What Is the Relation Between Potential Energy and Force?

Potential energy describes how the energy of a system changes with configuration. The force associated with that energy is determined by how rapidly potential energy changes when position changes. In short, a conservative force points in the direction of decreasing potential energy.

This relation lets you calculate a force from a potential-energy function and understand why a system tends to move toward some configurations rather than others. It applies to conservative forces; it does not turn every force in a free-body diagram into a potential-energy force.

The force–potential relation

In one dimension, the force corresponding to a potential energy function $U(x)$ is

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

The minus sign matters: a positive slope in $U(x)$ means the force points toward decreasing $x$, while a negative slope means the force points toward increasing $x$.

In three dimensions, the corresponding equation is

$$\mathbf{F}=-\nabla U,$$

where $\nabla U$ is the gradient of potential energy. The gradient points in the direction of the steepest increase in $U$, so the negative gradient points in the direction of the steepest decrease.

The units provide a useful check. Potential energy is measured in joules, and a joule per metre is a newton. Thus, differentiating energy with respect to position gives the units of force.

Connection to work

For a conservative force, the work done as an object moves from position $A$ to position $B$ is the negative change in potential energy:

$$W_{A\to B}=-\Delta U=-(U_B-U_A).$$

If the force does positive work, the system’s potential energy decreases. If potential energy increases, the conservative force does negative work. This is the work–energy connection behind the derivative formula: over a small displacement $dx$, $dW=F_x\,dx$ and $dW=-dU$, so $F_x=-dU/dx$.

The zero of potential energy is a choice of reference. Adding a constant to $U$ changes neither its derivative nor the force. For example, with $y$ measured upward, gravitational potential energy near Earth can be written as $mgy$ or $mgy+C$; either gives the same force, $F_y=-mg$.

Worked example: an ideal spring

For a spring with stiffness $k$, measured from its relaxed position, the potential energy is

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

Differentiate with respect to $x$ and reverse the sign:

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

Suppose $k=200\,\text{N/m}$ and the spring is stretched $0.10\,\text{m}$ to the right. Then

$$F_x=-(200\,\text{N/m})(0.10\,\text{m})=-20\,\text{N}.$$

The negative sign means the spring force points left, back toward its relaxed position. At this displacement, the stored potential energy is

$$U=\frac{1}{2}(200)(0.10)^2=1.0\,\text{J}.$$

If the spring moves from $x=0.10\,\text{m}$ to $x=0$, its potential energy falls from $1.0\,\text{J}$ to zero. The spring does $+1.0\,\text{J}$ of work, consistent with $W=-\Delta U$.

What the relation says about equilibrium

An equilibrium position occurs where the force is zero, which in one dimension means $dU/dx=0$. A strict local minimum of $U$ is a stable equilibrium in the usual one-dimensional conservative setting: sufficiently small perturbations remain near the equilibrium, and in common smooth cases the force tends to restore the system. A strict local maximum is unstable; in the usual smooth, nondegenerate case, a small displacement produces a force away from the maximum. A zero slope alone does not establish stability; inspect the potential nearby. Non-strict or degenerate extrema may require closer analysis.

For the spring, $U(x)=\tfrac12kx^2$ has a minimum at $x=0$, so its equilibrium there is stable. A zero slope alone does not establish stability; you must also inspect the shape of the potential nearby.

Common mistakes

• Dropping the minus sign. The force is not the positive slope of potential energy. A spring stretched to the right has a positive displacement but a force to the left.
• Confusing force with potential energy. Force is measured in newtons; potential energy is measured in joules. They are related by a spatial derivative, not interchangeable quantities.
• Applying the formula to every force. Kinetic friction and air resistance are generally nonconservative. Their work depends on the path, so they cannot usually be represented by one position-only potential-energy function.
• Assuming total force always comes from potential energy. The gradient gives the conservative force associated with $U$. If friction or another nonconservative force also acts, include it separately when finding the net force.
• Treating the reference level as physical. The absolute value assigned to potential energy is arbitrary in ordinary mechanics; differences and derivatives carry the relevant information.

When several conservative forces act

If several conservative interactions are present, add their potential energies to obtain the total potential energy, $U_{\text{total}}=U_1+U_2+\cdots$. The associated total conservative force is then $-\nabla U_{\text{total}}$, which is the sum of the individual conservative forces. This is useful for analyzing a system with, for example, both gravitational and spring potential energy.