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Chapter 1
Basics
1.1 Introduction
In this chapter there is a review of some basic engineering science terms and, as within this book and your studies you will come across many equations, the manipulation of equations and the units in which quantities are specified. Engineers make measurements to enable theories to be tested, relationships to be determined, values to be determined in order to predict how components might behave when in use and answers obtained to questions of the form ‘What happens if?’ Thus, there might be measurements of the current through a resistor and the voltage across it in order to determine the resistance. Thus, in this introductory chapter, there is a discussion of the measurement and collection of data and the errors that can occur.
1.2 Basic terms
Mass
The mass of a body is the quantity of matter in the body. The greater the mass of a body the more difficult it is to accelerate it. Mass thus represents the inertia or ‘reluctance to accelerate’. It has the SI unit of kg.
Density
If a body has a mass m and volume V, its density ρ is:
- Density has the SI unit of kg/m3.
- 3 Relative density
- Relative density is by what factor the density of a substance is greater than that of water and is thus defined as:
- Since relative density is a ratio of two quantities in the same units, it is purely a number and has no units.
Example
If the density of water at 20°C is 1000 kg/m3 and the density of copper is 8900 kg/m3, what is the relative density of copper?
Relative density of copper = 8900/1000 = 8.9
4 Force
We might describe forces as pushes and pulls. If you pull a spring between your hands we can say that your hands are applying forces to the ends of the spring. If there is an unbalanced force acting on an object it accelerates. Force has the SI unit of the newton (N).
5 Weight
The weight of a body is the gravitational force acting on it and which has to be opposed if the body is not to fall. The weight of a body of mass m where the acceleration due to gravity is g is mg. Weight, as a force, has the SI unit of N.
Example
What is the weight of a block with a mass of 2 kg if the acceleration due to gravity is 9.8 m/s2?
Weight = mg = 2 x 9.8 = 19.6 N
- It has the SI unit of N/m2, this being given the special name of pascal (Pa).
1.3 Manipulating equations
The following are basic rules for manipulating equations:
- Adding the same quantity to, or subtracting the same quantity from, both sides of an equation does not change the equality.
- Multiplying, or dividing, both sides of an equation by the same non-zero quantity does not change the equality.
In general, whatever mathematical operation we do to one side of an equation, provided we do the same to the other side of the equation then the balance is not affected.
The term transposition is used when a quantity is moved from one side of an equation to the other side. The following are basic rules for use with transposition:
- A quantity which is added on the left-hand side of an equation becomes subtracted on the right-hand side.
- A quantity which is subtracted on the left-hand side of an equation becomes added on the right-hand side.
- A quantity which is multiplying on the right-hand side of an equation becomes a dividing quantity on the left-hand side.
- A quantity which is dividing on the left-hand side of an equation becomes a multiplying quantity on the right-hand side.
Suppose we have the equation F = kx and we want to solve the equation for x in terms of the other quantities. Writing the equation as kx = F and then transposing the k from the left-hand side to the right-hand side (or dividing both sides by k) gives
Example
Determine L in the equation
Squaring both sides of the equation gives us the same as multiplying both sides of the equation by the same quantity since 2 is the same as √(L/10):
Hence, we have L = 40. We can check this result by putting the value in the original equation to give 2 = √(40/10).
Brackets are used to show terms are grouped together, e.g. 2(x + 3) indicates that we must regard the x + 3 as a single term which is multiplied by 2. Thus when removing brackets, each term within the bracket is multiplied by the quantity outside the bracket. When a bracket has a + sign in front of the bracket then effectively we are multiplying all the terms in the bracket by +1. When a bracket has a - sign in front of the bracket then we are multiplying all the terms in the bracket by -1. When we have (a - b) (c + d) then, following the above rule for the removal of brackets, each term within the first bracket must be multiplied by the quantity inside the second bracket to give:
a(c + d) + b(c + d) = ac + ad + bc + bd
As an example, consider the following equation for the variation of resistance R of a conductor w...
