Unit 4 · Topic 07

Force and potential energy

A function U(x) contains local information about a conservative force. Its negative slope gives the force direction and helps classify equilibria.

Unit 4Force and potentialOpen navigation

Concept 01

Force points toward lower U

Essential The minimum you should retain

In one dimension, Fx=−dUdx F_x=-dU/dx .

UnderstandInterpret and connect

If U increases with x, force points toward -x; if U decreases, force points toward +x.

DeepenFormulation and conditions

The minus sign expresses the physical tendency toward lower potential energy.

ExploreConnections for further study

The slope magnitude gives the local magnitude of the conservative force.

Mathematical relation

Force and motion from U(x)

Fx=−dUdx,K=E−U,v=2(E−U)m F_x=-\frac{dU}{dx},\quad K=E-U,\quad v=\sqrt{\frac{2(E-U)}m}
Represents

Conservative force as negative slope and allowed speed in a 1D energy diagram.

Physical interpretation

Force points toward lower U and only U≤E U\le E is allowed.

DeepenVariables, conditions, and checks

Variables

F_x
conservative force; usual unit: N
U
potential energy; usual unit: J
E
total mechanical energy; usual unit: J
v
speed; usual unit: m/s

Conditions of application

  • One-dimensional conservative system.
  • The speed expression requires E≥U.

Dimensional check

J/m=N; the square root gives m/s.

Errors it helps prevent

  • Using F=dU/dx.
  • Allowing U>E with real K.

F_x is the negative slope

A parabola shows force toward the minimum and zero force at equilibrium.F_x > 0F_x = 0F_x < 0
  • U=x²

On the left the slope is negative and F points toward +x; on the right the slope is positive and F points toward -x.

Concept 02

Recovering familiar forces

Essential The minimum you should retain

For U=(1/2)kx², the derivative gives F_x=-kx.

UnderstandInterpret and connect

For U=mgy with +y upward, it gives F_y=-mg.

DeepenFormulation and conditions

The additive constant in U disappears upon differentiation.

ExploreConnections for further study

Different potential references produce exactly the same force.

Worked example

Equilibrium from U(x)

A particle has U(x)=4x-x², in joules when x is in metres.

Given
  • U(x)=4x-x² J
Target

Find and classify equilibrium.

  1. Principle

    Fx=−dUdx F_x=-dU/dx .

  2. Derivative

    dU/dx=4-2x, so F_x=2x-4.

  3. Equilibrium

    F_x=0 gives x=2 m.

  4. Classification

    U''=-2<0: U has a local maximum and equilibrium is unstable.

Conclusion

Equilibrium is at x=2 m and is unstable.

Concept 03

Equilibrium condition

Essential The minimum you should retain

At equilibrium, F_x=0 and therefore dU/dx=0.

UnderstandInterpret and connect

A local minimum is stable equilibrium and a local maximum is unstable equilibrium.

DeepenFormulation and conditions

A locally flat region represents neutral equilibrium in the ideal model.

ExploreConnections for further study

Not every zero-slope point is stable; the local shape of U must be examined.

Concept 04

Smooth potentials near equilibrium

Essential The minimum you should retain

Near a smooth minimum, U often looks locally parabolic.

UnderstandInterpret and connect

This approximation produces a force nearly proportional to displacement.

DeepenFormulation and conditions

Curvature measures how stiffly the system is confined around equilibrium.

ExploreConnections for further study

That is why spring-like behaviour appears near many stable equilibria even when the interaction is not literally a spring.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Using F_x=dU/dx.

A conservative force obeys Fx=−dUdx F_x=-dU/dx .

Classifying every point with dU/dx=0 as stable.

Inspect the local shape: a minimum is stable, a maximum unstable, and a flat region neutral.