Potential Energy and Conservative Forces

Overview

This page develops the ideas of:

• Potential energy: energy stored due to position or configuration
• Conservative forces: forces for which work done depends only on initial and final positions

These ideas are central to energy methods in mechanics and extend the overview in Work, Energy and Power.

Common H2 applications:

• objects raised or lowered in gravitational fields
• springs and elastic systems
• motion with negligible friction
• conservation of mechanical energy

Why It Matters

Potential energy allows position-dependent forces to be handled with energy methods instead of detailed force-by-force motion analysis.

Definition

Potential energy is energy stored because of position or configuration. It is meaningful when work done by a force can be related to a change in stored energy.

For a conservative force:

$$W_{\text{cons}}=-\Delta E_p$$

This means a conservative force doing positive work reduces potential energy, while work done against a conservative force increases potential energy.

Key Representations

$$\Delta E_p=mg\Delta h$$ $$E_p=\frac{1}{2}k{x}^2$$ $$W_{\text{cons}}=-\Delta E_p$$ $$E_k+E_p=\text{constant}$$

Scalar and Vector Distinction

Be precise.

Vector Quantities

• Force: $\vec{F}$
• Displacement: $\vec{s}$
• Velocity: $\vec{v}$
• Acceleration: $\vec{a}$

Scalar Quantities

• Work: $W$
• Potential energy: $E_p$
• Kinetic energy: $E_k$
• Height
• Distance

Potential energy is a scalar quantity.

What Is Potential Energy?

Potential energy is energy associated with:

• position in a force field
• configuration of a system
• deformation of an elastic object

Examples:

• lifted object
• stretched spring
• compressed spring

Only changes in potential energy are physically important.

Gravitational Potential Energy

Near Earth’s Surface

For height change $\Delta h$ in a uniform gravitational field:

$$\Delta E_p=mg\Delta h$$

If zero level is chosen conveniently:

$$E_p=mgh$$

where:

• $m$ = mass
• $g$ = gravitational field strength
• $h$ = height above chosen reference level

Important Notes

• Reference level is arbitrary.
• Only differences in potential energy matter.
• Valid when $g$ is approximately constant near Earth’s surface.

See also Gravitational Fields

Elastic Potential Energy

For a Hooke’s law spring:

$$F=kx$$

where:

• $k$ = spring constant
• $x$ = extension or compression

Stored elastic potential energy:

$$E_p=\frac{1}{2}k{x}^2$$

This equals the area under the force-extension graph.

Conservative Forces

Definition

A force is conservative if the work it does between two points is independent of path taken.

Work depends only on:

• initial position
• final position

Examples

• gravitational force
• elastic spring force
• electrostatic force

Consequences

For a conservative force:

• energy can be stored as potential energy
• mechanical energy can be conserved (if no non-conservative forces act)

Non-Conservative Forces

A force is non-conservative if work done depends on path length or route taken.

Examples:

• friction
• drag
• air resistance

These forces usually transfer mechanical energy into:

• thermal energy
• sound
• internal energy

Path Independence

Conservative Case

Lifting an object vertically or moving it along a ramp to the same height gives the same change in gravitational potential energy:

$$\Delta E_p=mg\Delta h$$

Same start and end heights → same energy change.

Non-Conservative Case

With friction, longer path means more energy lost.

Work Done and Potential Energy Change

For a conservative force:

$$W_{\text{cons}}=-\Delta E_p$$

Meaning:

• if conservative force does positive work, potential energy decreases
• if potential energy increases, external work must usually be supplied

Gravitational Example

Object falls downward.

Gravity does positive work:

$$W_g>0$$

So:

$$\Delta E_p<0$$

Potential energy decreases.

Spring Example

Released stretched spring pulls block forward.

Spring force does positive work on block, so elastic potential energy decreases.

Mechanical Energy Conservation

If only conservative forces act:

$$E_k+E_p=\text{constant}$$

That is:

$$(E_k+E_p{)}_i=(E_k+E_p{)}_f$$

Useful for:

• falling objects
• roller coasters (idealised)
• pendulums (neglecting losses)
• spring-mass systems

See Energy Forms and Conservation

Worked Examples

Example 1: Gain in Gravitational Potential Energy

A $3.0\text{ kg}$ object is raised by $2.5\text{ m}$.

$$\Delta E_p=mg\Delta h$$ $$=3.0(9.81)(2.5)$$ $$=73.6\text{ J}$$

Example 2: Spring Energy

A spring of constant $200{\text{ N m}}^{-1}$ is stretched by $0.10\text{ m}$.

$$E_p=\frac{1}{2}k{x}^2$$ $$=\frac{1}{2}(200)(0.10{)}^2$$ $$=1.0\text{ J}$$

Example 3: Falling Object Speed

A ball drops from height $5.0\text{ m}$, neglect air resistance.

Loss in GPE = gain in KE:

$$mgh=\frac{1}{2}m{v}^2$$

Mass cancels:

$$9.81(5.0)=\frac{1}{2}{v}^2$$ $$v=9.90{\text{ m s}}^{-1}$$

Example 4: Friction Present

A block slides down rough slope.

Then:

$$\text{loss in GPE}\ne \text{gain in KE}$$

because some energy is dissipated by friction.

Need to include work done by friction.

Graph Interpretation (Advanced Useful View)

Gravitational Field Near Earth

If:

$$E_p=mgh$$

then gradient with respect to height gives force magnitude:

$$\frac{d{E}_p}{dh}=mg$$

General Statement

For one-dimensional conservative systems:

$$F=-\frac{d{E}_p}{dx}$$

H2 students should mainly understand sign meaning:

• force acts toward decreasing potential energy.

Common Exam Pitfalls

1. Using Wrong Reference Level

Absolute value of $E_p$ depends on chosen zero level.

Only changes matter.

2. Assuming Mechanical Energy Conserved with Friction

Not true unless losses negligible.

3. Wrong Height Change

Use vertical height difference, not path length.

4. Mixing Force and Energy Units

• Force in N
• Energy in J

5. Forgetting Squared Term in Spring Energy

$$E_p=\frac{1}{2}k{x}^2$$

not $\frac{1}{2}kx$.

Summary

• Potential energy is stored energy due to position/configuration.
• Near Earth:

$$\Delta E_p=mg\Delta h$$

• Spring energy:

$$E_p=\frac{1}{2}k{x}^2$$

• Conservative forces have path-independent work.
• For conservative forces:

$$W=-\Delta E_p$$

• If only conservative forces act:

$$E_k+E_p=\text{constant}$$

Links

• Work, Energy and Power
• Kinetic Energy and Work-Energy Theorem
• Energy Forms and Conservation
• Power and Efficiency
• Forces
• Dynamics
• Gravitational Fields
• Vectors