# Classical mechanics

## Contents

### Linear motion

 $$\begin{array}{l} x &\text{position} &\mathrm{m} \\ v &\text{velocity} &\mathrm{m \cdot s^{-1} } \\ a &\text{acceleration} &\mathrm{m \cdot s^{-2} } \\ t &\text{time} &\mathrm{s} \end{array}$$
$$\begin{array}{l} x = x_0 + v_0 \, t + \frac{a t^2}{2} \end{array}$$

### Circular motion

 $$\begin{array}{l} \theta &\text{angle} &\mathrm{rad} \\ \omega &\text{angular velocity} &\mathrm{rad \cdot s^{-1} } \\ \alpha &\text{angular acceleration} &\mathrm{rad \cdot s^{-2} } \\ t &\text{time} &\mathrm{s} \\ \mathbf{x} &\text{position} &\mathrm{m} \\ \mathbf{v} &\text{tangential velocity} &\mathrm{m \cdot s^{-1} } \\ \mathbf{a} &\text{centripetal acceleration} &\mathrm{m \cdot s^{-2} } \end{array}$$
$$\begin{array}{ll} \theta = \omega \, t \\ \mathbf{x} = \langle \, r \, \cos \theta , \, r \, \sin \theta \, \rangle && \lvert\mathbf{x}\rvert = r \\ \mathbf{v} = \langle \, - \omega r \, \sin \theta, \, \omega r \, \cos \theta \, \rangle && \lvert\mathbf{v}\rvert = \omega \, r \\ \mathbf{a} = - \omega^2 \mathbf{x} && \lvert\mathbf{a}\rvert = \omega^2 r \\ \end{array}$$

### Simple harmonic motion

 $$\begin{array}{l} x &\text{displacement} &\mathrm{m} \\ v &\text{acceleration} &\mathrm{m \cdot s^{-1} } \\ a &\text{acceleration} &\mathrm{m \cdot s^{-2} } \\ F &\text{restoring force} &\mathrm{N} \\ m &\text{mass} &\mathrm{kg} \\ k &\text{constant} &\mathrm{N \cdot m^{-1} } \\ A &\text{amplitute} &\mathrm{m} \\ \omega &\text{angular velocity} &\mathrm{rad \cdot s^{-1} } \\ t &\text{time} &\mathrm{s} \\ \phi &\text{phase shift} &\mathrm{rad} \\ T &\text{period} &\mathrm{s} \\ f &\text{frequency} &\mathrm{Hz} \\ E &\text{energy} &\mathrm{J} \end{array}$$
$$\begin{array}{l} x = A \, \cos(\omega \, t + \phi) \\ v = -\omega \, A \, \sin(\omega \, t + \phi) \\ a = -\omega^2 x \\ F = ma = -kx \\ \omega = \sqrt{k \, / m} = 2 \pi / \, T \\ f = 1 / \, T \\ E = E_k + E_p = mv^2 / 2 + kx^2 / 2 = kA^2 / 2 \end{array}$$

### Pendulum

 $$\begin{array}{l} \theta &\text{angle} &\mathrm{rad} \\ \omega &\text{angular velocity} &\mathrm{rad \cdot s^{-1} } \\ \alpha &\text{angular acceleration} &\mathrm{rad \cdot s^{-2} } \\ \mathbf{g} &\text{gravitational acceleration} &\mathrm{m \cdot s^{-2} } \\ L &\text{radius} &\mathrm{m} \\ \omega^2 &\text{angular frequency} &\mathrm{rad \cdot s^{-1} } \\ T &\text{period} &\mathrm{s} \\ f &\text{frequency} &\mathrm{Hz} \end{array}$$
$$\begin{array}{l} \alpha = -\omega^2 \theta \\ T = 2 \, \pi / \omega = 2 \, \pi \sqrt{L / g} \\ \omega^2 = g / L \end{array}$$

### Newton's laws of motion

 $$\begin{array}{l} \mathbf{F} &\text{force} &\mathrm{N} \\ m &\text{mass} &\mathrm{kg} \\ \mathbf{a} &\text{acceleration} &\mathrm{m \cdot s^{-2} } \end{array}$$
1. A body acted on by no $\mathbf{F}_{net}$ moves with constant velocity.
2. $\mathbf{F}_{net} = m \, \mathbf{a}$
3. If object $A$ exerts a force $\mathbf{F}$ on object $B$, then object $B$ exerts a force $-\mathbf{F}$ on object $A$.

### Gravitation

 $$\begin{array}{l} F &\text{force} &\mathrm{N} \\ G &\text{Newton's constant} &\mathrm{m^3 \cdot kg^{-1} \cdot s^{-2} } \\ m &\text{mass} &\mathrm{kg} \\ r &\text{distance} &\mathrm{m} \\ \mathbf{g} &\text{gravitational acceleration} &\mathrm{m \cdot s^{-2} } \\ \mathbf{F_g} &\text{weight} &\mathrm{N} \end{array}$$
$$\begin{array}{ll} F = \frac{G \, m_1 \, m_2}{r^2} && G = 6.67384 \cdot 10^{-11} \, \mathrm{m^3 \cdot kg^{-1} \cdot s^{-2} } \\ \mathbf{F_g} = m \, \mathbf{g}&& \mathbf{g} = 9.80665 \, \mathrm{m \cdot s^{-2} } \end{array}$$

### Friction

 $$\begin{array}{l} \mathbf{F_f} &\text{friction force} &\mathrm{N} \\ \mathbf{F_s} &\text{static friction force} &\mathrm{N} \\ \mathbf{F_n} &\text{normal force} &\mathrm{N} \\ \mu_k &\text{kinetic coefficient of friction} & \\ \mu_s &\text{static coefficient of friction} & \end{array}$$
$$\begin{array}{l} \lvert\mathbf{F_f}\rvert = \mu_k \, \lvert\mathbf{F_n}\rvert \\ \lvert\mathbf{F_s}\rvert \leqslant \mu_s \, \lvert\mathbf{F_n}\rvert \end{array}$$

### Hooke's law

 $$\begin{array}{l} \mathbf{F} &\text{restoring force} &\mathrm{N} \\ k &\text{spring constant} &\mathrm{N \cdot m^{-1} } \\ \mathbf{x} &\text{displacement} &\mathrm{m} \\ \end{array}$$
$$\begin{array}{l} \mathbf{F} = -k \, \mathbf{x} \end{array}$$