3. A Study of Basic Physics and Mathematics

Physics is a natural science, mostly concerned with matter and how matter interacts. The study of physics is very broad, and incorporates mainly Mechanics (Or “Newtonian” Physics) and Atomic Physics. Physics also incorporates Chemistry – with studies such as radioactive materials – and other branches of science, such as Astronomy or Engineering.

Perhaps the most basic form of Physics for many is Mechanics, often referred to "Newtonian" Physics after Sir Isaac Newton in honour of his contributions to the discipline.

Most physics relies closely on mathematics and so an understanding of advanced mathematics, particularly trigonometry, mechanics, algebra, logarithms, working with graphs and integration and differentiation are extremely important to physicists. In fact, almost all forms of mathematics can crop up in physics.

In mathematics, similar lines of data often have to be recorded written and this can tedious to write or translate. For this reason, mathematicians often use shorthand symbols to explain or illustrate an argument. These are important to understand; as without them equations can often appear meaningless. A key of symbols is recorded in this articles glossary.

To start this introduction to physics, a few basic mathematical rules will be examined: firstly, algebra. Algebra is one of the main branches of mathematics, and can range from very basic equations to large, complex structures. Algebra is the study of quantity and relationships usually between independent variables. Simple algebra is found in the following form.

Let x=3 and y=2 and 4x + 2y = z then z=16

We can prove this by replacing x and y with their variable sums. As x is equal to 3 and there are 4 lots of x, the sum of 4x is 12. As with y, the sum of 2y is 4. The product of 12 and 4 is 16, so z is 16. If we change the input variables x and y, our output, z, is changed by a proportional amount.

Algebra in physics works in the same way. In mechanics, an objects force is directly proportional to the mass of the object multiplied by the acceleration acting on the object. This tells use that Force (F) is directly equal to Mass (m) times Acceleration (a). Hence:

F=ma

As with the equations above, the equation F=ma relies on 2 input variables, mass and acceleration, and provides an output, the force of the object. Using this data, we can also re-arrange formulas to change an outcome. Let us use the formula F=ma again. If F is equal to mass times acceleration then Force divided by either mass or acceleration must be equal to both acceleration or mass respectively.

Hence if F=ma, a=F/m and m=F/a

Another important aspect of physics is the use of constants. We know, for example, that the diameter of a circle is directly proportional to the circle’s perimeter. This tells us that D is directly proportional to P. This proportionality is constant, so therefore r is directly equal to a constant multiplied by P. This constant is approximately 3.14172, but to save time, mathematicians simply write the symbol pi (π).

So we can alter the inputs so find the necessary outputs and use symbols to show direct proportionalities. To use a more complex example, let us use the equation for the gravitational attraction between 2 objects: mass M and mass m. The equation is as follows.

F=GmM/r² where G is a constant 0.0000000000667 *

*In practice, this is written as 6.67 x 10^-11

Or, more simply, the gravitational force between two objects (F) is proportional to the sum of the mass of the two objects divided by the distance (r) separating the two objects squared (Note: distance is measured from the centre of mass of each object). As you can see, writing F=GmM/r² is much quicker and simpler.

We can also rearrange this equation to find out other inputs. For example, to calculate r we could rearrange in the following way:

F=GmM/r²

F r²=GmM

r²=GmM/F

r=√GmM/F

So the radius r is equal to the square root of the sum of the two masses m and M multiplied by the universal gravitational constant G and divided by the force F.

AAn important aspect of physics that can be implemented in many situations is the use of base units, sometimes referred to as SI units. The principle behind using base units is that every single measurable quantity is measure one or more of 7 base units, which are:

Length - Measured in meters (m)
Mass - Measure in Kilograms (Kg)
Time - Measured in seconds (s)
Electric Current - Measured in Amperes (A)
Temperature - Measured in Degrees Kelvin (K)
Luminosity - Measured in Candela (cd)
Amount of substance - Measured in Moles (mol)

In physics, all measurements can be measured using a combination of these 7 base units. Lets use Force, for example. We know that Force is directly equal to mass multiplied by acceleration. Acceleration is a measurement of how velocity changes over time, so acceleration is equal to speed divided by time. Speed is a measurement of distance over time, so hence:

F=ma

m = Mass, measured in Kilograms (Kg)

a = Acceleration, measured in velocity/time which is velocity per second

velocity = Distance over time which is equal to meters per second or ms^-1

Hence, acceleration = ms^-1 multiplied by s^-1 which is equal to ms^-2

So therefore Force (F) is measured by Kilogram, meters per second squared, equally written as:

F ≡ Kg m s^-2   ('≡' - is the same as/equivalent to)

A force on an object is measured in newtons. One newton indicates one unit of force. As we can see, the base units for the measurement of newtons is 'Kg m s^-2'. This means that one newton is the forced required to increase the velocity of a body of mass 1 Kg by an accelerate of 1 ms^-2. Using simple equations such as this, we can find out what units are measured in and even derive new equations by comparing base units.

There is one other benefit to deriving base units in that it allows you to test whether an equation is homogenous. If we had an equation of Force equals Speed (F=v) then we would know this equation is not correct as the base units are different. For an equation to work it the base units must be homogenous. However, be wary; not all homogenous equations are correct. Take this equation for example.

F ≠ 2 ma   ('≠' - is not equal to)

We know that force is proportional to mass multiplied by acceleration and so, in respects to base units, this equation is homogenous. However, the additional constant '2' changes the equation's output by a factor of two and gives a force that is too large. Remember, an equation must be homogenous to be correct, but an equation is not necessarily correct if it is homogenous.

Using these basic principles and a bit of common sense, you can tackle most practical physics simply using suitable equations. Knowing when and how to apply certain equations is vital to understanding different problems and these simple tips will prepare you for even some of the most complex of physics. As with mathematics, theories can become very complex and hard to grasp but, at the end of the day, the process of mathematics boils down to addition, subtraction, multiplication and division among various numerical values.

For more information or quick reminders on mathematical terminology, meanings of symbols or values of constants, please refer to the glossary.

Please also note the capitalisation of certain variables. A capital 'F' denotes a force, whilst a lowercase 'f' denotes a frequency. As with 'a' showing acceleration, whilst 'A' shows an Amplitude - a measurement of current in an electrical circuit. Voltage is similarly measured in Volts (V) whilst velocity is recorded as a lowercase 'v', usually written in italic form as a lowercase and uppercase V are very similar.