Examples of Forces

2.4. Examples of Forces#

2.4.1. Gravitation#

2.4.1.1. Universal Law of Gravitation#

The force $\vec{F}_{12}$ exerted by a mass $m_2$ at $P_2$ on a mass $m_1$ at $P_1$ is described by Newton’s Universal Law of Gravitation,

\[\vec{F}_{12} = G \, m_1 \, m_2 \dfrac{P_2 - P_1}{|P_2 - P_1|^3} \ ,\]

or

\[\vec{F}_{12} = G \, m_1 \, m_2 \dfrac{\hat{r}_{12}}{|\vec{r}_{12}|^2} \ ,\]

where $\vec{r}_{12} = (P_2 - P_1)$ is the vector pointing from point $P_1$ to point $P_2$, $r_{12} = |\vec{r}_{12}|$ is its magnitude, and $\hat{r}_{12} = \frac{\vec{r}_{12}}{|\vec{r}_{12}|}$ is the unit vector in the same direction. The universal gravitational constant $G$ is

\[G = 6.67 \cdot 10^{-11} \frac{N \, m^2}{kg^2}\]

and is considered a constant of nature.

Principle of Superposition of Causes and Effects (PSCE). Principle of superposition holds, i.e. the force acting on a mass $m$ placed in $P$ due to a set of $N$ masses $\{ m_k \}_{k=1:N}$ placed in $P_k$ is the sum of individual forces $\vec{F}_{k}$,

(2.3)#\[\vec{F} = \sum_{k=1}^N \vec{F}_{k} = G \, m \, \sum_{k=1}^N m_k \dfrac{P_k - P}{|P_k - P|^3} \ .\]

2.4.1.2. Gravitational Field#

The gravitational field generated by a set of masses $\{ m_k \}_{k=1:N}$ located at $P_k$ is a vector field associating a vector with physical dimensions $\frac{\text{[force]}}{\text{[mass]}}$ to each point in space $P$, that can be thought as the force per unit-mass acting on a test mass $m$ placed in $P$, whose expression directly follows from (2.3)

\[\vec{g}(P) = \dfrac{\vec{F}}{m} = G \, \sum_{k=1}^N m_k \dfrac{P_k - P}{|P_k - P|^3} \ .\]

Given the gravitational field $\vec{g}(P)$, the gravitational force experienced by a system of mass $m$ at $P$ can be written as

\[\vec{F}_g = m \vec{g}(P)\]

Gravitational Potential Energy. Gravitational potential of a system of 2 masses reads

\[V(P) = - G \, m \, m_1 \frac{1}{|P - P_1|} \ ,\]

as it can be easily shown evaluating its gradient,

\[\begin{split}\begin{aligned} \nabla V(P) & = - G \, m \, m_1 \, \hat{x}_k \, \dfrac{\partial}{\partial x_k} \dfrac{1}{|P - P_1|} = \\ & = - G \, m \, m_1 \, \hat{x}_k \, \left( - \dfrac{1}{|P - P_1|^2} \right) \dfrac{\partial}{\partial x_k} |P - P_1| = \\ & = G \, m \, m_1 \, \hat{x}_k \, \left( \dfrac{1}{|P - P_1|^2} \right) \dfrac{x_k - x_{1,k}}{|P - P_1|} = \\ & = G \, m \, m_1 \, \dfrac{x_k - x_{1,k}}{|P - P_1|^3} \, \hat{x}_k= \\ & = G \, m \, m_1 \, \dfrac{P-P_1}{|P - P_1|^3} \ . \end{aligned}\end{split}\]

Potential energy stored in a system of $N$ point masses $\{ m_k \}_{k=1:N}$ coincides with the work needed to build the system - a common choice to set the arbitrary additional constant of the energy is setting it equal to zero when masses are at infinite distances -, namely

\[V(P_k) = \sum_{\{i,k\}, i \ne k} G \, m_i \, m_k \frac{1}{|P_i - P_k|} \ ,\]

summing over different unordered pairs, i.e. $\{ 1, 2 \}$ and $\{2,1\}$ are the same pair and thus considered only once, or

\[V(P_k) = \frac{1}{2} \sum_{(i,k), i\ne k} G \, m_i \, m_k \frac{1}{|P_i - P_k|} \ ,\]

summing over different ordered pairs, i.e. $(1,2)$ and $(2,1)$ are different pairs.

2.4.1.3. Gravitational Field Near Earth’s Surface#

Within a limited domain near Earth’s surface, it is common to approximate Earth’s gravitational field as a uniform field, directed along the local vertical toward the center of the Earth, with intensity $g = G \frac{M_E}{R_E^2}$.

This model can be derived by approximating the position vector relative to the Earth’s center $P - P_E \sim R_E \hat{r}$ and the unit vector identifying the direction from a point in the domain to the Earth’s center with the local vertical $\hat{r}_{12} \sim - \hat{z}$,

\[\vec{g}(\vec{r}) = - G \dfrac{M_E}{R_E^2} \hat{z} = - g \hat{z} \ .\]

The gravitational force experienced by a body of mass $m$ near Earth’s surface is thus

\[\vec{F}_g = - m g \hat{z} \ ,\]

commonly referred to as weight.

Gravitational Potential Energy. It can be shown that the gravitational potential near Earth’s surface becomes

\[V(P) = m \, g \, z_P \ .\]

2.4.1.4. Gravitational field of a continuous mass distribution#

Mass density field $\rho(\vec{r}_0)$ for $\vec{r}_0 \in V_0$ produces the gravitational field in $\vec{r}$,

\[\vec{g}(\vec{r}) = \int_{\vec{r}_0 \in V_0} d \vec{g}(\vec{r}, \vec{r}_0) = - \int_{\vec{r}_0 \in V_0} G \rho(\vec{r}_0) \dfrac{\vec{r} - \vec{r}_0}{\left|\vec{r} - \vec{r}_0\right|^3} \ .\]

By direct computation, the gravitational potential $\phi(\vec{r})$, s.t. $\vec{g} = \nabla \phi$, reads

\[\phi(\vec{r}) = \int_{\vec{r}_0 \in V_0} G \rho(\vec{r}_0) \dfrac{1}{|\vec{r}-\vec{r}_0|}\]

2.4.1.4.1. Gauss’ law for the gravitational field#

The flux of the gravitational field produced by mass distribution $\rho(\vec{r}_0)$ in volume $V_0$ thorugh a closed surface $\partial V$ reads

\[\begin{split}\begin{aligned} \oint_{\vec{r} \in \partial V} \vec{g}(\vec{r}) \cdot \hat{n}(\vec{r}) & = - G \oint_{\vec{r} \in \partial V} \int_{\vec{r}_0 \in V_0} \rho(\vec{r}_0) \dfrac{\vec{r} - \vec{r}_0}{\left|\vec{r} - \vec{r}_0\right|^3} \cdot \hat{n}(\vec{r}) = \\ & = - G \int_{\vec{r}_0 \in V_0} \rho(\vec{r}_0) \oint_{\vec{r} \in \partial V} \dfrac{\vec{r} - \vec{r}_0}{\left|\vec{r} - \vec{r}_0\right|^3} \cdot \hat{n}(\vec{r}) \end{aligned}\end{split}\]

The inner integral can be written as the solid angle of the surface $\partial V$ as seen by the point $\vec{r}_0$, whose value is

\[\begin{split} \oint_{\vec{r} \in \partial V} \dfrac{\vec{r} - \vec{r}_0}{\left|\vec{r} - \vec{r}_0\right|^3} \cdot \hat{n}(\vec{r}) = 4 \pi \begin{cases} 1 & \quad \text{if $ \vec{r}_0 \in V$} \\ \theta(\vec{r}_0, \partial V) & \quad \text{if $ \vec{r}_0 \in \partial V$} \\ 0 & \quad \text{if $ \vec{r}_0 \notin V \cup \partial V$} \\ \end{cases} \end{split}\]

Thus, net contributions of the flux of the gravitational field $\vec{g}(\vec{r})$ through $\partial V$ come only from points $\vec{r}_0$ inside $V$, $\vec{r}_0 \in V$.1 Thus the flux becomes

\[ \oint_{\vec{r} \in \partial V} \vec{g}(\vec{r}) \cdot \hat{n}(\vec{r}) = - G \int_{\vec{r}_0 \in V_0 \cap V} 4 \pi \rho(\vec{r}_0) \]

or, setting $\rho(\vec{r}_0) = $ in all the points $\vec{r} \in V$, $\vec{r} \notin V_0$, and changing the name of the dummy integration variable $\vec{r}_0 \rightarrow \vec{r}$,

\[ \oint_{\vec{r} \in \partial V} \vec{g}(\vec{r}) \cdot \hat{n}(\vec{r}) = - G \int_{\vec{r} \in V} 4 \pi \rho(\vec{r}) \ . \]

If the gravitational field $\vec{g}(\vec{r})$ is regular enough for the divergence theorem to hold, it follows

(2.4)#\[ \oint_{\vec{r} \in \partial V} \vec{g}(\vec{r}) \cdot \hat{n}(\vec{r}) = \int_{\vec{r} \in V} \nabla \cdot \vec{g}(\vec{r}) = - G \int_{\vec{r} \in V} 4 \pi \rho(\vec{r}) \ , \]

or, for the arbitrariety of the volume $V$,

\[- \nabla \cdot \vec{g} = 4 \pi G \rho \ .\]

Introducing the gravitational potential $\phi(\vec{r})$, whose gradient equals the gravitational field $\nabla \phi = \vec{g}$ by definition, a Poisson equation for the gravitational potential follows

(2.5)#\[- \nabla^2 \phi = 4 \pi G \rho \ .\]

2.4.2. Elastic Actions: Linear Springs#

todo

1: Contributions from points outside $\partial V$ are identically zero; contributions from surface $\partial V$ are zero if volume mass density $\rho(\vec{r}_0)$ is regular enough, i.e. it contains Dirac’s $\delta$ representing surface distribution that would have non-negligible contributions in integration over $V$.