If the constant of gravitation (G), Planck's constant (h) and the velocity of light (c ) be chosen as fundamental units. What is the dimension of the raidus of gyration?

Explanation

Let radius of gyration `[K] prop [h]^(r )[c]^(y)[G]^(z)` by substituting the dimension of
`[K]=[L],[h]=[ML^(2)T^(-1)],[c]=[LT^(-1)]`,
`[G]=[M^(-1)L^(3)T^(-2)]`
`[L]=[ML^(2)T^(-1)]^(x)[LT^(-1)]^(y)[M^(-1)L^(3)T^(-2)]^(z)" ...(1)"`
`therefore M^(0)L^(1)T^(0)=M^(x-z)L^(2x+y+3z)T^(-x-y-2z)`
By comparing the powers of M, L, and T on both sides
`x-z=0" ...(2)"`
`2x+y+3z=1" ...(3)"`
`-x-y-2z=0" ...(4)"`
`x=z rArr (3) and (4)` we get
`{:(2z+y+3z=1),(-z-y-2z=0),(bar(" "z+z=1)):}`
`2z=1`
`z=(1)/(2)`
`x=(1)/(2), z=(1)/(2)`
Substitute in (3)
`2xx(1)/(2)+y+3xx(1)/(2)=1`
`y=(-3)/(2)`
we get, `x=(1)/(2), y=(-3)/(2), z=(1)/(2)`
So dimensions of radius of gyration are `[G^((1)/(2))C^((-5)/(2))G^((1)/(2))]`.