Conservative and Non Conservative Forces

Definitions of Conservative and Non-Conservative Forces

When solving a physics problem using energy, it is important to know if a force acting on an object is a conservative force or a non-conservative force. Let's define these forces so that we can better understand how they relate to the energy and work of a system. When the work done by a force is independent of the path taken, this force is a conservative force. This means that for a conservative force, the work done is only dependent upon the starting and ending positions, not the path taken to get there.

Gravity is an example of a conservative force. The force of gravity works on a ball kicked from the ground into the air, and the work done on the ball by gravity depends only on the vertical change in height. When the ball reaches the ground again, the net work done by gravity is zero. If there are only conservative forces acting in a system, the total mechanical energy, the sum of the kinetic and potential energies, is constant. This means that the kinetic energy transforms to potential energy and vice versa without any loss of mechanical energy.

Work done by gravity, a conservative force, on a ball kicked in the air only depends on the ball's change in height. Work done by air resistance, a non-conservative force, depends on the path taken, Pixabay

When the work done by a force is dependent on the path taken, this force is a non-conservative force. Air resistance is an example of a non-conservative force. When air resistance acts on the ball kicked into the air, it works against the ball as the ball goes into the air and as it falls back to the ground, causing the ball to slow down and thus lose kinetic energy. This results in less mechanical energy. This doesn't mean that the law of energy conservation is incorrect, we just need to think about other forms of energy. When air resistance does negative work on the ball, some kinetic energy transforms into thermal energy as the ball and the surrounding air get warmer. A force that decreases the mechanical energy in a system is called a dissipative force. All dissipative forces are non-conservative forces.

In the article, "Potential Energy and Energy Conservation", we discuss how the potential energy in a system comes from conservative forces doing work. There is potential energy in a system only when conservative forces are doing work. The work done by a conservative force is equal to the negative change in potential energy, \(W=-\Delta U\), and the change in kinetic energy is equal to the total work done in a system, \(W_{net} = \Delta K\). The total work done in a system is made up of the work done by conservative forces and the work done by non-conservative forces so that \(W_{net} = W_c + W_{nc} = \Delta K\). If we substitute the work done by the conservative forces into this equation, we get:

$$ \begin{aligned} W_c + W_{nc} &= \Delta K \\ -\Delta U + W_{nc} &= \Delta K \\ W_{nc} &= \Delta K + \Delta U \end{aligned} $$

We see from this equation that the change in kinetic energy and potential energy, or the change in total mechanical energy, is equal to the work done by non-conservative forces acting on objects in the system.