Potential energy

A force is called a conservative force if the work that this force performs does not depend on the trajectory that is taken to get from $\vec{r}_1$ to $\vec{r}_2$. The gravitational force, the Coulomb force, a spring force, and the Lorentz force are conservative forces. The drag force is not a conservative force. For every conservative force, a potential energy can be defined by first defining a reference position $\vec{r}_0$. Then the potential energy at $\vec{r}$ is minus the work the conservative force performs when an object is moved from $\vec{r}_0$ to $\vec{r}$. The potenial energy depends on the reference position. If a new reference position $\vec{r}_0'$ is chosen, the potential energy changes by a constant equal to minus the work the conservative force performs to move the object from $\vec{r}_0$ to $\vec{r}_0'$.

Gravitational force:
Consider a mass $m_1$ located at $\vec{r}=0$. The work required to move a mass $m_2$ from $\vec{r}_0$ to $\vec{r}$ in the graviational field of $m_1$ is,

$$W = \int\limits_{\vec{r}_0}^{\vec{r}} -\frac{Gm_1m_2}{|\vec{r'}|^2}\hat{r}\cdot d\vec{r'}.$$

The gravitational force is a conservative force and we can choose the path to take mass $m_2$ from $\vec{r}_0$ to $\vec{r}$. First we move mass $m_2$ along a the surface of a sphere of radius $|\vec{r}_0|$. Since the force is in the radial direction and motion along the surface of the sphere is perpendicular to the radial direction, no work is performed. The mass $m_2$ is moved until it is on the line between $\vec{r}=0$ and the point of interest $\vec{r}$. The expression for the work now becomes a one-dimensional integral,

$$W = \int\limits_{r_0}^{r} \frac{Gm_1m_2}{r'^2}dr'= Gm_1m_2\left(\frac{1}{r}-\frac{1}{r_0}\right),$$

where $r_0 = |\vec{r}_0|$ and $r = |\vec{r}|$. It is conventional to choose a reference position $\vec{r}_0$ far from the origin so that the $\frac{1}{r_0}$ term can be neglected.

$$E_{pot} = - W = -\frac{Gm_1m_2}{r}\hspace{1cm}\text{[J]}.$$

Near the surface of the earth, the force on a weight mass $m$ [kg] is $-\frac{Gmm_{\text{earth}}}{r_{\text{earth}}}\approx -9.81 m$. The change in potential energy of the weight when it is a distance $h$ [m] above the surface of the earth is $\Delta E_{pot}=mgh$ [J], where $g=$ 9.81 [m/s²] is the acceleration of gravity at the earth's surface.

Coulomb force:
The Coulomb force between two charges $q_1$ and $q_2$ has a similar form to the graviational force.

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

The Coulomb for is also a conservative force and the corresponding potential energy is,

$$E_{pot} = \frac{q_1q_2}{4\pi\epsilon_0 r} \hspace{1cm}\text{[J]}.$$

Linear spring force:
A linear spring exerts a conservative force that is proportional to the displacement of the spring from its equilibrium position $x=0$. The force is $F = -kx$, where $k$ is the spring constant. Integrating to find the work performed by this The potential energy is,

$$E_{pot} = \frac{kx^2}{2} \hspace{1cm}\text{[J]}.$$

Lorentz force:
The magnetic field exerts a force that is perpendicular to the motion of the charged particle so $\vec{F}_{\text{magnetic}}\cdot d\vec{r}=0$ and the magnetic force does not change the potential energy of the particle. The electric force is a conservative force $\vec{F} = q\vec{E}$. If the electric field is constant in some region, and a charge particle is moved from position $\vec{r}_0$ [m] to position $\vec{r}$ [m], then the change in potential energy is,

$$\Delta E_{pot} = -q \vec{E}\cdot (\vec{r}-\vec{r}_0) \hspace{1cm}\text{[J]}.$$

Drag force:
The drag force is a frictional force that always points in the opposite direction as the velocity $\vec{v}$ of an object. This means that it is not a conservative force and a potential energy cannot be defined for a drag force.

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