2.3.4.3. Non-Conservative Forces

Table of Contents

What makes a force non conservative

In the study of energy, some forces allow us to define a clean potential energy, while others do not. A non conservative force is a force for which the work done depends on the path taken, not only on the starting and ending positions.

If an object moves from point A to point B, a conservative force gives the same work no matter which route is followed. A non conservative force can give different amounts of work for different routes between the same two points.

Friction is the most common example. If you slide a box across a floor, the longer the path, the more energy is lost. Air resistance is another important example. These forces usually transform mechanical energy into other forms, such as thermal energy or sound.

A force is non conservative if its work depends on the path taken.
For a non conservative force,
$$W_{nc} \neq \text{a function of only initial and final position}.$$

Work and mechanical energy

Mechanical energy is the sum of kinetic energy and potential energy:

$$E_{\text{mech}} = K + U$$

When only conservative forces act, mechanical energy remains constant. When non conservative forces act, mechanical energy usually changes.

The key relation is

$$W_{nc} = \Delta E_{\text{mech}} = \Delta K + \Delta U$$

This means that the work done by non conservative forces equals the change in the system's mechanical energy.

If friction does negative work, then the mechanical energy decreases. The lost mechanical energy is not destroyed. It is converted into other forms of energy, often internal energy.

For motion involving conservative and non conservative forces,
$$W_{nc} = \Delta K + \Delta U$$
If $W_{nc} < 0$, mechanical energy decreases.
If $W_{nc} > 0$, mechanical energy increases.

Friction as a non conservative force

Suppose a block slides on a rough horizontal surface. Kinetic friction opposes the motion. If the friction force has magnitude $f_k$ and the block moves a distance $d$, then the work done by friction is

$$W_f = -f_k d$$

The negative sign appears because friction acts opposite to the displacement.

If the block starts with kinetic energy and no change in potential energy occurs, then

$$W_f = \Delta K$$

so friction reduces the kinetic energy. The missing mechanical energy appears mainly as thermal energy in the block and the floor.

This is why moving objects on rough surfaces slow down unless some external force continues to supply energy.

Air resistance and drag

Air resistance is also non conservative. Unlike simple kinetic friction, drag often depends on speed. At low speeds it may be approximately proportional to velocity, and at higher speeds often proportional to the square of speed. Because the force changes with motion and depends on the path through the fluid, its work is not described by a simple potential energy function.

A falling object in air does not keep all the mechanical energy predicted for ideal free fall. Some of the gravitational potential energy is transformed into thermal energy in the air and the object.

Energy dissipation

A central effect of non conservative forces is dissipation. Dissipation means that mechanical energy is converted into less easily recovered forms, especially heat.

For example, when a box slides to rest,

$$K_i + U_i + W_{nc} = K_f + U_f$$

If the only non conservative force is friction, then friction makes the final mechanical energy smaller than the initial mechanical energy.

In many practical situations, it is useful to write an energy balance that includes thermal energy:

$$K_i + U_i = K_f + U_f + E_{\text{thermal}}$$

This shows clearly that the total energy is still conserved, even though mechanical energy is not.

Non conservative forces do not violate conservation of total energy.
They reduce or increase mechanical energy by converting energy between mechanical and other forms.

Comparing conservative and non conservative forces

The distinction becomes clearer in a direct comparison.

Property	Conservative force	Non conservative force
Work depends on path	No	Yes
Work over closed path	Zero	Usually nonzero
Potential energy can be defined	Yes	No simple general potential energy
Mechanical energy conserved by itself	Yes, if only these forces act	No, mechanical energy changes
Common examples	Gravity, spring force	Friction, air resistance

For a closed path, a conservative force does zero net work. But with friction, moving an object around a loop and returning to the starting point still requires energy, and friction removes that energy as heat.

Closed path idea

Imagine pushing a box around a rough track and bringing it back to where it started. The initial and final positions are the same, so the change in potential energy is zero. If the speed also ends as it began, then the mechanical energy of the box is unchanged. But friction has still done negative work throughout the trip, so an external agent must have done positive work to keep the box moving. That supplied energy becomes thermal energy.

This is one of the clearest signs that friction is non conservative.

For a conservative force over any closed path,
$$W = 0$$
For a non conservative force, such as friction, over a closed path,
$$W \neq 0$$
in general.

Simple example

A block of mass $m$ slides a distance $d$ down a rough incline. Gravity and the normal force act, and friction opposes the motion. If the block drops through a vertical height $h$, then the change in gravitational potential energy is

$$\Delta U = -mgh$$

If the friction force is $f_k$, its work is

$$W_f = -f_k d$$

Using the energy relation,

$$W_f = \Delta K + \Delta U$$

$$-f_k d = \Delta K - mgh$$

and therefore

$$\Delta K = mgh - f_k d$$

This tells us that gravity increases the kinetic energy, but friction reduces that increase.

Visualizing a rough incline

Rendering scientific sketch…

Block sliding on a rough incline

When non conservative forces increase mechanical energy

Non conservative forces do not always reduce mechanical energy. Sometimes they increase it. For example, if a person pushes a box and does work on it, the box can gain kinetic energy. The applied force from the person is often treated as a non conservative external force.

If an applied force does positive work, then

$$W_{nc} > 0$$

and the mechanical energy increases.

So non conservative forces are not defined by whether they remove energy, but by the fact that their work is not determined only by position.

Main takeaway

Non conservative forces change mechanical energy by transferring energy into or out of the mechanical forms, kinetic and potential. Friction and air resistance are the main examples in elementary mechanics. They make the work depend on the path and often convert useful mechanical energy into thermal energy.

The most important equation for non conservative forces is
$$W_{nc} = \Delta K + \Delta U = \Delta E_{\text{mech}}$$
Mechanical energy is not conserved when non conservative forces act alone on a system, but total energy is always conserved.

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