Potential Energy, Gravitational Potential

Potential energy

In physics, potential energy is the energy held by an object because of its "position" of a physical quantity. For example, the higher an object is, the more gravitational potential energy it has. The more stretched or compressed a spring is, the more elastic potential energy it has. The higher temperature an object is, the more thermal potential energy it has.

Potential energy is tightly related to work. If you want to raise up some object high, you need to do work. If an object is high, it has the potential to accelerate and accumulate kinetic energy, of which any change is the work done on the object, according to the work-energy principle.

Potential and conservative force

Any potential energy is associated with what is called the conservative force. "Conservative" means the work done by the conservative force only depends on the initial and final "positions" of the object, not its "trajectory", or path. Friction force is a typical non-conservative force because the longer the path, the more negative work friction does.

We can define a scalar function called "potential" $U$ that only depends on the relative "position" of the object, so that the work done by the conservative force is the difference of the potential:

$$W_c=-\Delta U=U_{initial}-U_{final}$$

If the potential increases, then the conservative force does negative work; if the potential decreases, then the conservative force does positive work. The following example helps explain.

Gravitational potential and gravitational force

The gravitational potential energy is associated with the gravitational force. Suppose you move a boulder from the same elevation to a mountain top. No matter how you move it, either rolling over a perfect incline plane, a rough road up and down, or using an elevator, the gravitational force does the same amount of work:

$$W_g=-mg\Delta h$$

The gravitational force did negative work, because the force and the displacement have opposite directions.

On the surface of Earth, the gravitational potential is

$$U_g=mgh$$

where $m$ is the mass of the object, $g$ is the gravitational acceleration, $h$ is the height of the object (relative to a referenced height).

If we set the ground level as $h=0$, then

$$h_{initial}=0, h_{final}=\Delta h\Rightarrow U_{initial}=0, U_{final}=mg\Delta h$$ $$\Delta U=U_{final}-U_{initial}=mg\Delta h=-W_g$$

If we set the mountain top level as $h=0$, then

$$h_{initial}=-\Delta h, h_{final}=0\Rightarrow U_{initial}=-mg\Delta h, U_{final}=0$$ $$\Delta U=U_{final}-U_{initial}=mg\Delta h=-W_g$$

In general, just as there is gravitational force between any two objects, there is a gravitational potential between any two objects:

$$U_g=-\frac{Gm_1m_2}{r}$$

where $G\approx 6.6743\times 10^{-11} m^3/(kg\cdot s^2)$ is the gravitational constant, $m_1$ and $m_2$ are the masses of two objects, and $r$ is their distance.

From gravitational potential energy to kinetic energy

Let us consider the pulley experiment we did earlier. Before releasing the pulling weight, both the PAScar and the pulling weight stay at rest. Therefore, their kinetic energies are both zero. If we define when an object rests on the ground, it has 0 gravitational potential, then the gravitational potentials of the car and the pulling weight are $m_1gl$ and $m_2g\Delta h$, where $l$ and $\Delta h$ are their heights above the ground.

After releasing the pulling weight and when the pulling weight just lands, the PAScar and the pulling weight have the same speed $v$, because of the tightening of the rope, so their kinetic energies are $\frac{1}{2}m_1v^2$ and $\frac{1}{2}m_2v^2$, respectively. The gravitational potential of the car does not change, and the gravitational potential of the pulling weight becomes zero.