Conservative and Non-Conservative Forces: Definition, Potential Energy, Examples

Conservative and Non-Conservative Forces: When you throw a ball straight up, it travels a bit upwards, slows down, comes to a temporary stop and falls back down. Throughout its motion, it is under the action of two distinct forces, gravity and air resistance. One of them is conservative, and the other is non-conservative. What do we mean by the terms conservative and non-conservative? If we could somehow turn off the air resistance, which, by the way, is the non-conservative force in this story, the ball would return with the same speed or kinetic energy with which it was imbued.

In other words, under the action of gravity alone, the conservative force, the ball would return with the same kinetic energy with which it was thrown. Thus we have a notion of what a conservative force is. It returns what it takes. It gives back the kinetic energy it took from the ball. The ball came to a temporary stop, did it not? So, gravity took away the balls kinetic energy in its upward motion and returned it in its downward motion. And it returned all of it! Are there any special conditions under which conservative forces behave the way they do? What is the difference between conservative forces and non-conservative forces? Let’s find out.

What is a Conservative Force?

A conservative force is a force that does zero work on an object that takes a round trip under the influence of the force.

\(\oint {\overrightarrow {\text{F}} } .\overrightarrow {{\text{ds}}} = 0\,\,\,\,\,\,(1)\)

In the above expression, \(\overrightarrow {\text{F}} \) denotes the conservative force, and the circle drawn over the integral symbol denotes a round trip or a closed loop. It’s the mathematical expression of a conservative force.

 Another way of stating it is, the work done by a conservative force on an object is independent of the path taken by the object. How are the two definitions equivalent? We can prove the second statement using the first statement. In other words, if a work done by a conservative force is zero in a round trip, it naturally ensures that the work done by that force on an object will be independent of the path taken by that object. Let’s see how.

Suppose an object is taken in a round trip from \({\text{A}}\) to \({\text{B}}\) and back to \({\text{A}}\) under the action of a conservative force \(\overrightarrow {\text{F}} .\) Let us denote the path from \({\text{A}}\) to \({\text{B}}\) by \(1\) and the path from \({\text{B}}\) to \({\text{A}}\) by \(2.\) A conservative force does zero work on an object undergoing a round trip. Let us denote the work done by the force \(\overrightarrow {\text{F}} \) on the object as it moves from \({\text{A}}\) to \({\text{B}}\) via path \(1\) as \({{\text{W}}_{{\text{AB}},1}}\) and the work done by the force \(\overrightarrow {\text{F}} \) as the object moves from \({\text{B}}\) to \({\text{A}}\) via path \(2\) as \({{\text{W}}_{{\text{BA}},2}}.\)

We know that the work done in a round trip is zero by a conservative force. So, mathematically,

\({{\text{W}}_{{\text{AB}},1}} + {{\text{W}}_{{\text{BA}},2}} = 0\,\,\,\,\,\,(2)\)

Now let us repeat the process. We will take the object from \({\text{A}}\) to \({\text{B}}\) and back to \({\text{A}}.\) This time, however, we will take the object from \({\text{A}}\) to \({\text{B}}\) via a different path. Let’s number this path \(3.\) Using the same process of reasoning, we will end up with the expression:

\({{\text{W}}_{{\text{AB}},3}} + {{\text{W}}_{{\text{BA}},2}} = 0\,\,\,\,\,(3)\)

where \({{\text{W}}_{{\text{AB}},3}},\) is the work done by the conservative force \(\overrightarrow {\text{F}} \) in moving the object from \({\text{A}}\) to \({\text{B}}\) via path \(3.\) Subtracting equation \(\left( 3 \right)\) from equation \(\left( 2 \right),\) we get:

\({{\text{W}}_{{\text{AB}},1}} + {{\text{W}}_{{\text{BA}},2}} – \left({{{\text{W}}_{{\text{AB}},3}}, +{{\text{W}}_{{\text{BA}},2}}} \right) = 0\)
\({{\text{W}}_{{\text{AB}},1}} – {{\text{W}}_{{\text{AB}},3}} = 0\)
Or,
\({{\text{W}}_{{\text{AB}},1}}{\text{=}}{{\text{W}}_{{\text{AB}},3}}\,\,\,\,\,\,\,(4)\)

From equation \(\left( 4 \right),\) we observe that the work done by the conservative force \(\overrightarrow {\text{F}} \) an object, as it is taken from \({\text{A}}\) to \({\text{B}}\) via paths \(1\) and \(3\) are equal. Did we choose any specific curve for path \(3\)? It was random, wasn’t it? The equation \(4\) must hold true then for any choice of path \(3.\) We can therefore conclude, the work done by force \(\overrightarrow {\text{F}} \) on an object, as the object is moved from \({\text{A}}\) to \({\text{B}}\) is independent of the path taken, which translates to our second definition of a conservative force. Work done by a conservative force is independent of the path taken. Alternatively, the work done by the force \(\overrightarrow {\text{F}} \) depends only on end states \({\text{A}}\) and \({\text{B}}.\)

What is Potential Energy?

Let us think back on our earlier ball-thrown-up problem. We stated that gravity gives back what it takes away. Isn’t throwing a ball from our hand and catching it as it falls back technically a round trip? Although the ball has travelled along a straight line, it has returned to its starting point. This makes the path of the ball a round trip. As gravity is a conservative force, it will do zero work on the ball in a round trip of the ball. As gravity does zero work on the ball in its round trip, and as we have turned off air resistance (or as physicists prefer to say it, “neglecting air resistance”), the ball will have the same kinetic energy, and in turn, speed, as it started with when it was thrown.

When the ball was thrown up, the force of gravity gradually took away all the ball’s kinetic energy as it reached its maximum height. This is evidenced by the fact that the ball comes to a temporary stop (zero kinetic energy). For just an instant, the ball hovers in the air before plummeting back into your outstretched hands. We know that, at the instant the ball comes to a temporary stop, the force of gravity has taken away all the ball’s kinetic energy. But we also know, gravity will return the ball’s kinetic energy once the ball returns to our hands. So, one way of looking at this scenario is that the force of gravity stores the ball’s kinetic energy as it travels up to its maximum height, to be returned to the ball later. This stored energy is known as potential energy. As the conservative force involved is gravity, this potential energy is known as gravitational potential energy.

Now, suppose we study the ball’s motion in terms of kinetic and potential energies. In that case, we say that, as the ball moves up, its kinetic energy is converted to its gravitational potential energy until it reaches its maximum height. Once the ball starts falling back down, its gravitational potential energy is converted back into its kinetic energy, until all its kinetic energy is returned at the point of projection of the ball. This is also known as the law of conservation of mechanical energy.

\({\text{Kinetic}}\,{\text{Energy}} + {\text{Potential}}\,{\text{Energy}} = {\text{Mechanical}}\,{\text{Energy}} = {\text{constant}}\,\,\,\,\,\,\left( 5 \right)\)

We gained a qualitative understanding of potential energy. But physics doesn’t stop here. We need a more rigorous statement of potential energy for all conservative forces. Let’s define potential energy.

Potential Energy Equation

Since potential energy is energy stored away by the conservative force, it is related to the work done by the conservative force. The change in potential energy is defined as the negative of the work done by the conservative force.

\(\Delta {{\text{U}}_{{\text{AB}}}} = \, – \int\limits_{\text{A}}^{\text{B}} {\overrightarrow {{\text{F}}.} } \overrightarrow {{\text{ds}}} \,\,\,\,\,\,\left( 6 \right)\)

The above equation can be interpreted as when an object moves from \({\text{A}}\) to \({\text{B}}\) under the action of the conservative force \(\overrightarrow {\text{F}} ,\) the change in its potential energy is equal to the negative of the work done by the force \(\overrightarrow {\text{F}} \) on the object. The differential form of the above equation can be written as

\({\text{dU}} = \, – \int {\overrightarrow {{\text{F}}.} \,} \overrightarrow {{\text{dr}}} \,\,\,\,\,\,\left( 7 \right)\)

The above equation gives us the incremental change in potential energy \({\text{dU}}\) for an incremental change in position of the object \(\overrightarrow {{\text{dr}}} \) under the action of the conservative force \(\overrightarrow {\text{F}} .\)

(Thought exercise: Now, where is this energy stored? The energy is said to be stored away in the conservative force field. For simplicity’s sake, we often say that the potential energy is stored by the object itself. Potential energy makes some problems easier to solve. It helps explain the working of the conservative force as an exchange between potential and kinetic energies. When an object slows down under the action of a conservative force, we say its kinetic energy is converted to its potential energy. When it speeds up, we say that its potential energy is converted back into its kinetic energy.)

Use it or lose it. If we don’t use an equation immediately, we will not grasp its importance. Let us use equation \(\left( 6 \right)\) in our ball throwing problem. Let a ball of mass \({\text{m}}\) be thrown from point \({\text{A}}\) with a speed \({\text{v}}.\) It reaches point \({\text{B}}\) at a height \({\text{h}}\) before falling back down. The gravitational force acting on the ball is \({\text{m}}\overrightarrow {\text{g}} \) acting in the vertically downward direction. As this force is constant in direction and magnitude. We can take it out of the integral symbol

\(\Delta {{\text{U}}_{{\text{AB}}}} = \, – \int\limits_{\text{A}}^{\text{B}} {{\text{m}}\overrightarrow {\text{g}} } .\overrightarrow {{\text{ds}}} \)
\( = \, – {\text{m}}\overrightarrow {\text{g}} .\int\limits_{\text{A}}^{\text{B}} {\overrightarrow {{\text{ds}}} } \)
\( = \, – {\text{m}}\overrightarrow {\text{g}} .\overrightarrow {\Delta {\text{s}}} \)

As acceleration due to gravity and displacement are opposite in direction, the dot product is negative. We get:

\(\Delta {{\text{U}}_{{\text{AB}}}} = {\text{m}}\left| {\overrightarrow {\text{g}} } \right|\left| {\overrightarrow {\Delta {\text{s}}} } \right|\,\,\,\,\,\,\left( 8 \right)\)
\(\Delta {{\text{U}}_{{\text{AB}}}} = \,{\text{mgh}}\)

The change in potential energy when the ball moves from \({\text{A}}\) to \({\text{B}}\) is \({\text{mgh}}{\text{.}}\) This is alternatively stated as the ball stores potential energy of \({\text{mgh}}{\text{.}}\) when it is raised by a height \({\text{h}}{\text{.}}\) Recall equation \(\left( 5 \right).\) If we use the delta \(\Delta \) operator or the change operator in equation \(\left( 5 \right),\) we will get:

\(\Delta {\text{KE + }}\Delta {\text{PE=}}\Delta {\text{ME=}}\Delta {\text{constant}}\)

As mechanical energy is constant, its delta or change is equal to \(0.\) So the above equation simplifies to:

\(\Delta {\text{KE}} + \Delta {\text{PE}} = 0\)

Or,

\(\Delta {\text{KE}} + \Delta{\text{U}} = 0\,\,\,\,\,\left( 9 \right)\)

Using equation \(\left( 9 \right)\) for the ball, we get:

\(\Delta {\text{K}}{{\text{E}}_{{\text{AB}}}} + \Delta {{\text{U}}_{{\text{AB}}}} = 0\)
\({\text{K}}{{\text{E}}_{\text{B}}} – {\text{K}}{{\text{E}}_{\text{A}}} + \Delta {{\text{U}}_{{\text{AB}}}} = 0\)

The ball comes to a temporary rest at \({\text{B,}}\) and it has a speed of \({\text{v}}\) at \({\text{A}}.\) Plugging in \(\left( 8 \right)\) in the above equation, we get:

\(0 – \frac{1}{2}{\text{m}}{{\text{v}}^2} + {\text{mgh}} = 0\)

Or,

\({\text{mgh=}}\frac{1}{2}{\text{m}}{{\text{v}}^2}\,\,\,\,\,\left({10} \right)\)

Thus, if we know the speed with which the ball is thrown, we can find out the maximum height it will reach and vice versa. Here we see the application of potential energy in a simple projectile problem.

Potential Energy as a State function

Recall equation (6).

\(\Delta {{\text{U}}_{{\text{AB}}}} = \, – \int\limits_{\text{A}}^{\text{B}} {\overrightarrow {\text{F}} } .{\overrightarrow {{\text{ds}}} } \,\,\,\,\,\left( 6 \right)\)

The above equation only describes the change in potential energy as the object is moved from \({\text{A}}\) to \({\text{B}}\) under the action of the conservative force \({\text{F}}.\) We only have a definition for the change in potential energy. How do we write the potential energy at \({\text{A}}\) and \({\text{B}},\) in other words \({\text{UA}}\) and \({\text{UB}}.\) Since we only have a definition for \(\Delta {\text{U,}}\) we arbitrarily set a point or state as having \(0\) potential energy (called the reference point), and every other potential energy is calculated with respect to this reference point.

For example, for small heights, we set the gravitational potential energy at sea level as \(0.\) If the object is raised to a height \({\text{h,}}\) it will have potential energy \({\text{U}} = {\text{mgh}}.\) If it is taken to a depth of \({\text{H}}\) below sea level, it will have a potential energy of \({\text{U}} = \, – {\text{mgH}}\) and so forth. By setting a reference point of zero potential energy, we can calculate the object’s potential energy at all other points in space.

We can see that once we set a reference potential energy level, the potential energy of a body depends solely on its position or state. Although we used gravitational potential energy to demonstrate this, it holds for all conservative forces. As potential energy depends only on the body’s position or state, it is a state function.

Potential Energy is defined for a system of particles or objects. When we talk about the gravitational potential energy of a ball raised to a height, we are referring to the potential energy of the earth-ball system as a result of the mutual force of attraction between the object and the earth. Gravitational potential energy for a system of particles/objects of comparable mass is defined slightly differently and has not been explored in this article.

Which Forces are Conservative?

Examples of conservative forces are the force of gravity, the electrostatic force, the spring force, the magnetic force. Any force that is constant in magnitude and direction is also a conservative force. Although there are no forces in nature that are constant in magnitude and direction (unless configured to create such forces), such a force is often encountered in problem-solving. So always remember, a force constant in magnitude and direction is conservative.

The conservative nature of the gravitational force is why the earth is in a stable orbit and doesn’t spiral into the sun.

What is a Non-Conservative Force?

A force that is NOT conservative is a non-Conservative force. The work done in a round trip is not zero for a non-Conservative force. The work done depends on the path taken. Examples of non-Conservative forces include friction, air drag and viscous force.

Let us see, by a simple example, how a non-Conservative force produces non-zero work in a round trip.

Let us look back on the body thrown up the problem. Earlier, we neglected air drag to study the conservative force of gravity. Now we look at how air drag affects the problem. Air resistance or air drag always acts opposite the direction of motion. When the body moves up, air drag acts downwards. The force and displacement are in opposite directions; the dot product is negative, so the work done by air drag \({{\text{W}}_{{\text{AB}},{\text{air}}\,{\text{drag}}}} < 0.\) When the body moves downwards, air drag acts upwards. The force and displacement are once again in opposite directions; the dot product is negative, so the work done by air drag \({{\text{W}}_{{\text{BA}},{\text{air}}\,{\text{drag}}}} < 0.\) The total work done by air drag as the body completes its round trip is therefore negative and non-zero.

In reality, vertical projectiles return at speed slower than the speed with which they were projected. This is because the non-Conservative force of air drag takes away some of the projectile’s kinetic energy and doesn’t give it back. But where does this energy go? We know, in the grand scheme of things, matter and energy are conserved. So, the lost kinetic energy of the projectile must have gone somewhere. This lost energy is dissipated as heat and sound in the air/atmosphere. This is why non-Conservative forces are also known as dissipative forces.

Summary

A conservative force is a force that performs zero work on an object undergoing a round trip. The work done by a conservative force depends only on the end states or endpoints and not the path. Conservative forces include the force of gravity, the electrostatic force, the magnetic force, and the spring force. The change in potential energy is defined as the negative of the work done by the conservative force. By setting the potential energy at a reference point as zero, we can evaluate the potential energy at any other point in space. A non-conservative force is a force that performs non-zero work on an object undergoing a round trip. The work done by a non-conservative force depends on the path taken. Examples of non-conservative forces include friction, air drag, viscous force.

FAQs on Conservative and Non-Conservative Forces

Q.1. What is a conservative force?
Ans: A conservative force is a force that performs zero work on an object undergoing a round trip. The work done by a conservative force depends only on the end states or endpoints and not the path.

Q.2. What are some examples of conservative forces?
Ans: Conservative forces include the force of gravity, the electrostatic force, the magnetic force, and the spring force.

Q.3. What is potential energy?
Ans: The change in potential energy is defined as the negative of the work done by the conservative force. Potential energy simplifies some mechanics problems by looking at them as an exchange between kinetic and potential energies.

Q.4. What is a non-conservative force?
Ans: A non-conservative force is a force that performs non-zero work on an object undergoing a round trip. The work done by a non-conservative force depends on the path taken. Examples of non-conservative forces include friction, air drag, viscous force.

Q.5. Why are non-conservative forces also known as dissipative forces?
Ans: Non-conservative forces are dissipative as they do negative work and take away energy from a system. If only conservative forces are present, energy is simply converted from potential energy to kinetic energy and vice versa. But non-conservative forces take away energy from the system and dissipate it as heat and sound.