Lesson 28 — Applications of Vectors in Physics

Vectors in mechanics

Fundamental principles

1. Static equilibrium: $\sum \vec F_i = \vec 0$ (object at rest or in uniform rectilinear motion).
2. Newton's 2nd law: $\sum \vec F_i = m \vec a$.
3. Decomposition: for a ramp of inclination $\theta$, gravity $\vec g = -g\hat\jmath$ projects to $-g\sin\theta$ parallel (downhill) and $-g\cos\theta$ normal.

Work and energy

• Work: $W = \vec F \cdot \vec d$.
• Power: $P = \vec F \cdot \vec v$.
• Kinetic energy: $K = \frac{1}{2} m |\vec v|^2$.
• Work-energy theorem: $W_{\text{total}} = \Delta K$.

Vector kinematics

• Position: $\vec r(t)$.
• Velocity: $\vec v(t) = d\vec r/dt$.
• Acceleration: $\vec a(t) = d\vec v/dt$.

For uniformly accelerated motion: $\vec r(t) = \vec r_0 + \vec v_0 t + \frac{1}{2} \vec a t^2$

Classical forces

Equilibrium in 2D

3 independent equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ (moment about any point $O$).