Conservative and Non Conservative Forces

It is a beautiful day outside, so you decide to go on a hike. You carry a heavy backpack up a strenuous mountain trail, and at the end of the day you come down the mountain and end up where you started. Although you are exhausted from hauling your backpack up and down the mountain, the total work you have done on the backpack is zero. This is because the gravity force pulling on your backpack is a conservative force. In this article, we will define conservative force and non-conservative force and discuss the differences between them. We will also list different conservative and non-conservative forces and go over some example problems.

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.

Differences between Conservative and Non-conservative forces

Let's discuss the differences between conservative and non-conservative forces a bit more. We discussed above how conservative forces are path-independent while non-conservative forces are path-dependent. Let's think about a box that is pushed up a rough incline. The box is then pushed back down to its starting position. Since the box ended up where it started, the box moved in a closed path. The total work done by conservative forces when the object moves in a closed path is always zero. The conservative force working on the box in our example is the force of gravity; the total change in height of the box is zero, so the gravitational potential energy is zero.

The pushing force and the friction force are examples of non-conservative forces working on the box, since they depend on the path taken. As shown in the image below, the pushing force does positive work on the box as it travels up and down the incline, while the friction does negative work on the box. The net work done by these forces is not zero once the box returns to its initial position. We can test whether a force is conservative by considering the total work done when the force moves the object in a closed path; if the net work is zero, we know that it is a conservative force.

The work done by non-conservative forces depend on the path taken while the work done by conservative forces does not depend on the path, StudySmarter Originals

Another difference between conservative and non-conservative forces is that the work done by a conservative force can be reversed. When conservative forces like gravity or the spring force work on an object, they store potential energy that can be converted to kinetic energy to later reverse the work done. When a non-conservative force such as friction works on an object, kinetic energy converts to thermal energy, and we can not get the dissipated thermal energy back. Thus, work done by a non-conservative force is irreversible.

List of Conservative and Non-Conservative Forces

We mentioned a few examples of conservative and non-conservative forces. The table below gives a few more examples of these forces that you will encounter. We will talk about the electric force more later.

Examples of Conservative and Non-Conservative Forces

Conservative and non-conservative forces show up in almost all physics problems, so let's get some practice using them.