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1. Physical quantities and units

We have defined physics as the study of the natural phenomena in the universe and showed that the principal aim of physicists is to study how the systems in the universe works. We also outlined that physicists carry out many experiments from which they make observations, analyse and develop them into laws and theories of nature, which often exist in mathematical form.

Physics therefore largely depends on the ability of physicists to take precise and accurate measurements of the real-world systems. Scientist across the world use the same standard measurements and measurement systems. The essence of making standard measurements is for scientists to determine the magnitude of physical quantities against the agreed references/ standards. For example, scientists agreed that the standard unit of mass is the kilogram (kg), which is equivalent to the mass of an international prototype, a platinum-iridium cylinder kept in France. So scientist determine the mass of physical objects by comparing with this standard. For instance, a 2 kg bag of rice has twice the mass of the prototype standard, and a 0.5 kg piece of meat has half the mass.

Physical quantities and units

Physical quantities are quantities that scientists can measure, for example mass, weight, volume, voltage, charge and area. When stating a physical quantity, we should state both the magnitude and the unit. Without the unit, it will be difficult to tell the standard measure used as the reference. For example, if you say the length of the string is 28, it will be difficult to deduce without remeasuring if the system of units used is the foot, yard, metre or mile, so we should never omit units of measurements.

As a scientist, you should also be able to make reasonable approximations for certain physical quantities when need arises. For instance, you can approximate the speed of sound in air as 300 m s-2, to calculate the distance travelled by sound to hit a mountain, given the time taken to hear the echo.

S.I. Units.

There are so many systems of units used in the world. For example, the degree fahrenheit, the degree celsius and the kelvin are all temperature units used in everyday life. Scientists across the world however agreed to use the same standard system of units for easy reference and to avoid conversion between units. “SI unit” is the short term for the standard International System of Units based on the basic physical units in table 1.0 below.

Basic physical quantities and units

Basic physical quantities are the physical quantities used to make all the derived physical quantities. Base quantities resemble the basic requirements in the confectionary industry, such as flour, milk, sugar, salt, water, and margarine, used to make confectionary products, ranging from doughnuts, cakes, biscuits and buns, which are the derived products. In science, there are seven basic physical quantities and their units are the SI base units.

Table 1.0 Basic physical quantities and base units

Base
quantity
Base
unit
Symbol of
the base unit
Masskilogramkg
Lengthmetrem
Timeseconds
TemperaturekelvinK
Electric currentampereA
Amount of substancemolemol
Luminous intensitycandelacd

Notes:

  1. The Cambridge and ZIMSEC syllabuses do not require students to know about luminous intensity and only A’ level students need to know about amount of substance. Inclusion of the two at this stage is for completeness only.
  2. We write all unit names with initials in lowercase as shown in Table 1.0, regardless of whether the unit was named after a scientist or not. In addition, all symbols or abbreviations of units should be written with lowercase initials, unless the units are named after scientists like the kelvin (K), ampere (A) or the derived units with special names shown in the last column of Table 1.1. An exception is only given to the unit ‘litre’, which can be written in both the lowercase or uppercase initials (l or L), because, though not named after a scientist, its lowercase initial (l)can easily be confused with the number 1.
  3. We insert space between the magnitude and unit of a physical quantity. For example, the mass of the meat is 2 kg, not 2kg. Exception is when writing percentage, for example, Sarah got 90% in the test, not 90 %.
  4. Units may be written in their plural form using the ‘s’ only when unit name is written, for example, the height of the gate is 3 metres, not 3 metre. Singular form is the only acceptable way of writing symbols of units, for example, the temperature of water rose by 20 K, not 20 Ks or the resistance of the wire is 100 Ω, not 100 Ωs. So never use an ‘s’ in front of a unit symbol, unless the ‘s’ is referring to the symbol for second as a unit.

Derived physical quantities and units

Derived physical quantities are the quantities formed by using the relationship between the derived quantity and the basic physical quantities. The relationships are usually products and or quotient operations. Derived units are the units of derived physical quantities. Some derived units have custom names and symbols.

How to derive units of derived quantities.

Use the definition or relationship between the derived physical quantity and the basic physical quantities. The following examples will help to explain this.

Example 1a: Deriving the units of velocity. Velocity is the rate of change of displacement.

Velocity = Displacement (m) / time (s), so the unit is m/ s which should be written as m s-1

Example 1b: Deriving the units of acceleration. Acceleration is the rate of change of velocity.

Acceleration= Velocity (m s-1) / time (s), so the unit is m s-2

Example 1c: Deriving the units of force in SI base units. From Newton’s second law of motion, force is the rate of change of momentum.

Force = mass (kg) x acceleration (m s-2), so the unit is kg m s-2. This unit has a special name called the newton (N).

Example 1d: Deriving the units of pressure in SI base units. Pressure is the force acting per unit area.

Pressure = Force (kg m s-2) / Area (m2), so the unit is kg m-1 s-2. This unit has a special name called the pascal (Pa).

Example 1e: Deriving the units of energy in SI base units. Energy is the ability to work.

Energy =Work done = Force (kg m s-2) x Distance (m), so the unit is kg m2 s-2. This unit has a special name called the joule (J)

Example 1f: Deriving the units of power in SI base units. Power is the rate of doing work.

Power = Work done (kg m2 s-2) / time (s), so the unit is kg m2 s-3. This unit has a special name called watt (W).

Note: Do not worry about recalling all definitions of physical quantities at this stage because you will study them as we proceed with the other syllabus areas.

Table 1.1: Examples of derived physical quantities and units

Derived physical
quantity
Derived unitSpecial name of
the derived unit
Speed and Velocitym s-1
Accelerationm s-2
Aream2
Volumem3
Densitykg m-3
Forcekg m s-2newton (N)
Pressurekg m-1 s-2pascal (Pa)
Energykg m2 s-2joule (J)
Powerkg m2 s-3watt (W )
Electric chargeA scoulomp (C)
Voltagekg m2 s-3 A-1volt (V)
ResistanceV A-1ohm (Ω)

Notes:

  1. When two or more unit symbols are combined to form a derived unit, a space must separate them, for example, 1 W= 1 kg m2 s-3, not 1kgm2s-3.
  2. Try not to use the solidus (/), but the negative index when writing reciprocal symbols, for example, you may write the units of speed as m s-1 not m/ s. Also avoid using the solidus more than once in a unit, for example, you may write the unit of acceleration as m s-2, not m/ s/ s.
  3. We should never use the symbols of units without a numerical value preceding. You may use the unit name in such cases, for example, a cubic metre of cement cost 20 dollars, not a m3 of cement cost 20 dollars. Another example is, there are 1000 mm in a metre, not there are many mm in a metre.

Prefixes used with the SI units

Some physical quantities are too small or too big and their magnitudes may be difficult to interpret, read or write. To simplify mathematical operations, scientists use standard forms or prefixes in table 1.2, which are multiples or sub-multiples of the SI units. Use of standard forms also helps in indicating the size of the error in each measurement (we will cover this in the next chapter).

Imagine how complex it would be to read, write or solve equations involving tiny numbers such as the rest mass of an electron (me), the rest mass of a proton (mp) and the Plank constant (h):

me = 0. 000 000 000 000 000 000 000 000 000 000 911 kg

mp = 0. 000 000 000 000 000 000 000 000 001 67 kg

h = 0. 000 000 000 000 000 000 000 000 000 000 000 663 J s

Thanks to the standard form notation which reduces the complexity of each of the quantities above to read:

me = 9.11 × 10-31 kg
mp = 1.67 × 10-27 kg
h = 6.63 × 10-34 J s

There are also big numbers, such as the Avogadro constant (NA), that we can also interpret easily in standard form.

NA = 602 000 000 000 000 000 000 000 mol-1

NA = 6.02 × 1023 mol-1 in standard form.

Table 1.2 Prefixes used with SI units.

PrefixSymbolMultiplying factor
picop10-12
nanon10-9
microμ10-6
millim10-3
centic10-2
decid10-1
kilok103
megaM106
gigaG109
teraT1012

Prefixes in table 1.2 are used in front of a unit to change its meaning. The prefixes substitute for the multiplying factors they represent.

Notes:

  1. When using prefix in table 1.2, no space should be put between the prefix and the symbol of unit it represents, for example, the power generated is 20 MW, not 20 M W or the diameter of a fine wire is 1 mm, not 1 m m. This rule is important to avoid confusing prefixes such as the milli (m) and units such as the metre (m), for example, a speed of 2 m s-1 (2 metres per second) should look different from a time interval of 2 ms (2 milliseconds) by not putting space between prefix m for milli and s for second.
  2. When a unit is used with a prefix as explained in part 1 above, treat the prefix and unit as one thing, for example, 1 dm3 means 1 (dm)3 or 1 dm x 1 dm x 1 dm = 10-1 m x 10-1 m x 10-1 m = 1 litre, not 1 d(m3), which may be the case in mathematics when not referring to the letters as units. Similarly, 1 cm3 = 1 (cm)3 = 1 cm x 1 cm x 1 cm = 10-2 m x 10-2 m x 10-2 m = 1 x 10-6m3, not 1 c(m3) or 1 x 10-2m3.
  3. Do not use compound prefixes, one prefix is sufficient, for example, 10-12 F= 1 pF, not 1 mnF.

Example 1g: Write the following quantities using appropriate prefixes;

i) 260 000 000 W,

ii) 0.000 000 000 97 m,

iii) 0.000 100 F,

iv) 20 000 m.

Solution. Look for prefixes that we can use for the resultant to be an easy quantity to work with or pronounce.

i) 260 000 000 W = 260 x 106 W = 260 MW

ii) 0.000 000 000 97 m = 97 x 10-12 m = 97 pm

iii) 0.000 100 F = 100 x 10-6 F = 100 μF

iv) 20 000 m = 20 x 103 m = 20 km

Homogeneity of physical equations

Units of physical quantities are used to check the homogeneity of physical equations. A physical equation is homogeneous if and only if all the terms in the equation have the same units. If the terms in the physical equation have different units, the equation is not homogeneous and is therefore not correct because we can not subtract, add or equate terms with different units.

Example 1h: Determine if the following equations are homogeneous.

i) v2 =u2+2as,

the equation of motion under uniform acceleration, where v is the final velocity, u is the initial velocity, s is the displacement, and a is the acceleration.

ii) p = ρgh,

the equation for the hydrostatic pressure p, where ρ is the density of the fluid, g is the acceleration due to gravity and h is the height of the fluid column.

Solution

Consider the left-hand side (LHS) terms and then the right-hand side (RHS) terms separately. Equate nothing because at an advanced level an equal sign is strictly used for its intended mathematical function, which is not suitable for equating terms that are not equal, including cases where identity sign is required.

i) Consider the LHS term;

The unit of v is m s-1 and so the unit of v2 is m2 s-2

Consider the RHS terms;

The unit of u is m s-1 and so the unit of u2 is m2 s-2

The unit of a is m s-2 and that of s is m, so the unit of as is m2 s-2

All the terms in the equation have the same units, so the equation

v2 =u2+2as is homogeneous.

ii) Consider the LHS;

The unit of p is kg m-1 s-2

Consider the RHS;

The unit of ρ is kg m-3, g is m s-2, h is m, and so the unit of ρgh is kg m-1 s-2

Term on the LHS has the same unit as the term on the RHS, so the equation is homogeneous.

Note: A physical equation can be homogeneous but yet wrong because of missing terms, extra terms with the same units or wrong coefficients. Using example 1h, part i, if we had erroneously written the equation as

v2 =as (with a missing term)

or v2 =u2+2as + s2/t2 (extra term with the same unit)

or v2 =2u2+as (wrong coefficients),

the equations would still be homogeneous but yet incorrect, because of the explanations highlighted in brackets.

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