Using Newton's law of gravitation, prove Kepler's III^(rd) law of planetary motion for circular orbits.

Using Newton's law of gravitation, prove Kepler's III^(rd) law of plan

1.	Using Newton's law of gravitation, prove Kepler's III^(rd) law of planetary motion for circular orbits.
Answer» Solution :PLANET of mass 'm' moves around the Sun of mass 'M' in a circular orbit of radius 'R' with orbital velocity v. `therefore(mv^2)/(r)=(GMm)/(r^2)` `v^(2)=(GM)/(r)""...(1)` But, Orbital speed `(v)=("circumference")/("period of revolution")=(2pir)/(T)` Squaring, `v^(2)=(4pi^(2)r^(2))/(T^2)""...(2)` Comparing of equations (1) and (2), we get `(GM)/(r)=(4pi^(2)r^(2))/(T^2)` In CROSS MULTIPLICATION, `(GMm)/(r^2)` `T^(2)PROPR^(3)`