Mechanics

Mechanics is the study of motion, forces, energy, and momentum — the foundational physics that describes how objects move and interact. This section covers everything from measurement techniques to gravitational fields and material properties.

Topics Covered

Quantities and Units

• SI base units — kilogram, metre, second, ampere, kelvin, mole
• Derived units — newton ($\text{kg}\,\text{m}\,\text{s}^{-2}$), joule ($\text{kg}\,\text{m}^2\,\text{s}^{-2}$), watt ($\text{kg}\,\text{m}^2\,\text{s}^{-3}$)
• Prefixes — nano ($10^{-9}$), micro ($10^{-6}$), milli ($10^{-3}$), kilo ($10^3$), mega ($10^6$), giga ($10^9$)
• Scalars and vectors — distinguishing quantities that have direction from those that do not; vector addition and resolution

Kinematics

• Equations of motion — $v = u + at$, $s = ut + \frac{1}{2}at^2$, $v^2 = u^2 + 2as$
• Free fall — acceleration due to gravity $g = 9.81\,\text{m/s}^2$; projectile motion; independence of horizontal and vertical components
• Motion graphs — displacement-time (gradient = velocity), velocity-time (gradient = acceleration, area = displacement)

Dynamics

• Newton’s laws of motion — inertia, $F = ma$, action-reaction pairs
• Weight and mass — $W = mg$; the distinction between gravitational field strength and acceleration
• Drag and terminal velocity — the balance of weight and drag; why objects reach a terminal speed
• Momentum — $p = mv$; conservation of momentum; impulse $I = \Delta p = F\Delta t$

Work, Energy, and Power

• Work done — $W = Fs\cos\theta$; the joule
• Kinetic energy — $KE = \frac{1}{2}mv^2$
• Gravitational potential energy — $PE = mgh$ (near Earth’s surface)
• Conservation of energy — energy cannot be created or destroyed; only transformed
• Power — $P = \frac{W}{t} = Fv$; the watt
• Efficiency — $\eta = \frac{\text{useful output}}{\text{total input}} \times 100\%$

Circular Motion and Oscillations

• Circular motion — centripetal acceleration $a = \frac{v^2}{r}$; centripetal force $F = \frac{mv^2}{r}$
• Simple harmonic motion — $a = -\omega^2 x$; period $T = 2\pi\sqrt{\frac{m}{k}}$ (mass-spring), $T = 2\pi\sqrt{\frac{l}{g}}$ (pendulum)
• Resonance — driving frequency equals natural frequency; amplitude increases dramatically

Gravitational Fields

• Gravitational field strength — $g = \frac{GM}{r^2}$ (radial); $g \approx 9.81\,\text{N/kg}$ near surface
• Gravitational potential — $V_g = -\frac{GM}{r}$; escape velocity
• Orbits — satellite motion; geostationary orbit conditions

Properties of Materials

• Density — $\rho = \frac{m}{V}$
• Hooke’s law — $F = k\Delta x$; spring constant; limit of proportionality and elastic limit
• Stress, strain, and Young’s modulus — $\sigma = \frac{F}{A}$, $\varepsilon = \frac{\Delta L}{L}$, $E = \frac{\sigma}{\varepsilon}$
• Stress-strain curves — elastic region, yield point, plastic deformation, ultimate tensile strength, fracture
• Energy stored — $E = \frac{1}{2}F\Delta x = \frac{1}{2}k(\Delta x)^2$

Study Tips

1. Resolve all vectors — in every mechanics problem, choose your axes and resolve forces and velocities into components. Never skip this step.
2. Check dimensional consistency — verify that your final answer has the correct units. If asked for energy, your answer must be in joules.
3. Practise projectile problems — separate horizontal (constant velocity) and vertical (constant acceleration) components. They are independent.
4. Understand the difference between conservation of momentum (always true in a closed system) and conservation of kinetic energy (only in elastic collisions).
5. Draw stress-strain curves — be able to label the elastic region, yield point, UTS, and fracture point, and distinguish between brittle, ductile, and polymeric materials.

How to Use These Notes

Follow the sidebar order. Each page provides physical principles, derivations, worked examples, and exam-style problems. Start with quantities and kinematics before moving to dynamics and energy.