2.3.4.2. Conservative Forces

Table of Contents

Idea of a Conservative Force

A conservative force is a force for which the work done between two points depends only on the starting point and the ending point, not on the path taken between them. This is the key property.

If an object moves from point $A$ to point $B$ under a conservative force, then every possible path connecting $A$ and $B$ gives the same work by that force.

A closely related statement is that the total work done by a conservative force around any closed path is zero. A closed path means the object starts and ends at the same position.

For a conservative force,
$$W_{A \to B} \text{ depends only on } A \text{ and } B$$
and equivalently,
$$\oint \vec{F} \cdot d\vec{r} = 0$$
for any closed path.

These two ideas are the signature of conservative forces.

Why Conservative Forces Matter

Conservative forces are important because they allow us to define potential energy. When a force is conservative, energy can be stored in position, not only in motion.

This means that as an object moves, energy can shift back and forth between kinetic energy and potential energy without being lost by the conservative force itself.

For example, when a ball rises upward, its kinetic energy decreases while gravitational potential energy increases. When it falls back down, the opposite happens.

Path Independence

To understand path independence, imagine lifting an object from the floor to a shelf. You could move it straight up, or move it sideways first and then upward. Gravity acts during the motion, but the work done by gravity depends only on the change in height.

So for gravity near Earth,

$$W_g = -mg(y_f - y_i)$$

where $y_i$ is the initial height and $y_f$ is the final height.

If the object starts and ends at the same height, then the net work done by gravity is zero, even if the path was complicated.

This is exactly what we mean by path independence.

Rendering scientific sketch…

Different paths, same work by gravity

Closed-Path Test

A very useful test for a conservative force is to imagine the object moving around a loop and returning to where it started. Since the initial and final positions are the same, the work done by a conservative force must be zero.

Gravity gives a simple example. If you carry an object around a route and bring it back to the same place, gravity has done zero total work.

This does not mean the force was zero during the motion. It means that the positive and negative contributions to the work cancel over the full closed path.

Relation to Potential Energy

A conservative force can always be associated with a potential energy function $U$. The work done by the force is equal to the negative change in potential energy:

For a conservative force,
$$W_{A \to B} = -(U_B - U_A) = -\Delta U$$

This minus sign is very important. If the conservative force does positive work, the potential energy decreases. If the conservative force does negative work, the potential energy increases.

So instead of calculating work directly along a path, we can often use potential energy differences.

Common Examples

The most common conservative forces in introductory physics are shown below.

Force	Conservative?	Typical potential energy
Gravitational force near Earth	Yes	$U = mgy$
Universal gravitational force	Yes	$U = -\dfrac{GMm}{r}$
Spring force	Yes	$U = \dfrac{1}{2}kx^2$
Kinetic friction	No	No single potential energy function

Gravity and spring forces are standard conservative forces. Friction is the classic example of a non-conservative force because its work depends on the path length, not just the endpoints.

Gravitational Force as a Conservative Force

Near Earth's surface, the gravitational force on a mass $m$ is approximately constant and downward:

$$\vec{F}_g = -mg\,\hat{j}$$

If the object moves from height $y_i$ to height $y_f$, the work done by gravity is

$$W_g = -mg(y_f - y_i)$$

and this can be written as

$$W_g = -(U_f - U_i)$$

with

$$U = mgy$$

So gravity is conservative because the work depends only on the change in height.

Spring Force as a Conservative Force

For an ideal spring, Hooke's law gives

$$F_x = -kx$$

where $x$ is displacement from equilibrium and $k$ is the spring constant.

The work done by the spring as the object moves from $x_i$ to $x_f$ is

$$W_s = \frac{1}{2}k x_i^2 - \frac{1}{2}k x_f^2$$

This depends only on the initial and final values of $x$, not on how the object moved between them.

The corresponding potential energy is

$$U = \frac{1}{2}kx^2$$

So the spring force is also conservative.

Rendering scientific sketch…

Spring force and displacement

Mathematical Condition in One Dimension

In one dimension, a force $F(x)$ is conservative if its work can be written in terms of a potential energy function $U(x)$ such that

$$F(x) = -\frac{dU}{dx}$$

This means the force points in the direction of decreasing potential energy.

For example, if

$$U(x) = \frac{1}{2}kx^2$$

then

$$F(x) = -\frac{d}{dx}\left(\frac{1}{2}kx^2\right) = -kx$$

which is the spring force.

Physical Meaning of the Minus Sign

The minus sign in $F = -dU/dx$ tells us that conservative forces push or pull objects toward lower potential energy.

If potential energy increases as $x$ increases, then the force points toward smaller $x$.

If potential energy decreases as $x$ increases, then the force points toward larger $x$.

This is why objects released from rest often move "downhill" in a potential energy landscape.

Conservative Versus Non-Conservative Forces

It is useful to compare conservative and non-conservative forces directly.

Property	Conservative force	Non-conservative force
Work depends only on endpoints	Yes	No
Work around a closed path	Zero	Usually not zero
Can define potential energy	Yes	No single general potential energy
Mechanical energy by this force alone	Conserved	Not conserved

Friction is non-conservative because the work it does usually depends on the total distance traveled. If you slide a box across a floor, a longer path means more energy lost as heat.

By contrast, gravity and spring forces can store and release energy without that path dependence.

Recognizing a Conservative Force in Problems

In beginner mechanics problems, a force is usually treated as conservative if it is gravity or an ideal spring force. When this happens, you should think in terms of potential energy and energy conservation.

A force is likely non-conservative if it causes energy dissipation into heat, sound, or deformation, as friction often does.

Still, the defining test is not the name of the force but the work property. The true question is whether the work depends only on endpoints.

Summary

A conservative force is a force whose work depends only on initial and final position. Equivalently, its work over any closed path is zero. Because of this, we can define a potential energy function for it.

Key facts for conservative forces:
$$W_{A \to B} = -\Delta U$$
$$\oint \vec{F} \cdot d\vec{r} = 0$$
$$F(x) = -\frac{dU}{dx} \quad \text{in one dimension}$$

Gravity and ideal spring forces are the main examples. These forces do not destroy mechanical energy, they transfer it between kinetic and potential forms.

Comments

Please login to add a comment.

Don't have an account? Register now!