Potential Energy

Author: Matthew Lewine (mlewine3)

Zachary Karpenko (zkarpenko) Spring 2019

Potential energy (referred as [math]\displaystyle{ U }[/math]) is the energy that some body possesses due to their position relative to other bodies, configuration or stresses within itself, electric charges, and other factors. It can be pictured as a stored energy which has the potential to do some work but hasn't done it yet. Changing the parameters that give a system a potential energy (e.g. distance between objects), [math]\displaystyle{ U }[/math] changes as well. The change in [math]\displaystyle{ U }[/math] can result in a change in [math]\displaystyle{ K }[/math] (Kinetic Energy).

Example 1: Imagine a rock on top of cliff. It has somehow been displaced to a height H from the surface of the Earth. Due to this position, we say that this rock has some potential energy: if thrown from up there, it has the potential to do some work as it falls (break the floor). In fact, by throwing the ball, H is being reduced, thus U is being reduced. The change in U is translated to changes in other forms of energy. This difference appears as an increase in kinetic energy: that's why the ball gains speed as it falls.

*Fig.1* The energy stored in a stretched rubber band is a form of elastic potential energy.

Example 2: The stretched rubber band, for example, has the potential to do some work once you release it.

When referring to [math]\displaystyle{ U }[/math] being at a minimum or maximum, it is important to note that these can be in absolutes. If a system has all potential energy, then all of its energy is stored and there is absolutely no kinetic energy. An example to help think of this is a stationary ball on top of a cliff. It is not moving whatsoever, but if it were to be pushed off of the cliff, the potential energy would begin to be transformed into kinetic energy as it falls. There is also a moment, just before the ball hits the ground and while its velocity is at its theoretical maximum, that virtually zero potential energy is present. All of the energy would be kinetic energy in this case.

Energy is conserved, so the total amount of energy in a system never changes, but it can change forms, such as from potential to kinetic energy, or the other way around. In non-ideal systems, so most realistic earthly systems, energy can be transformed into other forms such as thermal energy. An example of this would be a toy racecar rolling along a flat surface. It eventually slows down and stops due to friction, which results in a decrease in kinetic energy since that energy is transformed into thermal energy on the tires.

In terms of potential energy, its capacity for doing work is a result of its position in a gravitational field (gravitational potential energy), an electric field (electric potential energy), or a magnetic field (magnetic potential energy). It may have elastic potential energy due to a stretched spring or other elastic deformation.

It is interesting to note, that the universe naturally prefers lower potential configuration of systems. This peculiarity is partly why balancing a pencil on one finger is so hard; the pencil contains potential energy that can be released easily. As taught in many engineering classes, this can be explained by one of the thermodynamic laws, where the universe always increase randomness, or entropy.

The Main Idea

When we want to discuss Energy, the first step is always defining your system. In fact, the Energy Principle states: [math]\displaystyle{ \vartriangle \!\, Esys = Wsurr }[/math]. Up to this point, energy was always studied in mono-particular systems. However, most situations in life are more complex and involve multi-particular systems.

An example would be any object that is displaced, but still has stuff interacting and working in it. We can imagine the earth, on which work is done by the Sun and other planets. Above the surface of the earth, there is a ball which also interacts with it. Surrounding planets also do work on the ball. So how can we analyze the behavior of both the ball and the Earth when they interact with each other and the Sun?

See "image 1" at https://drive.google.com/drive/folders/0B5m-O1TAXl8GWkxYNGhwQm1xMkU?usp=sharing

This analysis is infinite and can be very difficult to handle. That's why sometimes you need to consider multi-particular systems for your analysis. We might want to fix earth+ball as a point to analyze the work done by the Sun on this system. But the problem is, we'll be neglecting the interaction happening within the system.

If the ball is the system, the Earth represents the surroundings (image 1). The Kinetic energy of the system (ball only) increases gradually due to the work done by the Earth. Imagine the ball is 1 kg and is released from rest 10 m above the surface. When the ball has fallen to 5 m, what is [math]\displaystyle{ K }[/math]?

See "image 3" at https://drive.google.com/drive/folders/0B5m-O1TAXl8GWkxYNGhwQm1xMkU?usp=sharing

[math]\displaystyle{ \vartriangle \!\, Eball = WEarth }[/math]

[math]\displaystyle{ \vartriangle \!\, Kball = WEarth }[/math]

[math]\displaystyle{ \vartriangle \!\, Kball = Fy\vartriangle \!\,y = -mg\vartriangle \!\,y }[/math]

[math]\displaystyle{ Kf = -(1 kg)(9.8 N/Kg)(-5 m) = 5 J }[/math]

What if we decide the choose the ball and the Earth as the system?

See "image 2" at https://drive.google.com/drive/folders/0B5m-O1TAXl8GWkxYNGhwQm1xMkU?usp=sharing

In this case, we consider there is nothing significant in the surroundings. As a result

[math]\displaystyle{ \vartriangle \!\, Eball = 0 }[/math]

[math]\displaystyle{ \vartriangle \!\, Kball + \vartriangle \!\, Kearth = 0 }[/math]

However, we know that the kinetic energy of the ball increased, and the kinetic energy of the Earth also increased. We also know the surroundings did zero work. But the change in kinetic energy of the system is bigger than zero and so different from the work done by the surroundings. This is happening since we are overlooking a kind of energy in the system related to the interaction between both bodies: it is potential energy.

In any systems containing two or more interacting bodies (stars interacting in a galaxy), there is energy associated with the interactions between pairs of particles inside the system.

So, since the ball+earth system contains interacting objects, its energy is:

[math]\displaystyle{ Esys = EEarth + Eball + Uball/earth }[/math]

Since the interaction force is gravity (dependent of distance), a change in potential energy should be associated with a change of the separation between objects. This can also mean a change of shape of the multi-particle system, such as a spring stretching or compressing.

Getting back to our example above, as the ball and Earth get closer, the kinetic energy of the system increases but it is compensated by a decrease in potential energy (interaction energy). In fact, since there are no surroundings:

[math]\displaystyle{ \vartriangle \!\, Eball = 0 }[/math]

[math]\displaystyle{ \vartriangle \!\, Kball + \vartriangle \!\, Kearth + \vartriangle \!\, Uball/earth = 0 }[/math]

A Mathematical Model

Taking ball+earth as the system makes the force exerted on the ball interior to the system: it is an internal force which makes internal work and so changes [math]\displaystyle{ U }[/math].

[math]\displaystyle{ \vartriangle \!\, Kball = Wearth }[/math]

[math]\displaystyle{ \vartriangle \!\, Kball + (- Wearth) = 0 }[/math]

Definition: for a multi-particle system we define the change of potential energy [math]\displaystyle{ \vartriangle \!\, U }[/math] to be the negative of the internal work:

[math]\displaystyle{ \vartriangle \!\, U = -Wearth }[/math]

For the system of the ball + Earth:

[math]\displaystyle{ \vartriangle \!\, U = -(Fy \vartriangle \!\, y) = -(-mg)\vartriangle \!\, y = \vartriangle \!\,(mgy) }[/math]

To conclude, the difference between a single-particle and a multi-particle one is that the second has pairwise potential energy on top of the particular energy.

Revised energy principle for a multiparticle system

[math]\displaystyle{ \vartriangle \!\, (E1 + E2 + E3 + ...) + \vartriangle \!\, (U12 + U13 + U23 + ...) = W }[/math]

with [math]\displaystyle{ E1 = E1rest + K1 }[/math]

An algebraic process can help us derive equations for all kinds of Potential energies with respect to the parameters that affect it.

A force is considered conservative if it is acting on an object as a function of position only.

We can relate work to potential energy using the equation

[math]\displaystyle{ U = -\int \vec{F}\cdot\vec{dr} = -W }[/math]

This says that the potential energy U is equal to thework you must do to move an object from an arbitrary reference point [math]\displaystyle{ U=0 }[/math] to the position [math]\displaystyle{ r }[/math].

The potential energy of a rock on the top of a cliff is equal to the work you've done to bring the rock up to this point.

If we take the derivative of both sides of this equation and obtain:

[math]\displaystyle{ \frac{-dU}{dx} = F(x) }[/math]

Which means that the force on an object is the negative of the derivative of the potential energy function U. Therefore, the force on an object is the negative of the slope of the potential energy curve. Plots of potential functions are valuable aids to visualizing the change of the force in a given region of space.

Let's apply this relationship. If the potential energy function U is known, the force at any point can be obtained by taking the derivative of the potential. Let's consider gravitational potential and elastic potential.

The potential energy function U of gravitational potential is [math]\displaystyle{ mgh }[/math], where [math]\displaystyle{ m }[/math] is mass, [math]\displaystyle{ g }[/math] is the gravitational constant, and [math]\displaystyle{ h }[/math] is some distance away from the reference point at which U = 0. Then the force is

[math]\displaystyle{ F = \frac{-d}{dh}mgh = -mg }[/math]

We can go the other way as well. We know the force of gravity is [math]\displaystyle{ -mg }[/math], and integrating with respect to h we obtain [math]\displaystyle{ U = mgh }[/math].

This process can be done with elastic potential as well, where the force [math]\displaystyle{ F = -kx }[/math] and the potential energy function is [math]\displaystyle{ U = \frac{1}{2}k^{2} }[/math]

Here are the potential energy functions for all forms:

Type	Equation	Variables
Gravitational Potential	[math]\displaystyle{ U = \frac{GMm}{r} }[/math] [math]\displaystyle{ U = mgh }[/math] close to Earth's surface	[math]\displaystyle{ G }[/math] is the gravitational constant, [math]\displaystyle{ M }[/math] and [math]\displaystyle{ m }[/math], and [math]\displaystyle{ r }[/math] is distance
Elastic Potential	[math]\displaystyle{ U = \frac{1}{2}k^{2} }[/math]	[math]\displaystyle{ k }[/math] is the spring constant
Electric Potential	[math]\displaystyle{ U = k\frac{Qq}{r} }[/math]	[math]\displaystyle{ k }[/math] is Coulomb's constant, [math]\displaystyle{ Q }[/math] and [math]\displaystyle{ q }[/math] are point charges, [math]\displaystyle{ r }[/math] is distance
Magnetic Potential	[math]\displaystyle{ U = -μ \cdot B }[/math]	[math]\displaystyle{ μ }[/math] is the dipole moment and [math]\displaystyle{ μ = IA }[/math] in a current loop and [math]\displaystyle{ I }[/math] is the current and [math]\displaystyle{ A }[/math] is the area

A Computational Model

A spring is like a rubber band, as you stretch it, it stores potential energy. Once released, this stored energy is converted to kinetic energy. The simulation at this link shows it https://trinket.io/embed/glowscript/fa7fbeaf3b [1]

It is interesting to see how potential(yellow) and kinetic(cyan) energy alter. When U is at a maximum, K is at a minimum. When K is at a maximum, U is at a minimum. This suggests an energy transfer: U is transformed into kinetic energy which causes the ball to move. Also total Energy is constant because there are no surroundings.

"See "Demo 3" at: https://drive.google.com/drive/folders/0B5m-O1TAXl8GWkxYNGhwQm1xMkU?usp=sharing"

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

This is a link to a video displaying an exercise resolved step by step by Dr. Greco [2]

Simple

Question

An object of mass 5 kg is held 10 meters above the Earth's surface. Relative to the surface, how much potential energy does this object have?

Solution

Using the equation [math]\displaystyle{ U = mgh }[/math] we can say [math]\displaystyle{ U = 5*9.8*10 = 490 }[/math] J.

Middling

Question

If it takes 4J of work to stretch a Hooke's law spring 10 cm from its unstretched length, determine the extra work required to stretch it an additional 10 cm.

Solution

The work done in stretching or compressing a spring is proportional to the square of the displacement. If we double the displacement, we do 4 times as much work. It takes 16 J to stretch the spring 20 cm from its unstretched length, so it takes 12 J to stretch it from 10 cm to 20 cm.

Formally: [math]\displaystyle{ W = \frac{1}{2}kx^{2}. }[/math] Given W and x we find k.
[math]\displaystyle{ 4 J = \frac{1}{2}k(0.1)^{2} }[/math]

[math]\displaystyle{ k =\frac{8}{0.1^{2}} = 800 }[/math] N/m.

Using [math]\displaystyle{ x = 0.2 }[/math] m, [math]\displaystyle{ W = \frac{1}{2}(800)(0.2)^{2} = 16 }[/math] J

Extra work: 16 J - 4 J = 12 J.

Difficult

Question

We have a point charge A of charge +Q at the origin. Let's say we want to move another charge B of +q, located 10m away from particle A, to a location 5m away from particle B. How much work does it require to move the particle B 5m closer to particle A?

Solution

We have a nonuniform electric field, so we need to integrate the potential energy function to find the amount of work needed. [math]\displaystyle{ W = \int_{10}^{5} \frac{-kQq}{r^2}dr= -kQq\int_{10}^{5}\frac{1}{r^2}dr = \frac{kQq}{10} }[/math]

Connectedness

Potential energy is the driving force behind voltage, or the electric potential difference, expressed in volts. In fact, the circuit has to have a potential in order to light stuff and do some electrical work. The energy use for this work is subtracted from this potential which explains the difference of potential known as V. As a computer engineering major, I recognize the importance of this concept, as without potential difference we would not have transistors or circuits to power our machines.

Nuclear potential energy also exists, and is the potential energy of the particles inside an atomic nucleus. Most nuclear elements like Pd and Ur have unstable nucleus due to a disproportion of nuclear particles like protons and neutrons. Unstable nucleus means a nucleus which is susceptible to change to another configuration: they have a potential. Elements used in nuclear reactions thus have kernels with a high potential energy. As we previously discussed, nature always try to go to the most stable state of energy, the state where it has the least potential energy. That's why most radioactive elements naturally decay and change composition to be in a more stable stage. The difference of energy between previous and next stage is the energy liberated in nuclear reactions as rays.

History

The term potential energy was introduced by the 19th century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concept of potentiality. His ideas have continued to be advanced into our understanding of potential and kinetic energy today.

Potential Energy

Contents

The Main Idea

A Mathematical Model

A Computational Model

Examples

Simple

Middling

Difficult

Connectedness

History

See also

Further reading

External links

References

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