Aslevels Physics Notes
Topic 1: Physical Quantities and Units
Physical Quantities
First and foremost Physicist are required to make very careful and
accurate observations and measurements. The characteristic of the
objects in nature (i.e. Quantities) that can be measured are known as
physical quantities. Examples of physical quantities are: length, mass, time,
weight, electric current, force, velocity and acceleration among others.
The quantities that cannot be measured are not physical quantities e.g.
happiness, love, friendship etc.
Two very important items need to be mentioned when stating a physical
quantity namely, the numerical value (magnitude) and the associated unit.
For instance, if the height of a boy is 1.75 m, then the physical quantity is
the height, the numerical value is 1.75 and the unit for height is meters (m).
S.I. Units
To be able to measure and compare the numerical values of a physical quantity, a standard size of that quantity is required. This standard size is known as the unit for that particular physical quantity. Subsequently, the different magnitudes of the same physical quantity are determined by comparing them to the standard size or the unit of the physical quantity. For example, if the length of a rod is 2 m, the physical quantity is length and the unit for length is metre (m).
Besides the meter (m), there are other units of length such as foot, yards
and miles. Although one unit can be converted into another unit by using a
conversion factor, scientists familiar with one system of unit find it
difficult and time consuming when they encounter other system of units.
As such, scientists have agreed to use a common system of units called ‘Le
SystΓ¨me International d'UnitΓ©s’ or ‘S. I. Units’ for short. The advantage
for using the S. I. system is that a physical quantity has only one unit.
Prefixes are used for multiples or sub multiples of the unit.
πΉh = πΉπππ π
And πΉv = πΉπ πππ
The force F has been resolved into two perpendicular (rectangular)
components Fh and Fv
Basic Quantities and Base Units
Let us a make simple analogy to understand the meanings of basic quantities
and base units. The basic materials used in constructing a building may be
stone, water, wood, steel or glass. Using some of the basic materials, a door
or roof of the building can be made. The building itself consists of doors,
walls, windows, floor, roof and stairs which individually are derived from
the above-mentioned basic materials.
Similarly, Physics involves the use of many physical quantities out of which
seven are arbitrarily chosen as physical quantities in S.I. The units for the
basic physical quantities are known as base units. Base Quantities are those
physical quantities on the basis of which other physical quantities can be
expressed. A base unit is a fundamental unit that is defined arbitrarily and
not by combinations of other units. Table 1.2 shows the seven basic physical
quantities and their associated base units.
Derived Quantities and Derived Units
Physical quantities other than the seven basic physical quantities are known
as derived quantities. A derived quantity is a combination of various basic
quantities. Units for derived quantities are known as derived units. Likewise
a derived unit is one which consists of two or more base units.
The derived unit for a derived quantity can be obtained from the
relationship between the derived quantity with the basic quantities.
Certain derived units have special names. For example, the S.I unit of force
is Newton (N).
When using units in calculations you should note that
- Names of units written in full begin with a lower case letter, despite being names of people (newton, hertz etc.)
- Symbols for units with special names begin with an upper case letter (N, Hz etc.)
- Full stops are not placed after unit symbols
- The symbol for a unit should remain unaltered in the plural (8J is correct; 8Js is incorrect)
Use of Units to check the homogeneity of physical equations
An equation is said to be homogeneous if the base units of every term in the
equation are identical. Each term is a group of one or more quantities
separated from other terms by a plus (+) or minus (-) sign or an equals (=)
sign.
Consider one of the equations of motion:
, where
π is the displacement, u is the initial velocity, π‘ is the time taken and π is
the uniform acceleration within the specific interval of time.
We need to check if the above-mentioned equation is homogeneous and,
consequently, we must find the base units of each term of the equation as illustrated below.
Two non-parallel vectors must be combined by use of
vector triangle (head – to – tail rule) or parallelogram
law. Each one of two vectors V₁ and V₂ is represented
in magnitude and direction by the side of a triangle. The
combined effect or resultant R, is given in magnitude
and direction by the third side of the triangle.
Two vectors may be added together to produce a single resultant. This
resultant behaves in the same way as the two individual vectors. It follows
that a single vector may be split up, or resolved, into two or more components. The combined effect of the components is the same as the original vector.
Notice that all the three terms of the equation have the same base units,i.e., m. ½ is a dimensionless constant (no unit). The equation π = π’π‘ +
1/2
ππ‘^2
is
therefore dimensional consistent or dimensional homogeneous or simply
homogeneous.
If one wants to be more mathematical in nature, the use of square brackets
can be used to show clearly that the base units of the respective terms
need to be found. This is means that instead of writing ‘Base units of π’π‘’,
we can simply write in symbolic forms ‘[π’π‘]’.
Alternative way to show that the aforementioned equation is dimensional homogeneous:
Since [π ] = [π’π‘] = [½ at² ] = m, the equation π = π’π‘ + ½ at² is dimensionally consistent or homogeneous.
Suppose that a student had written the following equation:
S=ut²+½ at
In terms of base units,
We can notice that the base units for the 3 terms in the equation are not
the same or homogeneous. It can be concluded that the equation S=ut²+½ at is wrong.
An equation which is not homogeneous must be wrong. On the other hand,
if the base units for the various terms in an equation are the same, it does
not imply that the equation is physically correct.
Note that some physical quantities have no units. Examples of these
quantities are: relative density, refractive index and strain among others.
All real numbers and some mathematical constants such as π also have no
units. They are sometimes referred to as dimensionless constants.
Finding the units of unknown quantities in an equation:
Additionally, base units are used to find units of unknown quantities in an
equation. The units of an unknown quantity in a physical equation can be
found by substituting known units into the defining equation.
Scalar and vector quantities:
A scalar quantity has a magnitude only. It is completely described by a
certain number and a unit.
Examples: Distance, speed, mass, time, temperature, work done, kinetic
energy, pressure, power, electric charge etc. are the examples of scalar
quantities.
A vector quantity has both magnitude and direction. It can be described
by an arrow whose length represents the magnitude of the vector and the
arrow-head represents the direction of the vector.
Examples: Displacement, velocity, moments (or torque), momentum, force,
electric field etc.
Vector addition
The addition of scalar quantities is very straightforward, and only requires
that the quantities being added have the same unit. The total of masses 9
kg, 5 kg, and 2 kg is simply 16 kg. The only situation in which vectors can be
added in this simple way is when they act along the same straight line.
The figure shows the effect of
adding two forces of magnitudes
30N and 20N which act along the
same line. The angle between
them is 0৹ when they act in the
same direction and 180৹ when
they are in opposite directions.
For all other angles between the
directions of the forces, the
combined effect, or resultant is some value between 10N and 50N
Vectors with different lines of action
Two non-parallel vectors must be combined by use of
vector triangle (head – to – tail rule) or parallelogram
law. Each one of two vectors V₁ and V₂ is represented
in magnitude and direction by the side of a triangle. The
combined effect or resultant R, is given in magnitude
and direction by the third side of the triangle.
The resultant may be found by means of a scale diagram.
Alternatively, having drawn a sketch of the vector
triangle, the problem may be solved using trigonometry
Resolution of vectors
Two vectors may be added together to produce a single resultant. This
resultant behaves in the same way as the two individual vectors. It follows
that a single vector may be split up, or resolved, into two or more components. The combined effect of the components is the same as the original vector.
Consider a force of magnitude F
acting at an angle π above the
horizontal. A vector triangle can
be drawn with a component Fh in
the horizontal direction and a
component Fv acting vertically.
Since the vector F and its
components form a right angled triangle, then
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