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MechanicsIntermediateStarter track

Concept module

Kepler's Third Law and Orbital Periods

Compare circular orbits around one source mass and see why larger orbits take longer: the path is longer, the circular speed is lower, and the same live model makes the period law visible without hiding the gravity-speed link.

Interactive lab

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

Time

0.00 s / 6.24 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s6.24 s

Kepler's Third Law and Orbital Periods

Keep the speed factor at 1.00 to match the circular-orbit condition. Move away from 1.00 to see the same gravity law bend the path inward or let it drift outward.

Graphs

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

Radius over time

Use this graph to confirm whether you are still in the circular case being described by the period law.

time (s): 0 to 6.24radius (m): 0 to 1.4

r(t)r_ref

Hover or scrub to link the graph back to the stage.time (s) / radius (m)

Controls

Adjust the physical parameters and watch the motion respond.

Source mass

M

4 kg

Changes the mass creating the inward gravity.

Reference orbit radius

r

1.4 m

Sets the dashed circular orbit whose period and speed are being compared.

Speed factor

\frac{v}{v _{c}}

1 x circular

1.00 means the actual speed equals the circular-orbit speed for the chosen mass and radius.

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.

Graph readingPrompt 1 of 1

With the source mass fixed and the orbit still circular, the period readout changes but the

T^{2} / r^{3}

readout stays fixed.

Try this

Move the radius slider while keeping speed factor at 1.00. The speed curve shifts and the period changes, but the law ratio stays tied to the same source mass.

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

4 kg

A heavier source mass supports a faster circular speed at the same radius, so the circular period gets shorter.

Graph: Actual speed and circular speedGraph: Gravity and turning requirementOverlay: Gravity vectorOverlay: Reference orbit

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 one prompt at a time. The clearest period idea depends on whether you are comparing circular cases, changing source mass, or breaking the circular condition.

Graph readingPrompt 1 of 1

With the source mass fixed and the orbit still circular, the period readout changes but the

T^{2} / r^{3}

readout stays fixed.

Try this

Move the radius slider while keeping speed factor at 1.00. The speed curve shifts and the period changes, but the law ratio stays tied to the same source mass.

Why it matters

It is the direct readout version of Kepler's third law for this one-source model.

Control: Reference orbit radiusControl: Speed factorGraph: Actual speed and circular speedOverlay: Reference orbitEquation

r

Equation

M

Guided overlays

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

5 visible

Overlay focus

Reference orbit

Shows the dashed circle for the chosen circular-orbit target.

What to notice

Kepler's law is about the circular case riding on this dashed orbit, not about paths that peel away from it.

Why it matters

It keeps the chosen circular year separate from the actual path when the speed factor is not matched.

Control: Reference orbit radiusControl: Speed factorGraph: Radius over timeEquation

r

Equation

\frac{v}{v _{c}}

Challenge mode

Build honest circular-year comparisons from the same live gravity model. The checks read the same orbit state, overlays, and graphs instead of a detached answer key.

0/3 solved

TargetCore

5 of 9 checks

Break the circular-year case

Starting from Baseline year, lower the speed enough that the path is no longer the circular orbit Kepler's law is describing.

Graph-linkedGuided start2 hints

Suggested start

Keep the source mass and radius near baseline first, then lower only the speed factor.

Pending

Open the Gravity and turning requirement graph.

Radius over time

Matched

Keep the Reference orbit visible.

Matched

Keep the Gravity vector visible.

Matched

Keep the Trajectory trail visible.

Matched

Keep orbit radius between 1.35 m and 1.45 m.

1.4 m

Pending

Keep speed factor between 0.87 and 0.89.

Pending

Keep actual speed between 1.45 m/s and 1.53 m/s.

1.69 m/s

Matched

Keep gravity acceleration between 2 m/s^2 and 2.08 m/s^2.

2.04 m/s²

Pending

Keep required centripetal acceleration between 1.55 m/s^2 and 1.61 m/s^2.

2.04 m/s²

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

At t = 0 s, the satellite is 1.4 m from the source mass. Its speed is 1.69 m/s, while the local circular speed is 1.69 m/s. Gravity supplies 2.04 m/s² inward and the current turn would need 2.04 m/s². The chosen speed matches the circular-orbit condition closely, so gravity keeps turning the path without pulling it inward or letting it drift outward.

Equation detailsDeeper interpretation, notes, and worked variable context.

Gravity from one source mass

In this bounded lab with displayed units using $G = 1$, the inward gravitational acceleration is set by source mass and radius.

g = \frac{M}{r ^{2}}

M

Source mass 4 kg

r

Reference orbit radius 1.4 m

Circular speed

Matching gravity to the turning requirement fixes the sideways speed for a circular orbit at the chosen radius.

v_{c} = \frac{M}{r}

M

Source mass 4 kg

r

Reference orbit radius 1.4 m

\frac{v}{v _{c}}

Speed factor 1 x circular

Circular period

The circular period grows strongly with radius and shortens for heavier source mass.

T_{c} = 2 π \frac{r ^{3}}{M}

M

Source mass 4 kg

r

Reference orbit radius 1.4 m

Kepler's third-law form for one source mass

At fixed source mass, the period-squared versus radius-cubed ratio stays fixed for circular orbits in this one-source model.

T_{c}^{2} = \frac{4 π ^{2}}{M} r^{3}

M

Source mass 4 kg

r

Reference orbit radius 1.4 m

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 3 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 4 of 50 / 5 complete

Gravity and Orbits

Earlier steps still set up Kepler's Third Law and Orbital Periods.

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: Circular Orbits and Orbital Speed.

Start from track beginning

Short explanation

What the system is doing

Kepler's third law is the timing view of the same circular-orbit balance you already met in the speed story. Around one source mass, larger circular orbits take longer because they have farther to travel and because the allowed circular speed is lower at larger radius.

This bounded lab keeps one source mass, one chosen reference orbit, one speed factor, one live path, and the linked radius, speed, and acceleration-balance graphs tied to the same state. That makes the period law honest: it is not a disconnected astronomy rule, but the time consequence of gravity setting the circular speed.

Key ideas

01For circular orbits around one source mass, the period follows $T_c = 2\pi\sqrt{r^3/M}$ in the displayed units with $G = 1$.

02At fixed source mass, a larger circular orbit has both a longer circumference and a lower circular speed, so the period grows faster than the radius alone.

03Kepler's third law describes the circular-match case. If the speed factor moves away from 1.00, the path stops being the circular orbit whose period the law is talking about.

Live period checks

Solve the exact state on screen.

Use the same source mass, reference orbit radius, and speed factor already on screen. These examples stay tied to the live bench instead of jumping to a detached answer key.

Live valuesFollowing current parameters

For the current source mass and chosen circular orbit radius, what period should one full circular orbit take?

M

Source mass

4 kg

r

Reference orbit radius

1.4 m

1. Start from the circular speed

For the circular case, the same gravity-turning balance gives

v_{c} = M / r = 4/1.4

2. Turn speed into one full-orbit time

A circular period is circumference divided by circular speed, so

T_{c} = 2 π r / v_{c} = 2 π r^{3} / M

3. Compute the live circular period

For this bench,

T_{c} = 5.2 s

, with

ω_{c} = 1.21 rad/s

and

T_{c}^{2} / r^{3} = 9.87 s^{2} / m^{3}

Current circular period

T_{c} = 5.2 s

The period comes from the same live circular condition: one full orbit is the circumference divided by the allowed circular speed.

Kepler checkpoint

Keep the same source mass and the circular match. If the chosen orbit radius doubles, should the period double, grow by less than double, or grow by more than double?

Prediction prompt

Answer before you try the inner and outer circular presets: is the longer year caused only by more distance around, or by both more distance and lower circular speed?

Check your reasoning

It grows by more than double.

A larger circular orbit is not just a bigger circle. The allowed circular speed is also lower at larger radius, so the period scales as

r^{3/2}

rather than directly as

r

Common misconception

A larger orbit takes longer only because the object has farther to go around, while the orbital speed stays about the same.

The path is longer, but that is only half of the story. The circular speed is also lower at larger radius because gravity is weaker there.

That combination is why the period grows strongly with radius: the object travels farther and it does so more slowly.

Quick test

Reasoning

Question 1 of 4

Answer from the linked orbit state, not from detached memorization. The aim is to read why larger circular orbits take longer and when the law does not apply.

Around the same source mass, the circular orbit radius doubles while the path stays circular. What must be true?

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 fixed source mass at the center, a moving satellite launched from the right side of a chosen circular reference radius, and optional overlays for the dashed reference orbit, the live radius line, the tangent velocity vector, the inward gravity vector, and the trajectory trail.

Changing source mass, reference orbit radius, or speed factor updates the same orbit path, circular-speed readout, circular-period readout, law-ratio readout, and linked graphs together. Compare mode overlays a second setup without switching to a separate orbit model.

The displayed units use a bounded one-source gravity model with $G = 1$ . A minimum sample radius keeps the stage and graphs finite and readable while preserving the correct inward and inverse-square trends.

Graph summary

The radius-history graph compares the live radius with the chosen circular reference radius over time so you can tell whether the path is still the circular case.

The speed-history graph compares the live speed with the circular speed required at the current radius, and the acceleration-balance graph compares gravity with the turning requirement $v^{2} / r$ on that same live path.

Keep the orbit-timing story moving

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.

Unbound thresholdMechanics

Kepler's Third Law and Orbital Periods

See each variable before you move it.

Reference orbit

Break the circular-year case

Gravity and Orbits

What the system is doing

Solve the exact state on screen.

For the current source mass and chosen circular orbit radius, what period should one full circular orbit take?

Around the same source mass, the circular orbit radius doubles while the path stays circular. What must be true?

Keep moving through the concept map.

Escape Velocity

Gravitational Potential and Potential Energy

Circular Orbits and Orbital Speed