Physics equations/06-Uniform Circular Motion and Gravitation/A:proofs
Three proofs are offered for the acceleration of uniform circular motion. A fourth proof involves Kepler's third law, which was used by Newton to conclude that the force law was an inverse-square law.
- The first uses geometry and for that reason represents a simplified version of how Newton presented the concept to the world.
- The second uses calculus; this proof is actually easier to learn than the geometrical proof.
- The inverse square law can be deduced from Kepler's third law. But the inverse square law has deep consequences that allow for Maxwell's equations to exist. This latter statement is not easy to establish, so for now we simply point out the geometric relationship between gravity and butterguns.
- An advanced discussion allows for curvilinear motion that is neither circular nor uniform; this proof uses methods associated with general relativity.
- 1 Geometrical proof of equation for uniform circular motion
- 2 CALCULUS-based proof of formula for uniform circular motion
- 3 Derivation of Kepler's third law from the inverse square law
- 4 CALCULUS-based (and very advanced) proof of formula for general case (curvilinear motion)
- 5 References
Geometrical proof of equation for uniform circular motion
Using the figure we define the distance traveled by a particle during a brief time interval, , and the (vector) change in velocity:
1 , and
2 (rate times time equals distance).
3 (definition of acceleration).
4 (taking the absolute value of both sides).
5 (by similar triangles). Substituting (2) and (4) yields:
6 , which leads to , and therefore:
See this quiz based on this proof.
CALCULUS-based proof of formula for uniform circular motion
Take the derivative twice to get first then :
Obtain the desired result by taking the magnitude of each vector (i.e. a = (ax2+ay2)1/2).
Derivation of Kepler's third law from the inverse square law
Kepler's third law for circular orbits states that when an object orbits a massive central body, the cube of the distance is proportional to the square of the period. Newton showed that the proportionality constant is proportional to M, the mass of the central body. This allows one to deduce the mass of a planet from the motion of its moons. A generalization to the case where the mass ratio is not large (or small) allows us to deduce the mass of both stars in a binary star system.
- , where m is the mass of the orbiting object, and M>>m is the mass of the central body, and r is the radius (assuming a circular orbit).
- , where m is the mass of the orbiting object, and M>>m is the mass of the central body, and r is the radius (assuming a circular orbit). After some algebra:
A reveral of these steps allows one to derive the inverse square law from Kepler's third law.