uncertainty as ± 2.5 mm for this experiment.
b) Using the reading furthest from the mean
In this case, the measurement of 61 mm is further from the average value than 66
mm therefore we can use this value to calculate the uncertainty in the mean.
Uncertainty in the mean diameter = 64 mm – 61 mm = 3 mm. Therefore, the diameter
of the metal canister is 64 mm ± 3 mm.
c) Using the resolution of the instrument
This is used if a single reading is taken or if repeated readings have the same value.
This is because there is an uncertainty in the measurement because the instrument
used to take the measurement has its own limitations. If the three readings
obtained above were all 64 mm then the value of the diameter being measured lies
somewhere between 63.5 mm and 64.5 mm since a metre rule could easily be read to
half a millimetre. In this case, the uncertainty in the diameter is 0.5 mm. Therefore,
the diameter of the metal canister is 64 mm ± 0.5 mm. This also applies to digital
instruments. An ammeter records currents to 0.1 A. A current of 0.36 A would be
displayed as 0.4 A, and a current of 0.44 A would also be displayed as 0.4 A. The
resolution of the instrument is 0.1 A but the uncertainty in a reading is 0.05 A.
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RULES FOR UNCERTAINTY (Absolute vs. Relative Uncertainties)
Quoting your uncertainty in the units of the original measurement – for example, 1.2 ± 0.1 g
or 3.4 ± 0.2 cm – gives the “absolute” uncertainty. In other words, it explicitly tells you
the amount by which the original measurement could be incorrect. The relative uncertainty
gives the uncertainty as a percentage of the original value. Work this out with:
Relative uncertainty = (absolute uncertainty ÷ best estimate) × 100%
So in the example above:
Relative uncertainty = (0.2 cm ÷ 3.4 cm) × 100% = 5.9%
The value can therefore be quoted as 3.4 cm ± 5.9%.
Adding and Subtracting Uncertainties
Work out the total uncertainty when you add or subtract two quantities with their own
uncertainties by adding the absolute uncertainties. For example:
(3.4 ± 0.2 cm) + (2.1 ± 0.1 cm) = (3.4 + 2.1) ± (0.2 + 0.1) cm = 5.5 ± 0.3 cm
(3.4 ± 0.2 cm) − (2.1 ± 0.1 cm) = (3.4 − 2.1) ± (0.2 + 0.1) cm = 1.3 ± 0.3 cm
Multiplying or Dividing Uncertainties
When multiplying or dividing quantities with uncertainties, you add the relative
uncertainties together. For example:
(3.4 cm ± 5.9%) × (1.5 cm ± 4.1%) = (3.4 × 1.5) cm2 ± (5.9 + 4.1)% = 5.1 cm2 ± 10%
(3.4 cm ± 5.9%) ÷ (1.7 cm ± 4.1 %) = (3.4 ÷ 1.7) ± (5.9 + 4.1)% = 2.0 ± 10%
Multiplying by a Constant
If you’re multiplying a number with an uncertainty by a constant factor, the rule varies
depending on the type of uncertainty. If you’re using a relative uncertainty, this stays the
same:
(3.4 cm ± 5.9%) × 2 = 6.8 cm ± 5.9%
If you’re using absolute uncertainties, you multiply the uncertainty by the same factor:
(3.4 ± 0.2 cm) × 2 = (3.4 × 2) ± (0.2 × 2) cm = 6.8 ± 0.4 cm
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A Power of an Uncertainty
If you’re taking a power of a value with an uncertainty, you multiply the relative
uncertainty by the number in the power. For example:
(5 cm ± 5%)2 = (52 ± [2 × 5%]) cm2 = 25 cm2± 10%
Or
(10 m ± 3%)3 = 1,000 m3 ± (3 × 3%) = 1,000 m3 ± 9%
Natural Logarithms
The absolute uncertainty in a natural log (logarithms to base e, usually written as ln or loge)
is equal to a ratio of the quantity uncertainty to the quantity.
ln(𝑥 ± ∆𝑥) = ln(𝑥) ±
𝐹𝑜𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒, ln[(95 ± 5) 𝑚𝑚] = ln(95 𝑚𝑚) ±
∆𝑥
𝑥
5 𝑚𝑚
= [4.543 ± 0.053] 𝑚𝑚
95 𝑚𝑚
Uncertainty in logarithms to other bases (such as common logs/logarithms to base 10,
written as log10 or simply log) is this absolute uncertainty adjusted by a factor (divided by
2.3 for common logs). Thus, for period T = (24 ± 4) × 103 s, taking the log of T with its
absolute uncertainty gives,
log 𝑇 = log(24 × 103 ) ±
4
= 4.38 ± 0.07
24 × (2.3)
EXAMPLES
1) An object with momentum (85 ± 2) Ns catches up with, and sticks to another object
with momentum (77 ± 3) Ns. Find the total momentum of the two objects and its
uncertainty after the collision.
2) A reading on a balance of the mass of an empty beaker is (105 ± 1) g. After some
liquid is poured into the beaker, the reading becomes (112 ± 1) g. Deduce the mass of
liquid added and its uncertainty.
3) A plane travels at a speed of (250 ± 10) ms-1 for a time of (18000 ± 100) s. determine
the distance travelled and its uncertainty.
4) Determine the value of the kinetic energy, and its uncertainty, of a cyclist of mass
(63 ± 1) kg when travelling with a speed of (12.0 ± 0.5) ms-1.
11
5) A builder wants to calculate the area of a square tile. He uses a rule to measure the
two adjacent sides of a square tile and obtains the following results:

Length of one side = 84 mm ± 0.5 mm

Length of perpendicular side = 84 mm ± 0.5 mm
Determine the uncertainty in the area of the tile.
6) A metallurgist is determining the purity of a sample of an alloy that is in the shape
of a cube by determining the density of the material. The following readings are
taken:

Length of each side of the cube= 24.0 mm ± 0.5 mm

Mass of cube = 48.23 g ± 0.05 g
Calculate;
i.
the density of the material;
ii.
the percentage uncertainty in the density of the material.
7) Calculate the volume of a cylindrical drink can and the percentage uncertainty for
the final value given that the base radius of the metal can is 33 mm with an
uncertainty of 1% and the length, L of the can = 115 mm ± 2 mm.
SCALARS AND VECTORS

A vector is a quantity that has direction as well as magnitude. Examples include
displacement, velocity, acceleration, force, momentum, pressure, etc.

A scalar is a quantity with magnitude only. Examples include mass, length, time, area,
volume, density, speed, work, energy, power, etc.
Combining vectors (vector addition/subtraction)
Adding or subtracting scalars is just like adding or subtracting numbers, as long as you
always remember to include the unit. For example, 3kg + 4 kg = 7 kg.
Adding vectors can be difficult, subtracting vectors can be even more difficult. You have
to take into account their magnitude and direction. Forces are vector quantities. When we
add two forces together, we get a single force called the resultant force. The resultant
force is not an actual force at all. It is the sum of all the forces acting on an object. The
forces that we add might be caused by different things, for example, one force could be a
gravitational force and the other could be an electric force it might seem impossible for a
force of 8 N to be added to a force of 6 N and get an answer of 2 N, but it could be
correct if the two forces acted in opposite directions on the object. In fact, for these two
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forces a resultant force can have any magnitude between a maximum of 14 N and a
minimum of 2 N, depending on the angle that the forces have with one another.
In order to find the resultant of these two forces, a triangle of forces is used, as shown in
the diagrams below.
The mathematics of finding the resultant can be difficult but if there is a right angle in
the triangle, things can be much more straightforward.
To subtract two vectors, say vector B from vector A, a triangle of vectors is used in which
negative vector B (-B) is added to vector A. this is shown in the figure below. Note that,
A + (-B) is the same as A – B.
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Resolution of vectors
1) Component method
A component of a vector its effective value in a given direction. Hence, the x – component
of a displacement is the displacement parallel to the x – axis caused by the given
displacement. A vector in 3 – dimensions may be considered as the resultant of its
component vectors resolved along any three mutually perpendicular directions. Similarly, a
vector in two dimensions may be resolved into two components acting along any two
mutually perpendicular directions. The figure below shows the vector 𝐹⃗ inclined at an angle
θ to the horizontal and its X and Y vector components ⃗⃗⃗⃗⃗
𝐹𝐻 and ⃗⃗⃗⃗⃗
𝐹𝑉 , which have magnitudes
⃗⃗⃗⃗⃗
𝐹𝐻 = 𝐹⃗ cos 𝜃 and ⃗⃗⃗⃗⃗
𝐹𝑉 = 𝐹⃗ sin 𝜃.
Therefore, the magnitude of the resultant, using Pythagoras theorem, is given by;
𝐹𝑅 2 = 𝐹𝑉2 + 𝐹𝐻2
⇒
𝑭𝑹 = √𝑭𝟐𝑽 + 𝑭𝟐𝑯
The angle, θ, which F makes with the x – axis (positive or negative) is given by;
𝐹𝑉
tan 𝜃 = ( )
𝐹𝐻
⇒ 𝜽 = 𝐭𝐚𝐧−𝟏 (
𝑭𝑽
)
𝑭𝑯
2) Parallelogram Method
The resultant of two vectors acting at any angle may be represented by the diagonal of a
parallelogram. The two vectors are drawn in magnitude and direction as the sides of the
parallelogram and the resultant is the diagonal. The direction of the resultant is away from
the origin of the two vectors as shown below.
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The resultant (FR), using the cosine formula, is given by
𝐹𝑅2 = 𝐹12 + 𝐹22 − 2𝐹1 𝐹2 cos(180 − 𝜃) − − − − − − − (1)
= 𝐹12 + 𝐹22 + 2𝐹1 𝐹2 cos 𝜃 − − − − − − − − − − (2)
𝑭𝑹 = √𝑭𝟐𝟏 + 𝑭𝟐𝟐 + 𝟐𝑭𝟏 𝑭𝟐 𝐜𝐨𝐬 𝜽
∴
𝐭𝐚𝐧 𝜶 =
𝑭𝟐 𝐬𝐢𝐧 𝜽
𝑭𝟏 + 𝑭𝟐 𝐜𝐨𝐬 𝜽
Since, cos(180 − 𝜃) = − cos 𝜃.

A
A
γ
R
α
β

B
By the Sine formula, the length of the side of the triangle is proportional to the sine of
the angle facing that side. Thus,
𝑨
𝑩
𝑹
=
=
− − − − − − − −(3)
𝐬𝐢𝐧 𝜶 𝐬𝐢𝐧 𝜸 𝐬𝐢𝐧 𝜷
And from equation (1),
𝐹𝑅2 = 𝐹12 + 𝐹22 − 2𝐹1 𝐹2 cos 𝛽
Where, 𝛽 = 180 − 𝜃
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Object on inclined plane/surface
From figure above, the force F is the force the sloping ground exerts on a stationary
object resting on it (equal and opposite to the weight of the object). F can be resolved into
two components; 𝑭 𝐬𝐢𝐧 ∅ is the force along the slope and is the frictional force that
prevents the object from sliding down the slope. 𝑭 𝐜𝐨𝐬 ∅ is the component at right angles to
the slope.
EXAMPLES
1)
The two vectors a at 30o and b at 135o have equal magnitudes of 10.0 m. Find
a) the x – component and the y – component of their vector sum, r;
b) the magnitude of r and the angle r makes with the positive direction of the x –
axis.
2)
In the sum A + B = C, vector A has a magnitude of 12.0 m and is angled 40o counter
clockwise from the +x – direction, and vector C has magnitude of 15.0 m and is
angled 20o counter clockwise from the – x – direction. What is the magnitude of B
and the angle (relative to +x) of B?
3)
Three players on a reality TV show are brought to the center of a large, flat field.
Each is given a meter stick, a compass, a calculator, a shovel, and (in a different
order for each contestant) the following three displacements:

A: 72.4 m, 32.0o east of north

B: 57.3 m, 36.0o south of west

C: 17.8 m due south
The three displacements lead to the point in the field where the keys to a new
Porsche are buried. Two players start measuring immediately, but the winner first
calculates where to go. What does she calculate?
4)
A kite of weight 4.8 N, as shown below, is being pulled by a force in the string of
6.3 N acting in a direction of 27o to the vertical.
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a) Resolve the force in the string into horizontal and vertical components;
b) Assuming that the kite is flying steadily, deduce the upward lift on the kite
and the horizontal force the wind exerts on the kite.
5)
Two forces F1 and F2 of magnitudes 80 kN and 50 kN respectively acts
simultaneously on a particle such that the angle between the two forces is 60o.
Determine the magnitude and direction of the resultant force acting on the
particle.
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