Physics · Gravity and Orbits

Circular Orbits and Orbital Speed

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Escape VelocityWhat if the speed is high enough to leave instead of circle?

Key takeaway

A circular orbit is the special case where gravity already matches the turning requirement v^2/r.
At the same radius, a heavier source needs a larger circular speed and gives a shorter period.
At the same source mass, a wider circular orbit moves more slowly by a square-root factor, travels farther, and therefore takes much longer to go around.

Common misconception

A satellite is not floating because gravity vanished. Orbit is the case where gravity is still present and is exactly what keeps the path turning inward.

Gravity is still acting all the time in orbit. It is the inward acceleration that keeps the path curved instead of straight.

Carry orbit ideas forward

Keep this idea moving

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Period lawGravity and Orbits

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.

Unbound launch thresholdGravity and Orbits

Escape Velocity

Launch outward from one bounded gravity source and see how source mass, launch radius, and total specific energy decide whether the object escapes or eventually returns.

Energy viewGravity and Orbits

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.

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Reference and support

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Equation snapshotTurning requirement · Circular-orbit speedShow 2 equations

Use these together: gravity has to match the turning requirement, and the circular-speed formula is what drops out.

Turning requirement
$a_{c} = \frac{v ^{2}}{r}$
Any path with speed $v$ at radius $r$ needs this inward acceleration to keep turning at that radius.
Circular-orbit speed
$v_{c} = \frac{M}{r}$
This is the one speed that makes gravity and the turning requirement match at the chosen radius.

Why it behaves this way

Explanation

A circular orbit is not a place where gravity disappears. It is continuous free fall: gravity keeps pulling inward, and the sideways speed is exactly right so the object keeps missing the central mass at the same radius.

This lab lets you change the source mass, the chosen reference radius, and the launch speed as a fraction of the circular value. Then you compare the live path with the radius, speed, and acceleration-balance graphs. The assumptions stay visible: one central source, circular-orbit targets, displayed G = 1 units, and no atmosphere or other forces. The key check is simple: for a circular orbit, gravity must exactly match the inward acceleration needed to keep turning at that radius.

Key ideas

01For a circular orbit at radius

r

, gravity itself provides the centripetal acceleration:

M / r^{2} = v^{2} / r

in the displayed units where

G = 1

02At one chosen radius, a speed below the circular value makes gravity stronger than the turning requirement, so the path bends inward. A speed above the circular value makes the turning requirement larger than gravity, so the path opens outward.

03At fixed radius, a heavier source needs a larger circular speed. At fixed source mass, a larger circular orbit needs a smaller speed by a square-root factor and a much longer period.

Worked examples

Solve the live orbit

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the stage and graphs as evidence. First find the circular speed for the chosen reference circle. Then compare the live gravity with the turning requirement v^2/r to decide whether the path should stay circular, bend inward, or open outward. Watch the scaling: doubling radius gives 1/sqrt(2) of the circular speed, not half.

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Example 1 of 2

Frozen valuesUsing frozen 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. Match gravity to the turning requirement

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. Calculate the live circular speed, angular speed, and period

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.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows one source mass at the center, one moving satellite launched from the right side of the chosen reference circle, and optional overlays for the dashed reference orbit, the current radius line, the tangent velocity vector, the inward gravity vector, and the trajectory trail.

Changing source mass, reference radius, or speed factor updates the same orbit path, readouts, and graphs together. The main comparison is whether the live path stays on the reference circle or peels inside or outside it because gravity and the turning requirement no longer match.

Graph summary

The radius-history graph shows whether the live orbit stays on the chosen reference radius or drifts inward or outward. The speed-history graph compares the actual speed with the circular speed required at the current radius.

The acceleration-balance graph compares the inward gravity with the turning requirement $v^{2} / r$ on the same live path. A circular orbit is the case where those two curves match.

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Step 3 of 5

Gravity and Orbits

Circular Orbits and Orbital Speed appears later in this track, so it is cleaner to start from the beginning first.

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