Editing Openai/68de7724-adbc-8001-9a00-362fe2600e44 (section)

==== 4️⃣ Step 4: Connection with circular motion (basic geometry) ====

A planet orbiting a star moves in a circle (approximately).
An object moving in a circle of radius rrr at speed vvv needs a “centripetal” acceleration:

a=v2ra = \frac{v^2}{r}a=rv2
Newton said: “The gravitational force provides exactly this centripetal force.”

So:

Gm1m2r2=m2v2r\frac{G m_1 m_2}{r^2} = m_2 \frac{v^2}{r}r2Gm1m2=m2rv2
Cancel m2m_2m2 (the orbiting body’s mass):

Gm1r2=v2r\frac{G m_1}{r^2} = \frac{v^2}{r}r2Gm1=rv2
Multiply by rrr:

Gm1r=v2\frac{G m_1}{r} = v^2rGm1=v2
This links speed of orbit vvv to mass of central object m1m_1m1 and radius rrr.