MIT 8.01 - Classical Mechanics & Newtonian Dynamics

A body remains at rest or continues to move in a straight line at a constant velocity unless acted upon by a net external force ($\sum \vec{F}=0⟹\vec{v}=\text{constant}$). This establishes the concept of inertial reference frames.

The net external force on an object equals the rate of change of its linear momentum: ${\vec{F}}_{\text{net}}=\frac{d\vec{p}}{dt}$. For constant mass $m$, this reduces to ${\vec{F}}_{\text{net}}=m\vec{a}$.

When two objects interact, they exert equal and opposite forces on each other: ${\vec{F}}_{A\to B}=-{\vec{F}}_{B\to A}$. Action-reaction pairs act on different bodies simultaneously.

For motion with constant acceleration $a$:
1. $v(t)=v_0+at$
2. $x(t)=x_0+v_{0}t+\frac{1}{2}a{t}^2$
3. $v^2=v_0^2+2a(x-x_0)$
4. $x-x_0=\frac{v_0+v}{2}t$

Acceleration directed radially inward toward the center of curvature for an object undergoing circular motion: $a_c=\frac{v^2}{r}={\omega }^{2}r$, where $v$ is linear speed, $r$ is radius, and $\omega$ is angular velocity.

The net work $W_{\text{net}}$ done by all forces on a particle equals the change in its kinetic energy $K$:
$W_{\text{net}}=\Delta K={\int }_{{\vec{r}}_1}^{{\vec{r}}_2}{\vec{F}}_{\text{net}}\cdot d\vec{r}=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2$

A force $\vec{F}$ is conservative if work done around any closed path is zero ($\oint \vec{F}\cdot d\vec{r}=0$). Potential energy $U$ is defined relative to conservative forces by $\vec{F}=-\nabla U$, or $\Delta U=-W_{\text{internal}}=-{\int }_{x_i}^{x_f}{F}_{x}dx$.

In a system where only conservative forces do work, total mechanical energy is conserved:
$E_{\text{mech}}=K+U=\text{constant}⟹K_i+U_i=K_f+U_f$
If non-conservative forces act: $W_{\text{nc}}=\Delta E_{\text{mech}}$.