POTENTIAL ENERGY: CHARACTERISTICS, TYPES, CALCULATION AND EXAMPLES - PHYSICAL - 2025

The potential energy is the energy that bodies under its own configuration. When objects interact, there are forces between them capable of doing work, and this ability to do work, which is stored in their arrangement, can be translated into energy.

For example, humans have harnessed the potential energy of waterfalls since time immemorial, first by turning mills and then at hydroelectric plants.

Niagara Falls: a huge reservoir of gravitational potential energy. Source: Pixabay.

On the other hand, many materials have a remarkable ability to do work by deforming and then returning to their original size. And in other circumstances, the arrangement of the electrical charge allows the storage of electrical potential energy, as for example in a capacitor.

Potential energy offers many possibilities to be transformed into other forms of usable energy, hence the importance of knowing the laws that govern it.

Origin of potential energy

The potential energy of an object has its origin in the forces that affect it. However, potential energy is a scalar quantity, while forces are vector. Therefore, to specify the potential energy, it is enough to indicate its numerical value and the selected units.

Another important quality is the type of force with which potential energy can be stored, since not every force has this virtue. Only conservative forces store potential energy in the systems on which they act.

A conservative force is one for which the work does not depend on the path followed by the object, but only on the starting point and the arrival point. The force that drives the falling water is gravity, which is a conservative force.

On the other hand, elastic and electrostatic forces also have this quality, therefore there is potential energy associated with them.

Forces that do not meet the aforementioned requirement are called non-conservative; Examples of these are in friction and air resistance.

Types of potential energy

Since potential energy always derives from conservative forces such as those already mentioned, we speak of gravitational potential energy, elastic potential energy, electrostatic potential energy, nuclear potential energy, and chemical potential energy.

Gravitational potential energy

Any object has potential energy as a function of its height from the ground. This seemingly simple fact illustrates why falling water is capable of driving turbines and eventually being transformed into electrical energy. The skier example shown here also shows the relationship of weight and height to gravitational potential energy.

Another example is a roller coaster car, which has higher potential energy when it is at a certain height above the ground. Once it has reached ground level, its height equals zero and all of its potential energy has been transformed into kinetic energy (energy of motion).

The animation shows the exchange between gravitational potential energy and kinetic energy, of an object moving on a roller coaster. The sum of both energies, called mechanical energy, is constant throughout the movement. Source: Wikimedia Commons.

Elastic potential energy

Objects such as springs, bows, crossbows, and rubber bands are capable of storing elastic potential energy.

By drawing the bow, the archer does work that is stored as potential energy of the bow-arrow system. When you release the bow, this energy is transformed into the movement of the arrow. Source: Pixabay.

The elasticity of a body or a material is described by Hooke's law (up to certain limits), which tells us that the force capable of exerting when it is compressed or stretched is proportional to its deformation.

For example in the case of a spring or spring, this means that the more it shrinks or stretches, the greater the force it can exert on an object placed at one end.

Electrostatic potential energy

It is the energy that electric charges have by virtue of their configuration. Electric charges of the same sign repel each other, so to place a pair of positive or negative charges in a certain position, an external agent must do work. Otherwise they would tend to separate.

This work is stored in the way the loads were located. The closer the charges of the same sign are, the higher potential energy the configuration will have. The opposite happens when it comes to loads of different signs; As they attract each other, the closer they are, the less potential energy they have.

Nuclear potential energy

Approximate representation of the Helium atom. In the nucleus the protons are represented in red and the neutrons in blue.

The atomic nucleus is made up of protons and neutrons, generically called nucleons. The former have a positive electrical charge and the latter are neutral.

Since they are agglomerated in a tiny space beyond imagination, and knowing that charges of the same sign repel each other, one wonders how the atomic nucleus remains cohesive.

The answer lies in other forces besides electrostatic repulsion, characteristic of the nucleus, such as the strong nuclear interaction and the weak nuclear interaction. These are very strong forces, far exceeding the electrostatic force.

Chemical potential energy

This form of potential energy comes from how the atoms and molecules of substances are arranged, according to the different types of chemical bonds.

When a chemical reaction takes place, this energy can be transformed into other types, for example by means of a cell or electric battery.

Examples of potential energy

Potential energy is present in daily life in many ways. Observing its effects is as easy as placing any object at a certain height and being certain that it can roll or fall at any time.

Here are some manifestations of the types of potential energy previously described:

-Roller coasters

-Cars or balls rolling downhill

-Bows and arrows

-Electrical batteries

-A pendulum clock

When one of the spheres at the ends is put into motion, the movement is transmitted to the others. Source: Pixabay.

-Swinging on a swing

-Jump on a trampoline

-Use a retractable pen.

See: examples of potential energy.

Calculation of potential energy

The potential energy depends on the work done by the force and this in turn does not depend on the trajectory, so it can be stated that:

-If A and B are two points, the work W _AB necessary to go from A to B is equal to the work necessary to go from B to A. Therefore: W _AB = W _BA, so:

-And if two different trajectories 1 and 2 are tried to join said points A and B, the work done in both cases is also the same:

W ₁ = W ₂.

In either case the object experiences a change in potential energy:

Well, the potential energy of the object is defined as the negative of the work done by the (conservative) force:

But since work is defined by this integral:

:

Note that the units of potential energy are the same as those of work. In the SI International System the unit is the joule, which is abbreviated J and is equal to 1 newton x meter, by the English physicist James Joule (1818-1889).

Other units for energy include cgs erg, pound-force x foot, BTU (British Thermal Unit), calories, and kilowatt-hours.

Let's see below some particular cases of how to calculate potential energy.

Calculation of gravitational potential energy

In the vicinity of the earth's surface, the force of gravity points vertically downwards and its magnitude is given by the equation Weight = mass x gravity.

Denoting the vertical axis with the letter “y” and assigning to this direction the unit vector j, positive up and negative down, the change in potential energy when a body moves from y = y _A to y = and _B is:

Calculation of elastic potential energy

Hooke's law tells us that the force is proportional to the deformation:

Here x is the strain and k is an eigen constant of the spring, indicating how stiff it is. Through this expression the elastic potential energy is calculated, taking into account that i is the unit vector in the horizontal direction:

Calculation of electrostatic potential energy

When you have a point electric charge Q, it produces an electric field that perceives another point charge q, and which does work on it when it is moved from one position to another in the middle of the field. The electrostatic force between two point charges has a radial direction, symbolized by the unit vector r:

Figure for example 1. Source: F. Zapata.

Solution

When the block is at a height h _A with respect to the floor, it has gravitational potential energy due to its height. When released, this potential energy is gradually converted into kinetic energy, and as it slides down the smooth curved ramp, its speed increases.

During the path from A to B, the equations of uniformly varied rectilinear motion cannot be applied. Although gravity is responsible for the movement of the block, the movement it experiences is more complex, because the trajectory is not rectilinear.

Conservation of energy in path AB

However, since gravity is a conservative force and there is no friction on the ramp, you can use the conservation of mechanical energy to find the speed at the end of the ramp:

The expression is simplified by noting that the mass appears in each term. It is released from rest v _A = 0. And h _B is at ground level, h _B = 0. With these simplifications, the expression reduces to:

Work done by rubbing in section BC

Now the block starts its journey in the rough section with this speed and finally stops at point C. Therefore v _C = 0. Mechanical energy is no longer conserved, because friction is a dissipative force, which has made a work on the block given by:

This work has a negative sign, since the kinetic friction slows the object, opposing its movement. The magnitude of the kinetic friction f _k is:

Where N is the magnitude of the normal force. The normal force is exerted by the surface on the block, and since the surface is totally horizontal, it balances the weight P = mg, therefore the magnitude of the normal is:

Which leads to:

The work that f _k does on the block is: W _k = - f _k.D = - μ _k.mg.D.

Calculation of change in mechanical energy

This work is equivalent to the change in mechanical energy, calculated like this:

In this equation there are some terms that vanish: K _C = 0, since the block stops at C and U _C = U _B also vanish, since these points are at ground level. The simplification results in:

The mass again cancels out and D can be obtained as follows:

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill.
2. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 2. Dynamics. Edited by Douglas Figueroa (USB).
3. Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed Prentice Hall.
4. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
5. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 1-2.