Question

if the time required by a planet to go round the sun is denoted by T and it's mean distance from the sun is denoted by D then T=D^n. Find n

Answer 1

When a planet goes round the sun the gravitational force and the centripetal force stay equal so as to keep the planet in the orbit. Therefore:-the centripetal force = mv^2/r whewre r= radius and gravitational force GMm/r^2
now these 2 are equal there fore

m(v^2)/D = GMm/(D^2). (M=mass of sun, m=mass of planet)
thus v= square root of (GM/D)
therefore time period(T)= 2*pi*D/square root of (GM/D)
T= 2*pi*(D*square root of D)/square root of (GM)
thus T is proportional to D^n where n=3/2