HomeConcept libraryMechanicsGravity and Orbits

MechanicsIntermediateStarter track

Concept module

Circular Orbits and Orbital Speed

See why a circular orbit needs the right sideways speed, how gravity supplies the centripetal acceleration, and how source mass and radius together set orbital speed and period on one bounded live model.

Interactive lab

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

Time

0.00 s / 7.63 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s7.63 s

Circular Orbits and Orbital Speed

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

Compares the live radius with the chosen reference circular-orbit radius over the same time window.

time (s): 0 to 7.63radius (m): 0 to 1.6

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

1.6 m

Sets the dashed reference circle used for the circular-orbit target.

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.

ObservationPrompt 1 of 1

Notice that the trail stays on the dashed reference circle only when the speed factor is very close to 1.00.

Try this

Leave the speed factor at 1.00, then scrub the radius graph. The live radius keeps returning to the same value because gravity and the turning requirement stay matched.

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

Strengthens the inward gravity everywhere, so the circular speed and circular angular speed both rise at the same reference radius.

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 thing to watch changes depending on whether the orbit is close to circular, too slow, too fast, or being compared against another setup.

ObservationPrompt 1 of 1

Graph: Radius over time

Notice that the trail stays on the dashed reference circle only when the speed factor is very close to 1.00.

Try this

Leave the speed factor at 1.00, then scrub the radius graph. The live radius keeps returning to the same value because gravity and the turning requirement stay matched.

Why it matters

This is the stable circular-orbit case the later formulas are describing.

Control: Speed factorGraph: Radius over timeOverlay: Reference orbitOverlay: Trajectory trailEquation

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

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

Only the right speed keeps the live path riding on this dashed circle.

Why it matters

It separates the chosen circular target from the actual path taken when the speed factor is too low or too high.

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

r

Equation

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

Challenge mode

Tune one bounded gravity source into orbit-balance targets. The checks read the same live orbit state, overlays, and graphs instead of a detached answer sheet.

0/3 solved

ConditionStretch

2 of 12 checks

Inner orbit, faster compare

Open compare mode and keep both setups circular with the same source mass, but make Setup B the smaller-radius orbit so it moves faster and finishes sooner.

Compare modeGraph-linkedGuided start2 hints

Suggested start

Keep both setups near speed factor 1. The comparison should be about radius, not about being too slow or too fast.

Pending

Open the Actual speed and circular speed graph.

Radius over time

Matched

Keep the Reference orbit visible.

Matched

Keep the Trajectory trail visible.

Pending

Stay in compare mode while editing Setup B.

Explore

Pending

Keep Setup A source mass between 3.95 kg and 4.05 kg.

Pending

Keep Setup B source mass between 3.95 kg and 4.05 kg.

Pending

Keep Setup A orbit radius between 1.95 m and 2.05 m.

Pending

Keep Setup B orbit radius between 1.05 m and 1.15 m.

Pending

Keep Setup A speed factor between 0.99 and 1.01.

Pending

Keep Setup B speed factor between 0.99 and 1.01.

Pending

Keep Setup B actual speed between 1.85 m/s and 1.97 m/s.

Pending

Keep Setup B period between 3.5 s and 3.75 s.

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

At t = 0 s, the satellite is 1.6 m from the source mass. Its speed is 1.58 m/s, while the local circular speed is 1.58 m/s. Gravity supplies 1.56 m/s² inward and the current turn would need 1.56 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 1.6 m

Turning requirement

Any path with speed v and radius r needs this inward acceleration to keep turning.

a_{c} = \frac{v ^{2}}{r}

r

Reference orbit radius 1.6 m

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

Speed factor 1 x circular

Circular-orbit speed

Matching gravity to the turning requirement gives the speed for a stable circular orbit at the chosen radius.

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

M

Source mass 4 kg

r

Reference orbit radius 1.6 m

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

Speed factor 1 x circular

Circular angular speed

The angular speed follows from $\omega = v/r$ once the circular speed is fixed.

ω_{c} = \frac{M}{r ^{3}}

M

Source mass 4 kg

r

Reference orbit radius 1.6 m

Circular period

At larger radius the period grows strongly, while heavier source mass shortens the period.

T_{c} = 2 π \frac{r ^{3}}{M}

M

Source mass 4 kg

r

Reference orbit radius 1.6 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 3 of 50 / 5 complete

Gravity and Orbits

Earlier steps still set up Circular Orbits and Orbital Speed.

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 Potential and Potential Energy.

Start from track beginning

Short explanation

What the system is doing

A circular orbit is not a place where gravity turns off. It is a special free-fall case where gravity already points inward and the sideways speed is exactly the value that keeps the radius from shrinking or growing.

This bounded lab keeps one source mass, one chosen reference radius, one speed factor, one live path, and the linked radius, speed, and acceleration-balance graphs on the same state. That makes the orbit condition honest: gravity supplies the centripetal acceleration, and the circular speed changes whenever you change source mass or radius.

Key ideas

01For a circular orbit at radius r, gravity itself is the centripetal acceleration: M/r^2 = v^2/r in the displayed units with G = 1.

02At one chosen radius, a speed below the circular value lets gravity bend the path inward, while a speed above the circular value lets the path open outward.

03The circular-orbit speed grows with source mass and falls with larger radius, while the circular period grows strongly with radius and shortens for heavier source mass.

Live orbit checks

Solve the exact state on screen.

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

Live valuesFollowing current parameters

For the current source mass and chosen reference orbit radius, what circular speed keeps the orbit circular?

M

Source mass

4 kg

r

Reference orbit radius

1.6 m

1. Start from the circular-orbit balance

Set gravity equal to the needed centripetal acceleration:

M / r^{2} = v_{c}^{2} / r

2. Solve for the circular speed

That gives

v_{c} = M / r = 4/1.6

in the displayed

G = 1

units.

3. Compute the live circular speed

So the circular-orbit speed is

v_{c} = 1.58 m/s

, with

ω_{c} = 0.99 rad/s

and

T_{c} = 6.36 s

for that same reference circle.

Required circular speed

v_{c} = 1.58 m/s

The circular speed here comes from the shared gravity-and-turning balance rather than from a separate orbit rule.

Orbit-balance checkpoint

Start from the right side of the dashed reference circle. If the speed factor is set below 1, should the next part of the path stay on the circle, drift outside it, or bend inside it?

Prediction prompt

Answer before you press the too-slow preset: is gravity now too weak, too strong, or just right for the turning requirement?

Check your reasoning

It bends inside the dashed circle because the chosen speed is too small for a circular orbit at that radius.

At fixed radius, lowering the speed lowers

v^{2} / r

. Gravity is then stronger than the inward acceleration needed for a circular turn, so the path curves inward.

Common misconception

A satellite in orbit is floating because gravity is basically absent there.

Orbit still needs gravity. The gravitational pull is the inward acceleration that keeps the path curved instead of straight.

What changes is not whether gravity exists, but whether the sideways speed matches the circular-orbit requirement at that radius.

Quick test

Reasoning

Question 1 of 4

Answer from the linked orbit state, not from detached formulas. Each question checks whether you can read circular-orbit balance honestly from the bench.

At one chosen circular-orbit radius, the actual speed is lower than the circular value. What should happen first?

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 of the stage, a moving satellite launched from the right side of a chosen 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, live readouts, 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. Hovering or scrubbing the graph previews the same instant on the orbit stage.

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 squared over r on that same live path.

Carry the orbit bridge forward

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.

Period lawMechanics

Circular Orbits and Orbital Speed

See each variable before you move it.

Reference orbit

Inner orbit, faster compare

Gravity and Orbits

What the system is doing

Solve the exact state on screen.

For the current source mass and chosen reference orbit radius, what circular speed keeps the orbit circular?

At one chosen circular-orbit radius, the actual speed is lower than the circular value. What should happen first?

Keep moving through the concept map.

Kepler's Third Law and Orbital Periods

Escape Velocity

Gravitational Potential and Potential Energy