Engineering Technician (EngTech) Mathematics and Scientific Principles for Engineering Technicians Syllabus

What is the difference between the mean, median and mode of a data set?

Mean is the arithmetic average $\bar{x} = \frac{\sum x}{n}$; median is the middle value when ordered; mode is the most frequently occurring value.

How is the range of a data set defined?

$$\text{range} = \text{largest value} - \text{smallest value}$$

Distinguish between a scalar and a vector quantity, with examples of each.

A scalar has magnitude only (e.g. mass, speed, energy, temperature). A vector has both magnitude and direction (e.g. force, velocity, acceleration, displacement).

Define the moment of a force about a point.

$$\text{moment} = F \times d$$ the force $F$ multiplied by the perpendicular distance $d$ from the pivot to the line of action of the force. Unit: newton metre ($N\cdot m$).

State the principle of moments for a body in equilibrium.

For equilibrium the sum of clockwise moments about any point equals the sum of anticlockwise moments about that point: $\sum M_{\text{cw}} = \sum M_{\text{acw}}$.

What are the two conditions for a rigid body to be in static equilibrium?

(1) The resultant force is zero: $\sum F = 0$ (in each direction). (2) The resultant moment about any point is zero: $\sum M = 0$.

How do you resolve a force $F$ acting at angle $\theta$ to the horizontal into components?

Horizontal component $F_{x} = F\cos\theta$; vertical component $F_{y} = F\sin\theta$.

State Newton's three laws of motion.

1st: A body stays at rest or moves at constant velocity unless acted on by a resultant force. 2nd: $F = ma$. 3rd: For every action there is an equal and opposite reaction.

What is the difference between speed and velocity?

Speed is a scalar — the rate of change of distance. Velocity is a vector — the rate of change of displacement in a given direction.

State the SUVAT equations of motion for constant acceleration.

$v = u + at$; $\quad s = ut + \frac{1}{2}at^{2}$; $\quad v^{2} = u^{2} + 2as$; $\quad s = \frac{(u+v)}{2}t$.

Define acceleration and give its SI unit.

Acceleration is the rate of change of velocity, $a = \frac{\Delta v}{\Delta t}$, with SI unit $\text{m}\cdot\text{s}^{-2}$.

What is the acceleration due to gravity near the Earth's surface?

Approximately $g = 9.81\,\text{m}\cdot\text{s}^{-2}$ directed downwards.

Define momentum and state its SI unit.

Momentum $p = mv$ (mass times velocity); SI unit $\text{kg}\cdot\text{m}\cdot\text{s}^{-1}$. It is a vector quantity.

State the principle of conservation of linear momentum.

In a closed system with no external resultant force, total momentum before a collision equals total momentum after: $m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}$.

Define the weight of an object and give its formula.

Weight is the gravitational force on a mass: $W = mg$, measured in newtons. It is distinct from mass, which is in kilograms.

Define work done by a constant force and give its formula and unit.

$$W = Fd\cos\theta$$ force times distance moved in the direction of the force. SI unit: the joule, $J$ ($1\,J = 1\,N\cdot m$).

Give the formulas for kinetic energy and gravitational potential energy.

Kinetic energy $E_{k} = \frac{1}{2}mv^{2}$; gravitational potential energy $E_{p} = mgh$.

State the principle of conservation of energy.

Energy cannot be created or destroyed, only transferred or converted from one form to another; the total energy of an isolated system remains constant.

Define power and give two formulas for it.

Power is the rate of doing work or transferring energy: $P = \frac{W}{t}$ and $P = Fv$. SI unit: the watt, $W$ ($1\,W = 1\,J\cdot s^{-1}$).

How is efficiency defined and expressed as a percentage?

$$\eta = \frac{\text{useful output energy (or power)}}{\text{total input energy (or power)}} \times 100\%$$

Define pressure and give its formula and SI unit.

$$p = \frac{F}{A}$$ force per unit area. SI unit: the pascal, $Pa$ ($1\,Pa = 1\,N\cdot m^{-2}$).

What is the formula for the hydrostatic pressure at depth $h$ in a fluid?

$$p = \rho g h$$ where $\rho$ is the fluid density, $g$ gravitational acceleration, and $h$ the depth below the surface.

Define density and give its formula and SI unit.

$$\rho = \frac{m}{V}$$ mass per unit volume. SI unit: $\text{kg}\cdot\text{m}^{-3}$.

State Pascal's principle (the basis of hydraulics).

A pressure applied to an enclosed fluid is transmitted equally and undiminished to every part of the fluid and the walls of its container, enabling force multiplication: $\frac{F_{1}}{A_{1}} = \frac{F_{2}}{A_{2}}$.

State Ohm's law and give its equation.

The current through a conductor is proportional to the voltage across it (at constant temperature): $$V = IR$$ where $V$ is voltage, $I$ current, $R$ resistance.

Give the formulas for electrical power in a resistive circuit.

$P = VI = I^{2}R = \frac{V^{2}}{R}$, measured in watts.

How do you calculate total resistance for resistors in series and in parallel?

Series: $R_{T} = R_{1} + R_{2} + R_{3} + \dots$. Parallel: $\frac{1}{R_{T}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \dots$

Define electric charge in terms of current and time.

$$Q = It$$ charge equals current times time. SI unit: the coulomb, $C$ ($1\,C = 1\,A\cdot s$).

What is the relationship between specific heat capacity and the heat energy needed to change a substance's temperature?

$$Q = mc\Delta T$$ where $m$ is mass, $c$ the specific heat capacity ($J\,kg^{-1}\,K^{-1}$), and $\Delta T$ the temperature change.

Define the coefficient of linear thermal expansion and give the expansion formula.

It is the fractional change in length per degree temperature rise, $\alpha$. The change in length is $$\Delta L = \alpha L_{0} \Delta T$$ where $L_{0}$ is the original length.

Distinguish between the tensile strength, stiffness (Young's modulus) and ductility of an engineering material.

Tensile strength is the maximum stress before fracture. Stiffness is resistance to elastic deformation, given by Young's modulus $E = \frac{\text{stress}}{\text{strain}} = \frac{\sigma}{\varepsilon}$. Ductility is the ability to be drawn into wire / undergo large plastic strain before fracture.