How to find total mechanical energy?

Mechanical energy is defined as the sum of an object’s energies due to its motion and position. The amount of mechanical energy an object possesses determines the amount of work it can do on other objects, or in other words, the amount of energy it is capable of transferring to other objects.

Given the above definition, we can be more specific about the types of energy that contribute to the total mechanical energy of an object:

• Kinetic energy $E_k$ is the energy associated with an object’s motion and is given by the formula $E_k = \dfrac{1}{2}mv^2$ where $m$ is the mass of the object and $v$ is its speed. Note that this is the macroscopic energy of motion of the particle, and does not include the microscopic energy of the movement of its particles which contributes to the object's internal energy.

The energy associated with an object’s position is called its potential energy. Two main types of potential energy will contribute to mechanical energy:

• Gravitational potential energy $E_p$ is the energy of an object due to its position in a gravitational field. For an object at a height $h$ above a reference point, its gravitational potential energy is calculated with the formula $E_p = mgh$, where $m$ is the mass of the object and $g$ is the gravitational field strength. At the Earth’s surface, the value for $g$ is 9.8 N kg$^{-1}$.

• Elastic potential energy $E_H$ is is the energy stored due to the deformation of an elastic object. Work can be done in changing the shape of an object, for example stretching or compressing a string, and energy is stored in the object as a result. This stored energy is released when the object returns to its original shape. The standard formula for elastic potential energy is derived from Hooke's Law: $E_H = \dfrac{1}{2}kx^2$. Where $k$ is the spring constant and $x$ is the displacement from the equilibrium position.

Another potential energy that can be considered to contribute to total mechanical energy is electric potential energy. Like gravitational potential energy, a charged object will have stored energy due to its position in an electric field. Because the idea of mechanical energy is normally applied to larger objects and not small charged particles, we will ignore it here.

Having discussed the different types of energy that contribute to the mechanical energy of an object, we can create a formula for total mechanical energy. In words, the formula is

$\hspace{3em}$ Mechanical Energy = Kinetic Energy + Potential Energy

or more specifically

$\hspace{3em}$ Mechanical Energy = Kinetic Energy + Gravitational Potential energy + Elastic Potential Energy

In the form of an equation, the total mechanical energy can be expressed as

$\hspace{3em} E_{tot}=E_k + E_p + E_H$

or

$\hspace{3em} E_{tot}=\dfrac{1}{2}mv^2+ mgh+ \dfrac{1}{2}kx^2$