Energy is the capacity to do work.

Energy

The mechanical energies are:

Kinetic energy \((E_k)\)
When a body is in motion, we say that it has kinetic energy. The amount of energy is given by: $$ E_k(v) = \frac{mv^2}{2},$$ where \(v\) is the velocity of the object and \(m\) its mass.
Gravitational Potential Energy \((E_p)\)
The energy assigned to a body in the Earth's gravitational field, near the surface, has its value given by: $$E_p(h) = mgh,$$ where \(m\) is the mass of the object being analyzed, \(g\) is the acceleration of gravity and \(h\) is the height where the body is, always in relation to a reference height.
Elastic Potential Energy \((E_e)\)
Is the energy that has an elastic material when compressed or stretched of \(\Delta x\) , and has value $$E_e(\Delta x) = \frac{k (\Delta x)^2}{2} ,$$ where \(k\) is the elastic constant of the material.
Mechanical Energy \((E_m)\)
It is the sum of all energies present in the system, i.e., $$ E_m = E_k + E_p + E_e .$$
Note that energy is a scalar.

Conservation Laws

The energy is just a number but can be used to describe transformations in physical behaviors of systems. This tactic works very well, so well that some people believe that energy is something that does exist in the real world, it does not exist, there is no instrument that measures energy. Some instruments measure some other physics quantities and then calculate the energy. The concept of energy is just a mathematical method developed to facilitate the calculations.

These numbers (energies) are conserved quantities, that is, do not increase or diminish for an isolated system.

It is important to know that:

  • Conservative forces \( \rightarrow \) there are no losses of mechanical energy.
  • Dissipative forces \( \rightarrow \) there are losses of mechanical energy (typically with heat and sound), that is: \(E_m\) is not conserved.

Conservation of Mechanical Energy

The mechanical energy of an object subject to the action of conservative forces is constant and does not change, this implies that $$E_m(A) = E_m(B),$$ where \(E_m(A)\) and \(E_m(B)\) are the mechanical energy between two states \(A\) and \(B\). No matter what path the system took to go from \(A\) to \(B\).

Energy conservation can also be applied to mechanical systems that are not conservative, by including work \(w_{ext}\) performed by non - conservative forces. The change in system energy is equal to the work done on the system: $$\Delta E_k + \Delta E_p = w_{ext}$$ or $$ E_{k_i} + E_{p_i} = E_{k_f} + E_{p_f} + w_{ext}.$$

Kinetic Energy Theorem (TEC)

The sum of the works done by all the forces acting on a particle, between any two points, is equal to the change in kinetic energy of the particle between these points. Mathematically: $$ \sum_i^n w_i = \Delta E_k $$