Consider a planet of mass (m), revolving round the sun. The time period (T) of revolution of the planet depends upon the radius of the orbit (r), mass of the sun (M) and the gravitational constant (G). Using dimensional analysis, verify Kepler' third law of planetary motion.

Answer

`T^(2)=(kr^(3))/(GM)`

Explanation

`Tmu(r)^(a)(m)^(b)(G)^(c)`
&`[G]=([F]L^(2))/(M^(2))=M^(-1)L^(3)T^(-2)`
`rArr[T]=k(L)^(a)(M)^(b)(M^(-1)L^(3)T^(-2))^(c) [k rarr "Proporsionality constant"]`
`rArr a+3c=0, -2c=1rArrc=-(1)/(2)`
`b-c=0`
`a=(3)/(2),b=-(1)/(2)`
`rArr T=(kr^(3/2))/(M^(1/2)(G)^(1/2)) rArr T^(2)=(K^(2)r^(3))/(mG)`