Definition of Mechanical Energy

The Energy Mechanical is just one of the many forms of energy that exist. An object being thrown upward with a certain speed to then fall with almost the same initial speed, a pendulum swinging from side to side reaching almost the same height, a spring that contracts and returns to its original shape, these are all clear examples of mechanical energy in action and its conservation. But, before talking about this, it is important to talk a little about Kinetic energy Y potential energy.

Kinetic energy

Kinetic energy is a type of energy that is associated with the state of movement of an object, that is, with its speed. The greater the speed at which a body moves, the greater its kinetic energy. When an object is at rest, its kinetic energy is zero. In classical mechanics the kinetic energy \(K\) of a body with mass \(m\) moving at a speed \(v\) is given by:

\(K=\frac{1}{2}m{{v}^{2}}\)

Let us imagine that we have a rock in our hand and we push it upwards, at first the rock will have certain speed as a consequence of our push, that is, it will have a certain amount of energy kinetics. As the rock ascends, it will slow down and therefore its kinetic energy will be less and less. You may have heard that “energy cannot be created or destroyed, it is only transformed”, so in this example of the rock, where has its kinetic energy gone? To answer this question it is necessary to talk about potential energy.

Potential energy

In general terms, potential energy is a type of energy that can be associated with the configuration or arrangement of a system of different objects that exert forces on each other. Returning to the previous example, the rock has a certain potential energy depending on its position with respect to a point of reference, which could well be our hand, because it is under the influence of the gravitational attraction of the Land. In this case the value of the potential energy will be given by:

\(U=mgh\)

Where \(U\) is the gravitational potential energy, \(m\) is the mass of the rock, \(g\) is the acceleration gravity of the Earth and \(h\) is the height at which the rock is with respect to our hand.

When we throw the rock up, its kinetic energy will be transformed into energy potential reaching a maximum value when the rock reaches a certain height and is slowed down by complete. As you can see, there are two ways to view this example:

1) When we throw the rock upwards, it slows down due to the strength gravity exerted by the Earth.

2) When we throw the rock upwards, it slows down because its kinetic energy is transformed into potential energy.

This here is of great importance because the evolution of the same system can be viewed in terms of forces acting or in terms of energy.

conservative forces

In the previous example it was mentioned that there is a potential energy associated with the gravitational force, but is this valid for any force? The answer to this question is no, and this is valid only for a type of force called "Conservative Forces", some examples of these would be gravity, the elastic force, the force electric etc.
A characteristic of conservative forces is that the mechanical work they do on a body to move it from one point to another is independent of the path it follows. said body from the initial point to the end, this is the same as saying that the mechanical work done by a conservative force in a closed path is equal to zero.

To visualize this let's go back to our previous example, when we throw the rock up, gravity will start to do a negative mechanical work (opposite to motion) on it causing it to lose kinetic energy and gain energy potential. When the rock reaches its maximum height it will stop and start to fall, now gravity will be doing a job positive mechanical on the rock which will manifest itself in a loss of potential energy and a gain of energy kinetics. The path of the rock ends when it reaches our hand again with the same kinetic energy with which it took off (in the absence of the resistance of the air).

In this example, the rock reached the same point from which it started, so we can say that it made a closed path. When the rock was going up, gravity did negative mechanical work and when the rock was falling, gravity did positive mechanical work. of the same magnitude as the previous one, therefore, the total work done by the gravitational force along the entire path of the rock was equal to zero. The forces that do not comply with this are called "Non-Conservative Forces" and some examples of these are friction and friction.

Another thing we can see in the example above is the relationship between kinetic energy, potential energy, and mechanical work. We can say that:

\(\text{ }\!\!\Delta\!\!\text{ }K=W\)

\(\text{ }\!\!\Delta\!\!\text{ }U=-W\)

Where \(\text{ }\!\!\Delta\!\!\text{ }K\) is the change in kinetic energy, \(\text{ }\!\!\Delta\!\!\text{ }U\) is the change in potential energy and \(W\) is the mechanical work.

Conservation of mechanical energy

As mentioned at the beginning, the mechanical energy of a system is the sum of its potential energy and its kinetic energy. Let \(M\) be the mechanical energy, we have:

\(M=K+U\)

The mechanical energy of a closed system in which only conservative forces (not friction or friction) interact is a quantity that is conserved as the system evolves. In order to see this, let us recall that we previously mentioned that \(\text{ }\!\!\Delta\!\!\text{ }K=W\) and \(\text{ }\!\!\Delta\!\ !\text{ }U=-W\), we can then say that:

\(\text{ }\!\!\Delta\!\!\text{ }K=-\text{ }\!\!\Delta\!\!\text{ }U\)

Suppose that at a point \(A\) our system has a kinetic energy \({{K}_{A}}\) and a potential energy \({{U}_{A}}\), subsequently our system evolves to a point \(B\) at which it has a kinetic energy \({{K}_{B}}\) and a potential energy \({{U}_{B}}\). According to the above equation, then:

\({{K}_{B}}-{{K}_{A}}=-\left( {{U}_{B}}-{{U}_{A}} \right)\)

Rearranging the terms of this equation a bit, we get:

\({{K}_{A}}+{{U}_{A}}={{K}_{B}}+{{U}_{B}}\)

But, if we look closely, we can see that \({{K}_{A}}+{{U}_{A}}\) is the mechanical energy of the system at point \(A\) and \ ({{K}_{B}}+{{U}_{B}}\) is the mechanical energy at point \(B\). Let \({{M}_{A}}\) and \({{M}_{B}}\) be the mechanical energies of the system at point \(A\) and at point \(B\), respectively, we can then conclude that:

\({{M}_{A}}={{M}_{B}}\)

That is, mechanical energy is conserved. It should be emphasized that this is valid only with conservative forces, since, in the presence of non-conservative forces, such as friction or friction, there is a dissipation of energy.

References

David Halliday, Robert Resnick & Jearl Walker. (2011). Fundamentals of Physics. United States: John Wiley & Sons, Inc.