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MechanicsIntroStarter track

Concept module

Gravitational Potential and Potential Energy

See one source mass create a negative potential well, compare how potential and potential energy change with distance, and connect the downhill slope of phi to the gravitational field on the same live model.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Drag the probe through the bounded gravity well. The potential map, contour circles, field arrow, force arrow, scan graphs, and worked examples all read the same one-source-mass model.

Graphs

Switch graph views without breaking the live stage and time link.

Potential and energy on the scan line

Shows how gravitational potential and the chosen probe's potential energy change along the current horizontal scan line.

probe x-position (m): -3.2 to 3.2potential / potential energy: -2 to 2

phiU on m_test

Hover or scrub to link the graph back to the stage.probe x-position (m) / potential / potential energy

Controls

Adjust the physical parameters and watch the motion respond.

Source mass

M

2 kg

Changes the source mass creating the well, so both phi and |g| scale from the same cause.

Probe x-position

x_{p}

1.6 m

Moves the probe left or right across both the stage and the linked scan graphs.

Probe y-position

y_{p}

1.2 m

Moves the probe to a different horizontal scan line so the graphs sample a different slice through the well.

Probe mass

m_{t e x t t es t}

1 kg

Changes the potential energy and probe force without changing the source-set potential or field.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ComparePrompt 1 of 5

Compare mode is most revealing when one setup doubles the distance and the other doubles the probe mass. One changes phi and g; the other changes only U and F.

Try this

Keep Setup A on Axis 1 m, make Setup B either Axis 2 m or Heavier probe, and compare the stage readouts with the two graphs.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

M

Source mass

2 kg

Changes the source creating the well, so the potential, field, force, and energy curves all scale from the same mass.

Graph: Potential and energy on the scan lineGraph: Field-slope link on the scan lineOverlay: Potential mapOverlay: Potential contoursOverlay: Field arrow

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the current prompt as a compact investigation cue. Each one points at one honest feature of the same source mass, probe location, and scan line already on screen.

ComparePrompt 1 of 5

Graph: Potential and energy on the scan line

Compare mode is most revealing when one setup doubles the distance and the other doubles the probe mass. One changes phi and g; the other changes only U and F.

Try this

Keep Setup A on Axis 1 m, make Setup B either Axis 2 m or Heavier probe, and compare the stage readouts with the two graphs.

Why it matters

It makes the difference between geometry changes and probe-response changes visible without leaving the same compact model.

Control: Probe x-positionControl: Probe y-positionControl: Probe massGraph: Potential and energy on the scan lineGraph: Field-slope link on the scan lineOverlay: Distance ringsOverlay: Force arrowOverlay: Field arrow

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

6 visible

Overlay focus

Potential map

Shows the gravitational well as a bounded scalar map that grows darker and deeper near the source mass.

What to notice

Farther from the source, the map lightens because phi rises back toward zero.

Why it matters

It makes the sign and depth of potential visible on one shared stage instead of leaving phi as an isolated number.

Control: Source massControl: Probe x-positionControl: Probe y-positionGraph: Potential and energy on the scan lineEquation

M

Equation

x_{p}

Equation

y_{p}

Challenge mode

Tune the same one-mass gravity well into compact targets. The checks read the live potential, energy, and field state instead of a detached answer sheet.

0/2 solved

TargetCore

5 of 8 checks

Double the energy, not the potential

Starting from Baseline diagonal, change only the probe mass so the potential energy doubles in magnitude while the potential and field stay fixed.

Graph-linkedGuided start2 hints

Suggested start

Leave the source mass and probe position alone. Only the probe mass should move.

Matched

Open the Potential and energy on the scan line graph.

Potential and energy on the scan line

Matched

Keep the Force arrow visible.

Matched

Keep the Field arrow visible.

Pending

Keep test mass between 1.95 kg and 2.05 kg.

1 kg

Matched

Keep potential between -1.01 and -0.99.

-1

Pending

Keep potential energy between -2.02 and -1.98.

-1

Matched

Keep field magnitude between 0.49 and 0.51.

0.5

Pending

Keep force magnitude between 0.99 and 1.01.

0.5

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At the probe (1.6 m, 1.2 m), a source mass of 2 kg gives gravitational potential phi = -1 relative to zero at infinity. The probe sits partway up the potential well, so phi is negative but not as deep as it is near the source. The local field magnitude is 0.5 and points down-left, which is the downhill direction on the potential landscape. For m_test = 1 kg, the potential energy is -1 and the force magnitude is 0.5. With a positive probe mass, the potential energy carries the same negative sign as the gravitational potential.

Equation detailsDeeper interpretation, notes, and worked variable context.

Source-to-probe distance

The current probe distance from the source mass sets the depth of the potential well and the field strength.

r = s q r t x_{p}^{2} + y_{p}^{2}

x_{p}

Probe x-position 1.6 m

y_{p}

Probe y-position 1.2 m

Potential from one source mass

The one-mass potential is negative when zero is defined at infinity, and its magnitude falls with distance.

p hi = - G df r a c M r

This bounded lab fixes G = 1 in the displayed units so the readouts emphasize sign, scaling, and geometry.

M

Source mass 2 kg

x_{p}

Probe x-position 1.6 m

y_{p}

Probe y-position 1.2 m

Potential energy of the probe mass

The probe mass scales the potential energy without changing the potential set by the source.

U = m_{t e x t t es t} p hi

M

Source mass 2 kg

m_{t e x t t es t}

Probe mass 1 kg

Field from potential

The gravitational field points downhill on the potential landscape.

v ec g = - nab l a p hi

y_{p}

Probe y-position 1.2 m

Horizontal scan-line link

Along the current horizontal slice, the x-component of the field is the negative slope of the potential graph.

g_{x} = - df r a c d p hi d x

x_{p}

Probe x-position 1.6 m

Inverse-square field trend

Potential weakens like 1/r, but field magnitude weakens faster like 1/r^2.

∣ v ec g ∣ = G df r a c M r^{2}

M

Source mass 2 kg

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 2 of 50 / 5 complete

Gravity and Orbits

Earlier steps still set up Gravitational Potential and Potential Energy.

1. Gravitational Fields2. Gravitational Potential and Potential Energy3. Circular Orbits and Orbital Speed4. Kepler's Third Law and Orbital Periods+1 more steps

Previous step: Gravitational Fields.

Start from track beginning

Short explanation

What the system is doing

Gravitational potential describes how much gravitational potential energy one kilogram of probe mass would have at a point relative to zero at infinity. For one source mass in this bounded lab, the potential is always negative, becomes more negative near the source, and rises back toward zero as the probe moves farther away.

This concept keeps the same one-source geometry as Gravitational Fields so the potential map, contour circles, live readout, scan graphs, worked examples, prediction prompts, and challenge checks all stay tied to one honest model. The field is not a separate rule pasted on top afterward: along the current scan line, the horizontal field component is the downhill slope of the potential graph.

Key ideas

01Gravitational potential phi is a scalar set by the source mass and distance, not by the probe mass.

02With zero defined at infinity, one isolated source mass produces negative potential everywhere in this bounded lab, and the well gets deeper as r gets smaller.

03Potential energy uses U = m_test phi, so changing only the probe mass rescales U and the force without changing phi or the source-set field.

Live potential-well checks

Solve the exact state on screen.

Solve the current probe state directly from the live controls. The same source mass, probe location, and probe mass already on screen drive the map, contour circles, scan graphs, and the substitutions below.

Live valuesFollowing current parameters

For the current source mass and probe position, what gravitational potential exists at the probe relative to zero at infinity?

M

Source mass

2 kg

x_{p}

Probe x-position

1.6 m

y_{p}

Probe y-position

1.2 m

1. Measure the live source-to-probe distance

From the origin to the probe,

r = s q r t x_{p}^{2} + y_{p}^{2} = 2, ma t h r m m

2. Apply the one-mass potential relation

With

G = 1

in the displayed units,

p hi = - M / r = - 2/2

3. Compute the current potential

That gives

p hi = - 1

at the probe.

Gravitational potential

p hi = - 1

Farther from the source, the potential has risen closer to zero, but it stays negative because zero is defined at infinity in this model.

Potential-well checkpoint

Start from Axis 1 m, then move the probe to Axis 2 m while keeping the source mass and probe mass fixed. Before you drag, predict what should happen to phi, U, and |g|.

Prediction prompt

Say the ratios first: which quantities should halve in magnitude, and which should fall to one quarter?

Check your reasoning

At twice the distance, the gravitational potential and the potential energy both drop to about half their original magnitude while staying negative, and the field magnitude drops to about one quarter.

For one source mass,

p hi p r o pt o - 1/ r

and

U = m_{t e x t t es t} p hi

, so doubling

r

halves both potential magnitudes. But

∣ v ec g ∣ p r o pt o 1/ r^{2}

, so the field falls more steeply to one quarter.

Common misconception

A negative gravitational potential means gravity is pointing outward or "backward."

The negative sign comes from the reference choice that potential is zero infinitely far away. Closer to the source, the potential is lower than that reference, so phi is negative.

The field still points inward because gravity follows the downhill direction of the potential landscape. A more negative potential means the probe sits deeper in the same inward gravity well, not that the force flips direction.

Quick test

Variable effect

Question 1 of 5

Answer from the live potential-well logic, not from detached formulas. Each question asks what the map, contours, or scan graphs must mean about gravitational potential and energy.

A probe moves from 1 m to 2 m away from one source mass on the same axis while the masses stay fixed. What must happen to the gravitational potential?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows one source mass fixed at the origin, a movable positive probe mass in a bounded two-dimensional gravity well, and optional overlays for a potential map, equal-potential contour circles, equal-distance rings, the local field arrow, the force arrow on the probe mass, and the horizontal scan line used by the graphs.

Dragging the probe changes the sampled location directly on the stage. The focused probe handle also responds to arrow keys for small position changes, and sliders provide the same controls for source mass, probe position, and probe mass.

Very near the source mass, the display uses a minimum sampling radius so the well depth, arrows, and graphs stay finite and readable. This keeps the visualization bounded while preserving the correct trend that potential becomes more negative and the field grows rapidly near the source.

Graph summary

The potential-energy-scan graph plots gravitational potential and the chosen probe's potential energy along the current horizontal scan line. Hovering the graph previews the same x-location on the stage.

The field-link graph plots the horizontal gravitational field component and the matching negative slope of the potential graph along that same scan line. Changing the probe mass rescales the energy curve, but the field-link graph remains a source-mass and geometry readout.

Connect fields, energy, and orbits

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Orbit bridgeMechanics

Gravitational Potential and Potential Energy

See each variable before you move it.

Potential map

Double the energy, not the potential

Gravity and Orbits

What the system is doing

Solve the exact state on screen.

For the current source mass and probe position, what gravitational potential exists at the probe relative to zero at infinity?

A probe moves from 1 m to 2 m away from one source mass on the same axis while the masses stay fixed. What must happen to the gravitational potential?

Keep moving through the concept map.

Circular Orbits and Orbital Speed

Escape Velocity

Electric Potential