Orbital Motion Formula

Orbital motion is motion of an object held by gravity in a curved path — often a closed path (ellipse or circle) — around a planet, moon, star, or other.

The Formula

GMmr2=mv2r\frac{GMm}{r^2} = \frac{mv^2}{r} so for a circular orbit v=GMrv = \sqrt{\frac{GM}{r}}

When to use: An orbit is like falling around a planet instead of straight down onto it.

Quick Example

A satellite stays in orbit because gravity provides the centripetal force needed to keep curving its path around Earth.

Notation

GG is the gravitational constant, MM is the central mass, mm is the orbiting mass, rr is orbital radius, vv is orbital speed, and TT is orbital period.

What This Formula Means

Motion of an object held by gravity in a curved path — often a closed path (ellipse or circle) — around a planet, moon, star, or other mass.

An orbit is like falling around a planet instead of straight down onto it.

Formal View

For a circular orbit, gravity supplies the centripetal force: GMm/r2=mv2/rGMm/r^2 = mv^2/r. This gives v=GM/rv = \sqrt{GM/r} and T=2πr3/(GM)T = 2\pi\sqrt{r^3/(GM)} for orbital period.

Worked Examples

Example 1

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A satellite orbits Earth (GM=4×1014GM = 4\times10^{14}) at radius r=8×106 mr = 8\times10^6\text{ m}. Find the period.

Answer

T≈7113 sT \approx 7113\text{ s}

First step

1
v=GM/r=5×107≈7071 m/sv = \sqrt{GM/r} = \sqrt{5\times10^7} \approx 7071\text{ m/s}.

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Example 2

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Two satellites orbit Earth at r1=7×106 mr_1 = 7\times10^6\text{ m} and r2=28×106 mr_2 = 28\times10^6\text{ m}. Find the ratio of their periods.

Example 3

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For a 500 kg500\text{ kg} satellite at r=7×106 mr = 7\times10^6\text{ m} around Earth (GM=4×1014GM = 4\times10^{14}), find total mechanical energy.

Common Mistakes

  • Thinking there is no gravity in orbit. - Fix this by naming the system, checking "Have I isolated one system and listed the external forces or torques acting on it before applying a law?", and attaching units or direction to the final statement.
  • Forgetting that lower orbits require higher orbital speed. - Fix this by naming the system, checking "Have I isolated one system and listed the external forces or torques acting on it before applying a law?", and attaching units or direction to the final statement.
  • Using orbital motion from a keyword alone - Signal words like force, push, pull only point to a possible model; the system must match too.
  • Substituting numbers before defining the system - A formula cannot repair a missing object, boundary, direction, medium, or circuit path.

Why This Formula Matters

Orbital Motion is central because forces explain changes in motion and balance. Students who can isolate a system and draw the interactions can avoid treating every force word as the same kind of cause.

Frequently Asked Questions

What is the Orbital Motion formula?

Motion of an object held by gravity in a curved path — often a closed path (ellipse or circle) — around a planet, moon, star, or other mass.

How do you use the Orbital Motion formula?

An orbit is like falling around a planet instead of straight down onto it.

What do the symbols mean in the Orbital Motion formula?

GG is the gravitational constant, MM is the central mass, mm is the orbiting mass, rr is orbital radius, vv is orbital speed, and TT is orbital period.

Why is the Orbital Motion formula important in Physics?

Orbital Motion is central because forces explain changes in motion and balance. Students who can isolate a system and draw the interactions can avoid treating every force word as the same kind of cause.

What do students get wrong about Orbital Motion?

Students often know a formula related to orbital motion but skip the recognition step: Have I isolated one system and listed the external forces or torques acting on it before applying a law? That leads to a correct-looking substitution attached to the wrong physical model.

What should I learn before the Orbital Motion formula?

Before studying the Orbital Motion formula, you should understand: gravity, gravitational field, centripetal force.

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