Key ideas
An object moving in a circle at constant speed is always accelerating toward the center. Gravity supplies that inward force for planets and satellites, which links circular motion to orbits.
- Centripetal acceleration is v² ÷ r and the net inward force is mv² ÷ r. It is not a separate force: it is the name for whatever force points toward the center.
- The speed can stay constant while the velocity changes, because the direction changes.
- Gravity falls off with the square of the distance from the center, so g = GM ÷ r².
- A circular orbit has v = √(GM ÷ r), which does not depend on the satellite’s mass. The escape speed is √2 times larger.
Key equations
a_c = v² ÷ rF_c = mv² ÷ rF = Gm₁m₂ ÷ r²g = GM ÷ r²v = √(GM ÷ r)v_esc = √(2GM ÷ r)T = 2πr ÷ v
Worked example
A satellite circles Earth 400 km above the surface. Take r = 6.37 × 10⁶ + 0.40 × 10⁶ = 6.77 × 10⁶ m and GM = 3.99 × 10¹⁴ m³/s². Then v = √(GM ÷ r) = √(5.89 × 10⁷) = 7.7 × 10³ m/s, and the period is T = 2πr ÷ v = 5.5 × 10³ s, about 92 minutes.
Common mistakes
- Drawing the centripetal force as an extra force on a free-body diagram, besides gravity or tension.
- Using the height above the surface instead of the distance from the center of the planet in the gravity formulas.
- Thinking astronauts float because there is no gravity. They float because they are in free fall around the Earth.
Practice
Test yourself with practice questions on this topic: Circular motion and gravitation