Potential Energy and Conservative Forces

Branch role: This page develops potential energy as a system/configuration quantity, derives near-Earth GPE, relates elastic energy to graph area, and explains $F_x=-\mathrm{d}U/\mathrm{d}x$.

Overview

This branch note explains potential energy as system energy associated with position or configuration, with emphasis on gravitational, elastic and electric examples used in H2 Physics.

Core Ideas

• Potential energy belongs to an interacting system and requires a chosen reference level.
• Conservative-force work can be represented by changes in potential energy.
• In one-dimensional uniform-field cases, the force is related to the negative gradient of potential energy.

Exam Relevance

Use this note when deciding a reference level, deriving or applying $\Delta U_g=mg\Delta h$, finding elastic energy from area, or interpreting $F_x=-\mathrm{d}U/\mathrm{d}x$.

What potential energy belongs to

Potential energy is associated with the configuration of interacting bodies, not with one completely isolated object. Examples include:

• object–Earth configuration: gravitational potential energy;
• charge configuration in an electric field: electric potential energy;
• deformed spring or elastic material: elastic potential energy.

The zero of potential energy is arbitrary. A change

$$\Delta U=U_f-U_i$$

is independent of adding the same constant to every value of $U$.

Conservative forces

A force is conservative if its work between two positions is independent of path. Equivalently, its total work around a closed path is zero.

For a conservative interaction,

$$W_{\mathrm{cons}}=-\Delta U.$$

Positive work by the conservative force corresponds to decreasing potential energy. Negative work corresponds to increasing potential energy.

Gravity, electrostatic force and an ideal spring force are conservative. Friction and drag are non-conservative because their work generally depends on the path and transfers mechanical energy into internal energy.

Near-Earth gravitational potential energy

Consider lifting a body of mass $m$ slowly from height $h_i$ to $h_f$. Slow lifting means negligible acceleration and negligible change in kinetic energy, so the applied upward force has magnitude $mg$.

The work done by the applied force is

$$W_{\mathrm{applied}}=mg(h_f-h_i).$$

This becomes the increase in gravitational potential energy of the object–Earth system:

$$\Delta U_g=mg\Delta h.$$

If $U_g=0$ is chosen at $h=0$,

$$U_g=mgh.$$

Figure: The equality $F_{\mathrm{lift}}=mg$ follows from zero acceleration during slow lifting; it is not true for every upward motion. The vertical change $\Delta h$ determines $\Delta U_g$, regardless of whether the body follows a vertical path or a ramp, provided $g$ is uniform and dissipative effects are handled separately.

The formula is valid near Earth’s surface over distances for which $g$ is approximately constant. The more general gravitational-potential treatment belongs in Gravitational Fields.

Worked example: arbitrary zero

A $2.00\mathrm{kg}$ mass moves from $1.50\mathrm{m}$ to $4.00\mathrm{m}$ above a floor:

$$\Delta U_g=(2.00)(9.81)(4.00-1.50)=49.1\mathrm{J}.$$

Choosing the tabletop rather than the floor as $U_g=0$ changes the individual $U_i$ and $U_f$ values but not $49.1\mathrm{J}$.

Elastic potential energy

The elastic potential-energy increase equals the work done in slowly deforming the material, provided dissipative losses are negligible:

$$\Delta U_s=\text{area under the loading force–extension graph}.$$

For a Hooke-law spring, $F_{\mathrm{applied}}=kx$, so from natural length to extension $x$:

$$U_s={\int }_0^{x}k{x}^{\prime }\mathrm{d}{x}^{\prime }=\frac{1}{2}k{x}^2.$$

Figure: The area rule is the general syllabus idea; the triangle $\frac{1}{2}k{x}^2$ is the special linear case. Extension $x$ is measured from natural length. The restoring force on an attached mass is $-kx$, directed toward the undeformed position.

Work by the spring

If extension changes from $x_i$ to $x_f$,

$$W_{\mathrm{spring}}=-\Delta U_s=-\frac{1}{2}k(x_f^2-x_i^2).$$

When a stretched spring moves toward natural length, $U_s$ decreases and the spring does positive work.

Electric potential energy: the required distinction

Electric potential energy is associated with the configuration of charges in an electric interaction. Like gravitational potential energy, it is defined through work by a conservative field and only differences are physically significant:

$$W_{\mathrm{electric}}=-\Delta U_e.$$

Detailed formulas involving electric potential are developed in Electric Fields. Here, the required distinction is conceptual:

• gravitational PE concerns masses and gravitational interaction;
• electric PE concerns charges and electric interaction;
• elastic PE concerns deformation of matter.

Force from a potential-energy graph

For a one-dimensional conservative interaction,

$$F_x=-\frac{\mathrm{d}U}{\mathrm{d}x}.$$

Figure: The force is the negative gradient of the potential-energy graph; here $C>0$ is a constant with force units. A straight line represents a uniform field because its gradient, and therefore the force, is constant. A steeper graph means a larger force magnitude; a stationary point has $F_x=0$.

Reading the sign

• If $U$ rises as $x$ increases, $\mathrm{d}U/\mathrm{d}x>0$ and $F_x<0$.
• If $U$ falls as $x$ increases, $\mathrm{d}U/\mathrm{d}x<0$ and $F_x>0$.
• The force points toward decreasing $U$.

Uniform gravitational example

Choose upward $x$. Near Earth,

$$U_g=mgx,$$

so

$$F_x=-\frac{\mathrm{d}{U}_g}{\mathrm{d}x}=-mg.$$

The negative sign correctly gives a downward gravitational force.

Hooke-law example

For

$$U_s=\frac{1}{2}k{x}^2,$$ $$F_x=-\frac{\mathrm{d}{U}_s}{\mathrm{d}x}=-kx.$$

Mechanical energy

When conservative interactions are represented through potential energy and no net non-conservative transfer changes the system’s mechanical energy,

$$E_k+U=\text{constant}.$$

If friction or an external drive matters, include its transfer explicitly rather than applying this equation blindly. See Energy Forms and Conservation.

Common errors

• Saying a body “contains GPE” without identifying the gravitational interaction/system.
• Treating $mgh$ as an absolute value independent of reference level.
• Using path length instead of vertical height difference in $mg\Delta h$.
• Assuming $F_{\mathrm{lift}}=mg$ when the body accelerates.
• Starting with $\frac{1}{2}k{x}^2$ before checking the force–extension relation is linear.
• Omitting the negative sign in $F_x=-\mathrm{d}U/\mathrm{d}x$.
• Calling the force equal to the graph gradient rather than its negative.

Summary

Interaction	Potential-energy relation	Force/work link
near-Earth gravity	$\Delta U_g=mg\Delta h$	$W_g=-\Delta U_g$
Hooke-law spring	$U_s=\frac{1}{2}k{x}^2$	$F_x=-kx$
any 1D conservative interaction	$U(x)$	$F_x=-\mathrm{d}U/\mathrm{d}x$

Links

• Work, Energy and Power
• Work and Force–Displacement Graphs
• Energy Forms and Conservation
• Gravitational Fields
• Electric Fields