Mechanical Work, Power, and Energy Principles

Mechanical Work

Mechanical work of a constant force is the product of the force by the displacement of its point of application. The formula is: $W = F \cdot \Delta r = F \cdot \Delta r \cdot \cos\theta$.

One Joule (1 J) is the work done by a force of 1 Newton (N) when its point of application moves 1 meter (m) in the same direction as the force. Thus, $1\text{ J} = 1\text{ N} \cdot 1\text{ m}$.

Power

If two forces do the same job, the one that completes it in less time is considered more effective. To account for the efficiency or speed with which a force does work, we introduce the concept of power. We distinguish between:

• Instant Power: The rate of doing elementary work at a certain time. $P = dW / dt = F \cdot dr / dt = Fv$.
• Average Power: For a steady job. $P = W / \Delta t$. Consequently, $W = Pt$.

Work and Kinetic Energy

At the end of the process, $\frac{1}{2}mv^2$ is what we call the kinetic energy of mass $m$ ($E_c$).

Work-Energy Theorem

The work done by the resultant of all forces acting on a body is invested in increasing its kinetic energy. $W_{AB} = \Delta E_c$ (Change in Kinetic Energy).

Work and Potential Energy

At point B, $mgh_B$ is called the gravitational potential energy at point B ($E_{pB}$).

Conservative Forces

A force is conservative if it meets the following conditions:

1. The work performed by the force when its point of application moves from A to B is equal to the potential difference between the points: $W_{AB} = E_{pA} - E_{pB}$.
2. The work performed in a closed path is zero. In fact, $W_{AA} = E_{pA} - E_{pA} = 0$.
3. The work to move its point of application from A to B does not depend on the path taken, but only on the initial and final positions: $W_{AB(I)} = W_{AB(II)}$.

Non-Conservative Forces

Forces that do not meet the above criteria are non-conservative. An instance of a non-conservative force is the force of friction ($W_{rozamiento}$). This force obviously depends on the path followed and is also called a dissipative force.

Examples of conservative forces include the constant or uniform force, and central forces, such as the elastic force ($\frac{1}{2}kx^2$).

Work and Mechanical Energy (Conservative and Non-Conservative Forces)

The work done by non-conservative forces between two points is equal to the change in the mechanical energy of the particle between those points: $W_{nc} = \Delta E_m$.

In a conservative field, the mechanical energy of a particle remains constant.