Forces

Five important forces to know are:

Gravitational force:
The gravitational force acting on a body of mass $m_1$ [kg] at position $\vec{r}_1$ [m] due to a body of mass $m_2$ [kg] at position $\vec{r}_2$ [m] is,

\[ \begin{equation} \large \vec{F} = -\frac{Gm_1m_2}{|\vec{r}_1-\vec{r}_2 |^2}\hat{r}_{2\rightarrow 1} \hspace{1cm}\text{[N]}. \end{equation} \]

Here $G$ = 6.6726×10-11 N m²/kg² is the gravitational constant and $\hat{r}_{2\rightarrow 1}=\frac{\vec{r}_1-\vec{r}_2}{|\vec{r}_1-\vec{r}_2|}$ is the unit vector pointing from $\vec{r}_2$ towards $\vec{r}_1$.

Coulomb force:
The Coulomb force acting on a particle of charge $q_1$ [C] at position $\vec{r}_1$ [m] due to a particle of charge $q_2$ [C] at position $\vec{r}_2$ [m] is,

\[ \begin{equation} \large \vec{F} = \frac{q_1q_2}{4\pi\epsilon_0 |\vec{r}_1-\vec{r}_2 |^2}\hat{r}_{2\rightarrow 1} \hspace{1cm}\text{[N]}. \end{equation} \]

Here $\epsilon_0$ = 8.854187817×10-12 F/m is the permittivity constant and $\hat{r}_{2\rightarrow 1}=\frac{\vec{r}_1-\vec{r}_2}{|\vec{r}_1-\vec{r}_2|}$ is the unit vector pointing from $\vec{r}_2$ towards $\vec{r}_1$.

Linear spring force:
A linear spring exerts a force that is proportional to the displacement $x$ [m] of the spring from its equilibrium position $x_0$.

\[ \begin{equation} \large F = -k(x - x_0) \hspace{1cm}\text{[N]}. \end{equation} \]

Here $k$ is the spring constant measured in units of N/m.

Lorentz force:
The force on a particle of charge $q$ [C] moving at velocity $\vec{v}$ [m/s] in an electric field $\vec{E}$ [V/m] and a magnetic field $\vec{B}$ [T] is,

\[ \begin{equation} \large \vec{F} = q(\vec{E}+\vec{v}\times\vec{B}) \hspace{1cm}\text{[N]}. \end{equation} \]

Drag force:
The drag force is a frictional force that points in the opposite direction as the velocity $\vec{v}$ of an object,

\[ \begin{equation} \large \vec{F}_{drag} = -b_1\vec{v} - b_2\vec{v}|\vec{v}|. \end{equation} \]

Here $b_1$ [N s/m] and $b_2$ [N s²/m²] are constants. For a low Reynolds number, the linear term $-b_1\vec{v}$ usually dominates whereas for a high Reynolds number, the quadratic term $-b_2\vec{v}|\vec{v}|$ dominates.

Questions