A planet moves around the sun in nearly circular orbit. Its period of revolution 'T' depends upon. (i) radius 'r' or orbit, (ii) mass 'm' of the sun and. (iii) The gravitational constant G show dimensionally that T^(2) prop r^(3).

A planet moves around the sun in nearly circular orbit. Its period of

1.	A planet moves around the sun in nearly circular orbit. Its period of revolution 'T' depends upon. (i) radius 'r' or orbit, (ii) mass 'm' of the sun and. (iii) The gravitational constant G show dimensionally that T^(2) prop r^(3).
Answer» Solution :Let `T=Kr^(a)M^(b)G^(c )"…(1)"` Where `K=a` dimensionless constant dimensions of the various quantities are [T] `=T,[r]=L, [M]=M` `[G]=(FR^(2))/(m_(1)m_(2))=(MLT^(-1)L^(2))/(MM)` `=M^(-1)L^(3)T^(-2)` Substituting these dimensionally in equation (1) we get, `[T]=[L]^(a)[M]^(b)[M^(-1)L^(3)T^(-2)]^(c )` `M^(0)L^(0)T^(1)=M^(b-c)=M^(b-c)=M^(b-a)L^(a+3c)T^(-2c)` Equating the dimensions of M, L and T, we get `b-c=0, a+3c=0,-2c=1` on solving, `a=(3)/(2), b=-(1)/(2), c=-(1)/(2)` `THEREFORE T=Kr^(3//2)M^(-1//2)G^(-1//2)` or `T^(2)=(K^(2)R^(3))/(MG)rArrtherefore T^(2) prop r^(3)`