Engineering Maths I — Formulae, Ohm's Law & Mechanical Advantage
Why Formulae Matter in Engineering
Engineers solve practical problems with mathematics. Whether they are designing a circuit to power a security alarm or calculating how much force a lever needs to lift a heavy component, they rely on tried-and-tested formulae to get reliable answers. The external exam includes short and extended responses in practical engineering contexts. When a calculation asks for working, show the formula, substitute correctly and include the unit.
Ohm’s law states that \(V = IR\), where V is voltage in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω). Rearranging gives \(I = \frac{V}{R}\) and \(R = \frac{V}{I}\).
Ohm’s Law — Voltage, Current and Resistance
In any simple electrical circuit, three quantities are linked. Voltage (V) is the “push” that drives electrons around the circuit. Current (I) is the rate of flow of charge, measured in amperes (A). Resistance (R) is the opposition to that flow, measured in ohms (Ω). The relationship is:
$$V = IR$$
If you know any two of the three quantities you can always find the third by rearranging the formula. Engineers use a triangle to help: write V at the top, I and R side by side at the bottom. Cover the quantity you want and read off the operation that remains.
Problem: A torch uses a 12 V battery and has a total circuit resistance of 6 Ω. Calculate the current flowing through the bulb.
Step 1 — Identify the formula. We need current (I), so rearrange Ohm’s law: \(I = \frac{V}{R}\)
Step 2 — Substitute the values with units. \(I = \frac{12\,\text{V}}{6\,\Omega}\)
Step 3 — Calculate and state the unit. \(I = 2\,\text{A}\)
The current flowing through the bulb is 2 amperes. Always write the unit — an answer of “2” on its own will not earn full marks for a calculate question.
Problem: A kettle connected to a 240 V mains supply draws a current of 8 A. Calculate the resistance of the heating element.
Step 1 — Identify the formula. We need resistance (R), so rearrange: \(R = \frac{V}{I}\)
Step 2 — Substitute. \(R = \frac{240\,\text{V}}{8\,\text{A}}\)
Step 3 — Calculate and state the unit. \(R = 30\,\Omega\)
The heating element has a resistance of 30 ohms. Notice how every step carries its units through — this is essential exam technique for calculate questions.
Electrical Power
Power is the rate at which a circuit transfers energy. It is measured in watts (W). Three equivalent formulae link power to voltage, current and resistance:
$$P = IV \qquad P = I^2 R \qquad P = \frac{V^2}{R}$$
Use whichever version matches the quantities you already know. For example, if you know current and voltage, use \(P = IV\). If you only know current and resistance, use \(P = I^2 R\).
Mechanical Advantage — Getting More Force Out
Mechanical advantage (MA) tells you how much a machine multiplies your applied force. It is a dimensionless ratio — it has no unit because it compares two forces in the same unit:
$$\text{MA} = \frac{\text{Load}}{\text{Effort}}$$
Load is the force the machine must overcome (for example, the weight of an object being lifted), measured in newtons (N). Effort is the force you apply to the machine, also in newtons (N). An MA greater than 1 means the machine makes the task easier by magnifying the force you put in. An MA less than 1 means the machine reduces force but increases speed — useful in different applications.
Mechanical advantage = Load ÷ Effort. If MA = 4, the machine multiplies your effort four times — a 50 N push produces 200 N of output force. Mechanical advantage has no unit because it is a pure ratio of two forces.
Velocity Ratio — How Far the Machine Moves
Velocity ratio (VR) is a separate ratio that compares the distances moved, not the forces. It tells you how many metres the effort must travel for every metre the load moves:
$$\text{VR} = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}}$$
A lever with VR = 3 means the effort end of the lever travels 3 m for every 1 m the load end moves. Like MA, VR is dimensionless. A common misconception is to confuse MA and VR — remember that MA is about force while VR is about distance.
Efficiency — How Well a Machine Uses Energy
In a perfect (ideal) machine there are no energy losses and MA equals VR exactly. Real machines always have friction and other losses, so some input energy is wasted as heat. Efficiency expresses how close a machine comes to the ideal:
$$\text{Efficiency} = \frac{\text{MA}}{\text{VR}} \times 100\%$$
An efficiency of 100% is physically impossible for a real machine. A well-lubricated lever might reach 80–95%. Efficiency must always be between 0% and 100%; if your calculation gives a value above 100% you have made an error.
Problem: A lever is used to lift a 600 N crate. The effort applied is 150 N. The effort end of the lever moves 2 m and the load end moves 0.4 m. Calculate (a) the mechanical advantage, (b) the velocity ratio, and (c) the efficiency of the lever.
(a) Mechanical advantage:
\(\text{MA} = \frac{\text{Load}}{\text{Effort}} = \frac{600\,\text{N}}{150\,\text{N}} = 4\)
(b) Velocity ratio:
\(\text{VR} = \frac{\text{Distance (effort)}}{\text{Distance (load)}} = \frac{2\,\text{m}}{0.4\,\text{m}} = 5\)
(c) Efficiency:
\(\text{Efficiency} = \frac{\text{MA}}{\text{VR}} \times 100\% = \frac{4}{5} \times 100\% = 80\%\)
The lever is 80% efficient. The 20% “lost” is dissipated as heat through friction at the pivot. Note how MA and VR are both dimensionless — no units are written alongside them.
Levers and Classes
Levers are one of the simplest machines. All levers have three elements: a load (the force being overcome), an effort (the force being applied) and a pivot (also called the fulcrum). The three classes of lever differ in where the pivot sits relative to the load and effort. A first-class lever has the pivot between load and effort (a see-saw, or a pair of pliers). A second-class lever has the load between the pivot and the effort (a wheelbarrow). A third-class lever has the effort between the pivot and the load (a pair of tweezers or a human forearm). All three classes follow the same MA and VR formulae.