Mechanics – Core Topics Overview

Mathematical Models in Mechanics

• Modelling cycle: real situation → assumptions → mathematical model → solve → interpret/validate.
• Typical assumptions: particle, rigid body, uniform rod, light/inextensible string, smooth surface, negligible air resistance.
• Quantities: scalars vs vectors; SI base units (m, kg, s); derived units – Newton $\text{N}$, Joule $\text{J}$, Pascal $\text{Pa}$.

Constant Acceleration (Kinematics)

• Applicable when acceleration $a$ is constant.
• SUVAT formulae:
– $v = u + at$
– $s = \frac{(u+v)}{2}t$
– $s = ut + \frac{1}{2} a t^2$
– $v^2 = u^2 + 2as$
• Graph links:
– Displacement–time: gradient = velocity.
– Velocity–time: gradient = acceleration; area = displacement.
– Acceleration–time: area = change in velocity.
• Vertical motion under gravity: take $g \approx 9.8\,\text{m s}^{-2}$ downward.

Vectors in Mechanics

• Component form: $\mathbf{a} = a<em>i\mathbf{i} + a</em>j\mathbf{j}$.
• Magnitude $|\mathbf{a}| = \sqrt{a<em>i^2 + a</em>j^2}$; direction $\theta = \tan^{-1}\left(\frac{a<em>j}{a</em>i}\right)$.
• Add/subtract vectors and scale component-wise.
• Velocity as a vector: $\mathbf{v} = \frac{d\mathbf{r}}{dt}$; apply SUVAT to each component for constant acceleration.

Dynamics of a Particle

• Free-body diagram: show all external forces (weight $mg$, tension $T$, normal reaction $R$, friction $F$, etc.).
• Newton’s 2nd law (vector): $\sum \mathbf{F} = m\mathbf{a}$.
• Two-dimensional motion: resolve forces along chosen perpendicular axes.
• Connected particles (light, inextensible string; smooth pulley): same magnitude of tension and acceleration.

Forces and Friction

• Resolving forces: use perpendicular axes, e.g. along/normal to an inclined plane.
• Incline components: down slope $mg \sin\theta$, normal $mg \cos\theta$.
• Friction: $F \le \mu R$; limiting (impending) value $F_{\max} = \mu R$.

Momentum and Impulse

• Momentum: $\mathbf{p} = m\mathbf{v}$.
• Impulse: $\mathbf{J} = \int \mathbf{F}\,dt = \Delta \mathbf{p}$ (for constant force $\mathbf{J} = \mathbf{F}t$).
• Conservation of linear momentum (isolated system): total $\mathbf{p}$ before = total $\mathbf{p}$ after.