Potential Energy

Conservative forces and potential functions

The work done by a conservative force depends only on the initial and final positions. It is independent of the route between them. Gravity near Earth's surface and the ideal spring force are standard examples. Kinetic friction is not conservative: a longer sliding path produces a larger magnitude of friction work, even when the endpoints are unchanged.

A conservative force $F_{c}$ defines potential-energy change through

Δ U = U_{f} - U_{i} = - W_{c} = - i \int f F_{c} \cdot d r .

Equivalently, the net work around any closed path is zero:

\oint F_{c} \cdot d r = 0.

Potential energy belongs to a system, not to one object in isolation. The gravitational potential energy $m g y$ belongs to the object-Earth system, and elastic potential energy to the spring-object system. As an object falls, it gains kinetic energy while the gravitational potential energy of the object-Earth system drops by the same amount.

Only differences in $U$ appear in force and energy equations. Adding a constant $C$ to every value, $U^{'} = U + C$ , changes no force and no observable energy difference. The zero may therefore be assigned at any convenient reference configuration. For near-surface gravity, taking $U = 0$ at $y = 0$ gives

U_{g} (y) = m g y, Δ U_{g} = m g (y_{f} - y_{i}) .

The result follows directly from the constant downward force $F_{g} = - m g \overset{y}{^}$ :

W_{g} = y_{i} \int y_{f} (- m g) d y = - m g (y_{f} - y_{i}) = - Δ U_{g} .

The approximation $U_{g} = m g y$ requires nearly constant $g$ over the height change. It is appropriate for laboratory, projectile, and ordinary engineering scales.

An ideal spring obeying $F_{x} = - k x$ , with $x = 0$ at its unstretched length, has

U_{s} (x) - U_{s} (0) = - 0 \int x (- k x^{'}) d x^{'} = \frac{1}{2} k x^{2} .

Thus $U_{s} = \frac{1}{2} k x^{2}$ under the reference $U_{s} (0) = 0$ . Compression and extension of equal magnitude store equal elastic potential energy. The force has opposite sign on the two sides of equilibrium because it points toward $x = 0$ .

The spring force is the negative slope of its parabolic potential. Equal extensions on either side of equilibrium have equal potential energy but opposite force directions.

Force from a potential function.

In one dimension, differentiating the defining integral gives

F_{x} = - \frac{d U}{d x} .

In three dimensions, the corresponding relation is

F_{c} = - \nabla U = - (\frac{\partial U}{\partial x} \overset{x}{^} + \frac{\partial U}{\partial y} \overset{y}{^} + \frac{\partial U}{\partial z} \overset{z}{^}) .

The force points in the direction of greatest decrease of potential energy. The slope, rather than the height of the graph, determines force. A high flat region has $d U / d x = 0$ and therefore no $x$ -directed force. A steep descending section has positive $F_{x}$ because $d U / d x < 0$ .

Equilibrium positions satisfy $F_{x} = 0$ , hence $d U / d x = 0$ . The local shape classifies one-dimensional equilibrium:

Stable equilibrium: $U$ has a local minimum. A small displacement produces a restoring force toward the equilibrium point.
Unstable equilibrium: $U$ has a local maximum. A small displacement produces a force away from the equilibrium point.
Neutral equilibrium: $U$ is locally constant. Small translations do not change the potential energy or produce a restoring force.

The second derivative gives the local test when it is nonzero:

\frac{d ^{2} U}{d x ^{2}}_{x_{0}} > 0 stable, \frac{d ^{2} U}{d x ^{2}}_{x_{0}} < 0 unstable .

The spring potential $U = \frac{1}{2} k x^{2}$ gives $d^{2} U / d x^{2} = k > 0$ , so $x = 0$ is stable.

Conservation of mechanical energy.

The work-kinetic-energy theorem states $Δ K = W_{net}$ . Separate the net work into conservative and nonconservative contributions:

Δ K = W_{c} + W_{nc} = - Δ U + W_{nc} .

Rearrangement gives the central accounting equation,

Δ (K + U) = W_{nc} .

When $W_{nc} = 0$ ,

E_{mech} = K + U = constant .

Mechanical energy is therefore conserved when only conservative forces act. The conserved quantity is the sum $K + U$ ; neither piece is conserved alone. A falling object trades potential energy for kinetic energy, and a projectile rising in vacuum trades kinetic for potential, each exchange exact.

With no nonconservative work, energy moves between kinetic and gravitational potential forms. The sum remains the same at the high and low points of the track.

An energy equation requires clear initial and final states. For an object moving under gravity and an ideal spring,

\frac{1}{2} m v_{i}^{2} + m g y_{i} + \frac{1}{2} k x_{i}^{2} = \frac{1}{2} m v_{f}^{2} + m g y_{f} + \frac{1}{2} k x_{f}^{2} .

Terms absent from a configuration are set to zero only after the reference choices are stated. For example, a spring term is absent only if no spring is deformed or if the initial and final spring extensions are both zero; a gravitational term need not be included when both configurations have equal height.

Energy accounting and reference levels

Friction and drag usually make mechanical energy decrease. The equation

Δ E_{mech} = W_{nc}

is still exact. For kinetic friction of constant magnitude $f_{k}$ over distance $d$ ,

W_{f} = - f_{k} d .

A system containing the moving object and surface records the mechanical decrease as an associated thermal-energy increase, approximately

Δ E_{th} = f_{k} d

when other transfers are negligible. A complete energy balance includes thermal energy, chemical energy, internal energy, and energy transferred across the system boundary by work or heating. Mechanical energy records the kinetic and selected potential terms within that larger account.

Friction transfers energy from organized mechanical motion to thermal energy in the block-surface system. Mechanical energy decreases, while total energy remains conserved for an isolated enlarged system.

External work changes the energy of a selected system. A person pushing a crate is external to a crate-only system, so the applied-force work is an external transfer. In a person-crate system, the force is internal and chemical energy decreases as the crate's kinetic energy and thermal energy change. Both system choices are valid; mixing their energy terms in one equation is not.

Power provides a rate form of this bookkeeping. If a motor raises a load at constant speed $v$ vertically, the mechanical output is $P = m g v$ . If the motor is not ideal, its input power exceeds that output. Energy conservation remains exact after the thermal and other internal transfers of the motor are included.

Reading a potential-energy diagram.

A graph of $U (x)$ contains both energetic and dynamical information. For a specified total mechanical energy $E$ , classical motion is possible only where

K = E - U (x) \geq 0.

The intersections $E = U (x)$ are turning points: $K = 0$ there, so the speed vanishes before the motion reverses. A horizontal line below a potential barrier confines the particle to one allowed region. A line above the barrier allows travel between the regions. The graph describes one coordinate; it does not display time directly, and equal horizontal distances do not imply equal travel times.

Near a stable equilibrium $x_{0}$ , a smooth potential can be expanded as

U (x) \approx U (x_{0}) + \frac{1}{2} \frac{d ^{2} U}{d x ^{2}}_{x_{0}} (x - x_{0})^{2} .

The missing linear term follows from $d U / d x = 0$ at equilibrium. Defining $k_{eff} = d^{2} U / d x^{2} ∣_{x_{0}} > 0$ gives a local restoring force

F_{x} \approx - k_{eff} (x - x_{0}) .

Small oscillations near any smooth stable minimum therefore have the same local form as a spring. The approximation fails when the displacement is large enough for higher derivatives of $U$ to matter.

Energy accounting protocol.

Build an energy equation in this order.

State the system and its initial and final configurations.
List the energy forms that change within that system.
Identify energy transferred across the boundary by external work or heating.
Write one equation with the signs fixed by those definitions.

A closed system obeys the broad form

Δ K + Δ U_{g} + Δ U_{s} + Δ E_{th} + Δ E_{int} = 0.

A system with external work $W_{ext}$ and heat transfer $Q$ defined positive into it has right side $W_{ext} + Q$ . Include thermal energy created by friction when total-energy conservation is claimed. Mechanical-energy conservation is the restricted case with zero nonconservative work.

Raising the potential reference by $h_{0}$ adds $m g h_{0}$ to every gravitational potential, and both sides of an energy equation gain the same constant, so force and prediction are untouched. Changing the system boundary is different: it moves terms between internal energy and external work, and must be applied the same way throughout one calculation. Evaluate every term between the same two states, so a valid potential difference is never paired with a friction distance or external-work interval taken from a different part of the record.

The ledger separates endpoint quantities from path-integrated transfers.

Quantity	Evaluation	State or path requirement
Kinetic energy	$K = \frac{1}{2} m v^{2}$	speed at each named state
Conservative potential	$Δ U = U_{f} - U_{i}$	same initial and final configuration
Nonconservative work	$W_{nc} = \int F_{nc} \cdot d r$	stated path interval
Thermal change	$Δ E_{th}$	included system and measurement window

Conservative work as an endpoint calculation.

For a conservative interaction, the work between two configurations is fixed by the potential-energy difference alone, with no need to reconstruct the route: every allowed route between the same endpoints gives the same conservative work. A direct lift and a long smooth ramp to the same height give equal gravitational work, even though their applied-force work and travel time differ.

Signs follow from the same relation. When potential energy rises from state i to state f, conservative work is negative: a particle climbing in gravity or stretching a spring feels negative conservative work, while an external agent may do positive work over the same displacement. The work-energy theorem sums all force works, so keeping the conservative and external terms separate avoids counting one transfer twice.

Reference-level changes leave physics unchanged.

The zero of a potential-energy function is a reference choice, not a place where the force vanishes. Replacing $U (y) = m g y$ by $U^{'} (y) = m g (y - y_{0})$ shifts every value by the constant $- m g y_{0}$ ; differences are untouched, so conservative work, force, turning points, and every speed prediction stay the same. The shift must reach every state in one calculation. Assigning $U = 0$ at the floor for the initial state and $U = 0$ at a table for the final state, with no account of the offset, invents an energy difference that is not there. The same holds for spring energy: any reference works as long as the constant offset is carried consistently.

Potential diagrams make the shift visible. Adding a constant slides the curve and any total-energy line up together, and the gap $E - U$ , the kinetic energy, is unchanged only when both move. Sliding the curve while holding a numerical energy line fixed describes a different state, not a new convention.

Energy diagrams, forces, and equilibrium

An energy diagram combines a potential curve $U (x)$ with a chosen total mechanical energy $E$ . The vertical gap $E - U (x)$ is kinetic energy. Classical motion is allowed where that gap is nonnegative and forbidden where the potential lies above the energy line. A turning point occurs where the gap closes: $E = U (x)$ and the speed is zero at that instant. The force need not be zero there; a nonzero slope can reverse the motion immediately after the stop.

Allowed intervals are read horizontally from the intersections of the energy line and potential curve. A particle in a well with energy below a neighboring barrier is confined to that well in the one-dimensional classical model. Raising the energy line past the barrier opens a connecting allowed interval. This is a statement about available kinetic energy, not a statement that the particle moves at constant speed: speed is greatest where $U$ is smallest within the allowed region.

The diagram requires one consistent reference. The numerical height of $E$ has no meaning by itself; only its separation from $U (x)$ matters. At two positions with the same potential value, the particle has the same kinetic energy and speed magnitude under mechanical-energy conservation, though its velocity directions may be opposite on the two sides of a turning point.

A horizontal total-energy line intersects a one-dimensional potential at turning points. Motion is allowed where the line lies above the curve, with speed set by the vertical gap; the barrier remains inaccessible until the energy level is raised above its peak.

The energy level and the potential slope answer different questions at the same position. Reading both prevents a turning point from being treated as an equilibrium.

Feature of $U (x)$	Energy statement	Force statement
$E = U$	$K = 0$ at a turning point	use $F_{x} = - d U / d x$
$d U / d x = 0$	no conclusion about $K$ by itself	equilibrium candidate
$d^{2} U / d x^{2} > 0$	local potential minimum	stable small-displacement response

Energy-diagram audit.

Read the diagram in order: fix the coordinate and reference, draw the total-energy line, mark its intersections with $U (x)$ as candidate turning points, then find the intervals where $E - U \geq 0$ . Compute a speed from the vertical gap only after those steps. Skipping them invites the standard error of reading a potential minimum as a turning point when the energy line sits well above it. Near a shallow intersection, the turning coordinate is only as sharp as the measured curve, so quoting it to more digits than the plot supports overstates the data.

Force from the slope of a potential.

The graph height is potential energy; its slope is force, $F_{x} = - d U / d x$ . The two readings are independent: a high flat plateau carries large potential energy and zero force, while a low but steep region exerts a large force. The negative sign pushes a particle downhill on the graph, toward lower potential energy. A slope read from two widely separated points can hide a nearby change in curvature, so close brackets or a local fit represent the derivative better.

Equipotential paths in two dimensions.

In two or three dimensions, an equipotential curve or surface consists of locations with the same potential energy. Moving exactly along an equipotential gives $Δ U = 0$ , so a conservative force does zero net work over that displacement. The force is perpendicular to an equipotential because $F_{c} = - \nabla U$ points in the direction of most rapid decrease of the potential. This is the multidimensional version of the one-dimensional slope rule.

Equipotential geometry separates path length from work. A long path winding around a contour can have zero conservative work if it stays at the same potential, whereas a short displacement crossing closely spaced contours can involve a large potential change and large work magnitude. The density of contour lines indicates the magnitude of the potential gradient on a map with a stated contour interval. Closely spaced contours correspond to a larger force magnitude than widely spaced contours for the same potential spacing.

The force direction is normal to the contour and points toward lower labelled potential. A particle may still have velocity along a contour, but the conservative force then has no instantaneous power because it is perpendicular to the velocity. If another force pushes the particle across contours, the conservative force does work whose sign follows the direction of the potential change.

Equipotential contours of a two-dimensional conservative-force field. The force is perpendicular to the contour and points toward lower potential; displacement along one contour has zero potential change and hence zero conservative work, regardless of the distance travelled along it.

Contour-work interpretation.

An object is displaced $4.0 m$ along a gravitational equipotential at a fixed height. Gravity does zero work despite the nonzero distance. A subsequent $1.5 m$ vertical descent changes potential by $- m g (1.5 m)$ , so gravity does positive work $+ m g (1.5 m)$ . The two segments differ because one follows a contour and the other crosses contours.

Conservative and dissipative energy accounting

An energy balance becomes reliable when conservative potential changes and nonconservative work are kept in separate columns. For a particle or stated system, the general mechanical relation is

K_{f} + U_{f} = K_{i} + U_{i} + W_{nc} .

Here $W_{nc}$ includes work by forces not represented by the selected potential, such as kinetic friction, drag, or an external drive. A negative nonconservative work reduces mechanical energy; a positive external push increases it. This equation does not assert that energy disappears when friction acts. It accounts for the mechanical subset while leaving thermal or other internal energy changes to the larger-system accounting.

An energy equation should not absorb friction into a potential simply because the friction magnitude is constant. Its work depends on the distance travelled and reverses sign with path direction, so a return path gives additional negative friction work rather than a recovery of the original mechanical energy. This path dependence sets friction apart from conservative gravity or an ideal spring.

Accounting decision procedure.

Sort the forces first: near-surface gravity and an ideal spring supply potential differences, while kinetic friction, drag, and a prescribed drive stay as work terms unless the system is enlarged to absorb their internal-energy changes. Fix the boundary before substituting, and let each transfer appear once. If gravity enters as $Δ U_{g}$ , it does not also enter as gravitational work; if friction enters as $W_{f}$ , there is no second “friction potential.” External work carries its own sign from the force and its point's displacement, positive for a lifting motor and negative for a brake on the moving object.

Quantitative potential-reference shifts.

Evaluate the same two configurations under two zeros to watch the cancellation directly. Floor-zero and table-zero potentials differ by a constant at every height, and the numerical total energy shifts by that same constant when quoted in the new convention, so differences, conservative work, and speeds are unchanged.

Force-potential consistency checks.

A proposed potential function and force law must agree through the negative-gradient relation. In one dimension, differentiating the potential gives the predicted force; integrating the negative force gives potential differences up to an arbitrary constant. These two operations are inverse checks. If a stated force is not the negative slope of the stated potential, the pair cannot describe the same conservative interaction over the interval.

A potential curve and its force graph must be related by negative slope. The parabolic potential has a linear restoring force crossing zero at the same equilibrium point; the opposite signs on either side of the minimum give the force direction toward lower potential.

Full energy solution with a dissipative segment.

A motion can contain conservative and dissipative stages within one energy method. Locate the interval where each nonconservative force acts and calculate its work only over that portion of the path. For a block launched by a spring, climbing smoothly, and then crossing a rough section, elastic and gravitational potential changes are conservative terms. Kinetic friction on the rough segment is a negative work term. The final kinetic energy follows from one complete account or from sequential stage balances that use the same system and reference choices.

The route must be physically feasible at every stage. If the available mechanical energy is exhausted on the smooth rise, the block never reaches the rough segment and its friction work is zero. If it reaches the rough segment but stops before its end, integrate friction only through the actual stopping distance. A negative final kinetic energy is not an energy prediction; it is evidence that the assumed final configuration is unreachable under the stated model.

A spring launch, smooth climb, and rough final segment require one energy account with distinct terms. Spring and gravitational potential changes are endpoint contributions; kinetic friction removes mechanical energy only on the labelled rough interval and determines whether the stated final point is reachable.

Consistency checks across the three descriptions.

Potential, force, and energy diagrams describe one model and must agree at every marked point: a stable equilibrium is a local minimum with zero force that confines a low enough energy line between turning points, and an unstable equilibrium is a local maximum with zero force and a barrier in place of a well. Two numerical checks back this up. Recomputing a potential difference under two zeros must return the same conservative work; a speed that shifts when only the reference zero moves means a constant was applied to one state and not the other. A friction stopping distance must land inside the rough interval it was derived for, or the final state moves to that boundary and the leftover kinetic energy is reported.

Effective potentials and multistage motion

When several conservative interactions act along one coordinate, their potential functions can be added to form an effective potential. The resulting curve contains the combined force information:

F_{x} = - \frac{d U _{eff}}{d x} .

For example, a vertical spring with a hanging mass has an elastic contribution and a gravitational contribution. A constant external conservative force can likewise add a linear term. The equilibrium position is found from the zero slope of the sum, not by separately setting each individual force to zero. Individual forces can be large at equilibrium while their vector sum is zero.

A local minimum of the effective potential is stable because a small displacement raises the potential and the negative slope produces a restoring force. A local maximum is unstable because a small displacement lowers the potential on either side and the force drives the object farther away. A flat region has zero force over the displayed interval and is neutral only within the model's resolution. Stability is therefore a local curvature statement, not a claim that the potential value is small or negative.

The total-energy line gives a further reading. A particle with energy slightly above a stable minimum is confined between nearby turning points and oscillates in the well. Raising the energy above an adjacent maximum opens an escape route. The barrier height is measured relative to the well minimum, so a vertical reference shift changes neither stability nor the threshold energy difference.

An effective potential combines conservative contributions into one curve. The left minimum is stable and confines low-energy motion between turning points, while the central maximum is unstable and becomes passable only when the total-energy line rises above the barrier.

Combined-potential equilibrium.

A vertical spring with upward coordinate $x$ has $U_{eff} = \frac{1}{2} k x^{2} + m g x$ . The equilibrium condition is $k x + m g = 0$ , so $x_{0} = - m g / k$ . The spring is stretched downward by this amount. Since $d^{2} U_{eff} / d x^{2} = k > 0$ , the equilibrium is stable. Adding a constant to this potential changes neither $x_{0}$ nor the restoring-force behaviour.

Conservation through a multi-stage motion.

Split a motion at each point where the force model or configuration changes: a spring acts during launch and then releases, a block runs smooth then rough, a motor cuts off at a marked position. Each stage inherits the exit state of the one before, and a force contributes only on the stage where it acts, with zero work elsewhere. That keeps a spring force from being carried past release and rough-surface friction from being applied to a smooth section. A single equation covers the whole motion only if every potential and work term is scoped to its interval.

A conservative stage passes mechanical energy through unchanged; a friction stage subtracts the friction work over its own path length. Because kinetic energy vanishes at a turning point, the current potential plus the accumulated nonconservative work fixes whether the particle reverses before reaching a later stage. A route can be energetically closed even when each local equation is correct, if the mechanical energy runs out before the next region begins.

Path-dependent work versus potential difference.

Potential difference is an endpoint quantity for a conservative interaction. Work by kinetic friction is a path quantity because it depends on the distance travelled along the contact. Two routes with the same initial and final heights therefore have the same gravitational potential change but can have different friction work if their lengths differ. The resulting final kinetic energies need not match, even though the conservative contribution is identical.

Comparing the two paths determines whether a potential-energy shortcut is valid. If the force can be written as the negative gradient of a single-valued potential over the region, use its endpoint difference. If its direction or magnitude depends on the route, speed, or history, retain it as a work term. Kinetic friction, drag, and applied forces with prescribed operating intervals belong in the second category. A constant force can be conservative, but a force of constant magnitude that follows the direction opposite motion is not.

A closed route provides a simple test. Conservative work sums to zero after return to the starting configuration. Friction work remains negative for each sliding leg, so it continues to reduce mechanical energy around the loop. This difference is not an optional convention; it determines whether one energy value can describe all routes to the same endpoint.

Two paths reach the same final height and therefore have the same gravitational potential change. The longer rough path has a larger negative friction-work contribution, so its final kinetic energy is smaller even though both routes share identical conservative endpoint data.

Path-dependence audit.

Before trading a work term for a potential difference, compare two routes between the same endpoints. If the contribution changes with the route's length, shape, or direction, it stays a path-work term, while the conservative part is still captured exactly by its endpoint difference. Over a closed cycle the split is stark: gravity and an ideal spring return what they took, but kinetic friction drains energy on every leg, so a particle can come home to the same potential with less kinetic energy than it started with.

Active-interval scoping.

A work term is bound to where its force acts: friction only while the surfaces slide, a spring only while it is deformed, an external drive or brake only over its stretch of the path. Noting each interval beside the energy equation keeps a correct potential curve from being paired with a dissipation or drive model that does not apply there.