M044 · Potential energy and turning points

One-Dimensional Potential Landscape

Can you predict a particle's motion just from a graph of its potential energy? Most of it. Draw a line at the total energy: the particle can only be where the landscape lies below that line, and it turns around exactly where the two meet. The curvature at the bottom of a valley gives only the frequency of small oscillations. A valley as flat as a fourth-power curve has zero curvature there, so that frequency comes out zero, yet the valley still traps the particle.

Reshape the well by changing its curvature, quartic term and tilt. Drag the release point, or set position and velocity together in phase space, and compare the allowed region, the exact turning points and the real period with the small-oscillation prediction.

Acquired time $0\,s$

Position $0\,m$

Velocity $0\,\mathrm{m\,s^{-1}}$

Conservative force $0\,N$

Potential energy $0\,J$

Kinetic energy $0\,J$

Mechanical energy $0\,J$

Dissipated energy $0\,J$

Current energy balance defect $0\,J$

Full-record maximum balance defect $0\,J$

Half-step displacement difference · first ten seconds $0\,m$

Actual integration step $0\,ms$

Nearest release-side minimum $0\,m$

Local undamped angular frequency $0\,\mathrm{rad\,s^{-1}}$

Allowed regions at release energy $1$

Release-region left boundary $0\,m$

Release-region right boundary $0\,m$

Conservative quadrature period $0\,s$

Last acquired full turn interval $\text{Await turns}$

Period quadrature resolution difference $0\,s$

Measured minus conservative period $0\,s$

Acquisition status $\text{Complete}$

Fine inspection time $0\,\mathrm{s}$

Signed viscous force $0\,\mathrm{N}$

Signed net force along guide $0\,\mathrm{N}$

Acquired numerical reversals $0$

Amber ruler span $0\,\mathrm{m}$

Amber tick interval $0\,\mathrm{m}$

Set the release, then record the motion

Can you predict a particle’s motion from the shape of its energy landscape?

Particle vectorsBlue is the conservative force, violet the viscous force and teal the velocity. The parallel display lanes line up with the particle position, and the force and speed scales stay fixed during a trial.Conservative force probesSmall blue arrows sample the given force field along the horizontal guide. They are virtual probes, drawn at the same force scale as the arrow on the particle.Energy-accessible intervalsGrey uses the release energy and teal the current numerical energy; amber marks the edges. The bands also cover separate wells that are not connected. They are not physical walls, and they do not promise that the particle will cross over.Metric rulerThe position, the equilibrium markers and the ruler share one fixed length scale. Particle size and hardware thickness are schematic.Particle mass

m

1.00\,kg

Quadratic stiffness

k

2.00\,N\,m^{-1}

Quartic confinement

q

1.00\,N\,m^{-3}

Potential tilt

f

0.00\,N

Release position

x_0

1.00\,m

Release velocity

v_0

0.00\,m\,s^{-1}

Observation window

t_{\mathrm{obs}}

24.00\,s

Requested step

h

6.00\,ms

Acquired record

t/t_{\mathrm{obs}}

0.00\,\%

Drag the purple release outline sideways, or drag the outline in phase space in both directions. The orange square at minus one metre sets the quadratic stiffness and leaves the other coefficients unchanged. Every drag has matching keyboard sliders. A purely quartic minimum is stable, but its linear small-oscillation frequency is zero. If both restoring coefficients are zero, acquisition stops and the last valid plot stays on screen.

The model is an ideal point particle moving along a horizontal guide in a given polynomial potential, with all quantities in SI units. The 3D force probes and the bands marking the allowed region are display aids; the Lab does not simulate rolling, a physical barrier or an actuator mechanism. The vertical curve is an energy graph, not a real track. Gray contours are drawn at the release energy, and the recorded motion is teal. Playback recomputes the motion from saved checkpoints. The half-step comparison covers at most the first ten seconds; energy drift is checked at every integration step.

Physics tutorial

The energy graph predicts a whole trajectory

BackgroundA potential-energy graph gives the energy stored at each position. Its slope gives the force, its flat points are equilibria, and a horizontal line at the total energy marks every position a classical particle can reach. The curve is not a track: the particle moves along the horizontal axis, and the vertical axis is energy.

Why it mattersOpenStax University Physics, section 8.4, uses potential-energy diagrams to find stable points and turning points. This Lab goes further: you can reshape the well, follow the motion in phase space, compute the period by integration and check how well the simulation conserves energy.

Start with the essentials

Focus question: Which predictions still hold when the bottom of the well is not shaped like a parabola?
One-sentence intuition: The turning points come from the full energy equation, but the small-oscillation frequency depends only on how sharply the well curves at its bottom.

Core mathematical model

A potential you can reshape

U(x)=\frac{kx^2}{2}+\frac{qx^4}{4}+fx

The quadratic and quartic coefficients are never negative. If at least one of them is positive, the well traps the particle. The tilt term moves the equilibrium to one side.

Force and stability

F(x)=-U\prime(x),\qquad U\prime(x_*)=0,\qquad \omega_0^2=\frac{U\prime\prime(x_*)}{m}

If the curvature at the minimum is positive, small oscillations are simple harmonic. If the curvature is zero, the higher-order terms decide whether the minimum is stable.

Allowed positions and turning points

E=\frac12mv^2+U(x),\qquad U(x)\le E,\qquad U(x_\pm)=E

The speed is zero at a turning point. A particle released at rest at the equilibrium has an allowed region that shrinks to a single point.

Period at finite amplitude

T(E)=\sqrt{2m}\int_{x_-}^{x_+}\frac{dx}{\sqrt{E-U(x)}}

The Lab evaluates this integral separately from the simulation. A cosine substitution removes the simple blow-up at each turning point, and the readout shows how much the result changes at half the resolution.

Energy drift and the small-oscillation reference

\delta E(t)=E(t)-E(0),\qquad x_{\mathrm{lin}}(t)=x_*+(x_0-x_*)\cos\omega_0t+\frac{v_0}{\omega_0}\sin\omega_0t

The reference curve holds only near the equilibrium. When the small-oscillation frequency is zero, the reference is motion at constant velocity; that is the linear approximation, and it does not say the real minimum is neutrally stable.

Common difficulties

Energy is not height

Typical misconceptionThe particle slides downhill along the plotted curve.

Better mental modelThe vertical axis is energy. The particle actually moves back and forth along the horizontal axis.

Force comes from the slope

Typical misconceptionThe force is largest where the potential energy is highest.

Better mental modelThe force depends on the slope at that point, not on the height. In a tilted well the equilibrium can even sit at a nonzero potential energy.

Zero curvature calls for higher-order terms

Typical misconceptionIf the small-oscillation frequency is zero, the minimum must be unstable.

Better mental modelA positive quartic term still traps the particle. It oscillates with a finite period that depends on the amplitude.

Run the experiment

01
Check a harmonic well
Select the Harmonic reference preset and press Acquire full record.
What to observe: The period computed between the turning points matches the textbook harmonic period and the time between repeated turns in the trajectory.
02
Reshape the well
Drag the orange stiffness handle, change Quartic confinement and compare the force with its linear prediction near the equilibrium.
What to observe: The release energy and the turning points change together, and the controls show the actual coefficient values.
03
Flatten the minimum
Select the Flat quartic minimum preset, then make the release displacement smaller.
What to observe: The small-oscillation frequency stays at zero, but the actual period gets longer as the release gets smaller.
04
Move the equilibrium
Select the Tilted landscape preset, then drag the release point in the phase-space plot in both directions.
What to observe: The equilibrium moves. Changing the starting velocity changes the allowed interval, but the potential itself stays the same.