Units, Dimensions & Mechanics

Physical quantities and SI units

Base quantities (SI)

• Since 2019, all SI base units are defined by fixing the values of fundamental constants (e.g. Planck constant for the kilogram, speed of light for the metre, elementary charge for the ampere, Boltzmann constant for the kelvin, Avogadro constant for the mole).
• Supplementary/derived units: radian (plane angle), steradian (solid angle); newton (N = kg·m/s²), pascal (Pa = N/m²), joule (J = N·m), watt (W = J/s), coulomb, volt, ohm, hertz.
• Prefixes: nano (10⁻⁹), micro (10⁻⁶), milli (10⁻³), kilo (10³), mega (10⁶), giga (10⁹).

Dimensions

The dimensional formula expresses a quantity in terms of base quantities [M], [L], [T] (and [A], [K] where needed).

Uses of dimensional analysis

1. Checking correctness of equations (principle of homogeneity — all terms must have the same dimensions).
2. Converting units between systems.
3. Deriving relations between quantities (e.g. pendulum period).

Limitations

• Cannot find dimensionless constants (like 2π).
• Fails for relations involving trigonometric, exponential or logarithmic functions and sums of terms.
• Cannot distinguish quantities with the same dimensions (work and torque).

Errors and significant figures

• Accuracy — closeness to true value; precision — closeness of repeated measurements.
• Systematic errors (instrumental, zero error, personal) and random errors.
• Absolute, relative and percentage errors.
• Propagation: for $Z=A\pm B$, absolute errors add; for $Z=AB$ or $A/B$, relative errors add; for $Z=A^n$, relative error multiplies by $n$.
• Significant figures: results of multiplication/division keep the least number of significant figures; addition/subtraction keep the least decimal places.

Kinematics

Projectile motion

Circular motion

• Angular velocity $\omega =v/r$; centripetal acceleration $a=\frac{v^2}{r}={\omega }^{2}r$; centripetal force $F=\frac{m{v}^2}{r}$.
• Banking of roads: $\tan \theta =\frac{v^2}{rg}$ (without friction) — see Geometric Design of Highways (superelevation).

Laws of motion

• First law — inertia; second law — $\vec{F}=\frac{d\vec{p}}{dt}=m\vec{a}$; third law — action–reaction.
• Momentum $p=mv$; impulse $J=F\Delta t=\Delta p$.
• Conservation of linear momentum — in the absence of external force.
• Friction: static $f_s\le {\mu }_{s}N$; kinetic $f_k={\mu }_{k}N$ (${\mu }_k<{\mu }_s$); angle of repose $\tan \theta ={\mu }_s$.

Work, energy and power

• $W=\vec{F}\cdot \vec{s}$; kinetic energy $\frac{1}{2}m{v}^2$; potential energy $mgh$ (gravitational), $\frac{1}{2}k{x}^2$ (spring).
• Work–energy theorem: net work = change in KE.
• Power $P=W/t=Fv$.
• Collisions: momentum conserved in all; kinetic energy conserved only in elastic collisions; coefficient of restitution $e=\frac{\text{relative velocity of separation}}{\text{relative velocity of approach}}$ (1 elastic, 0 perfectly inelastic).

Rotational motion