How to find Kepler's law from Newton's laws

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SUMMARY

The discussion focuses on deriving Kepler's law from Newton's laws of motion, specifically the equation T² = (4π²m)/(Mgr). Participants clarify the variables involved: M (mass stack, 0.3 kg), m (rubber bung, 0.0226 kg), T (time for one rotation), r (horizontal radius), and g (gravity, 9.8 N/kg). The relationship between gravitational force and centripetal force is established using F = Mgm/r² and F = mv²/r, leading to the conclusion that T² = (4π²r³)/Mg. A critical distinction is made between g (acceleration due to gravity) and G (gravitational constant).

PREREQUISITES
  • Understanding of Newton's laws of motion
  • Familiarity with gravitational concepts, specifically g and G
  • Basic knowledge of circular motion and centripetal force
  • Ability to manipulate algebraic equations
NEXT STEPS
  • Study the derivation of Kepler's laws from Newtonian mechanics
  • Learn about gravitational force calculations using F = G(m1m2)/r²
  • Explore the implications of circular motion in physics experiments
  • Investigate the differences between g and G in gravitational contexts
USEFUL FOR

Students in physics, educators teaching classical mechanics, and anyone interested in the relationship between gravitational forces and orbital motion.

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We were asked to do an experiment where we had to prove the equation:
T2=(4π2m)/Mgr
Where M is the mass of the mass stack in kg (0.3kg), m is the mass of the rubber bung in kg (0.0226kg), T is the time taken for one rotation in seconds, r is the horizontal radius of rotation in meters, and g is the strength of gravity (9.8Nkg-1).

And I tried to relate this to Newton's laws in my report by using:
F=Mgm/r2
and F=mv2/r

Which gave me:
T2=(4π2r3)/Mg

How do I reach the original formula using Newton's laws?
 
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Make sure you know there is a difference between "g" and "G".
g is the acceleration of gravity (about 9.8 m/s2)
G is the Gravitational Constant (6.67 × 10-11 m3/kg⋅s2)
 

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