If Gravitational constant #G#; Planck's constant #h# and velocity of light #c# are fundamental quantities; the dimension of radius of gyration will be??

1 Answer
Feb 27, 2018

Dimension of the Gravitational constant #G=[M]^-1[L]^3[T]^-2#
Dimension of Planck's constant #h=[M][L]^2[T]^-1#
Dimension of the velocity of light #c=[L][T]^-1#

Radius of Gyration will have units of length. Let the dimension of length in new system be

#[L]=[G]^x [h]^y [ c]^z#

Using first three dimensions we get

#[L]=([M]^-1[L]^3[T]^-2)^x ([M][L]^2[T]^-1)^y([L][T]^-1)^z#

#=>[M]^0[L]^1[T]^0=[M]^(-x+y) [L]^(3x+2y+z) [T]^(-2x-y-z)#

Comparing power of respective items on both sides we get

#-x+y=0# ......(1)
#3x+2y+z=1#......(2)
#-2x-y-z=0# .....(3)

Adding (2) and (3) we get

#x+y=1# .....(4)

Adding (1) and (4)

#2y=1#
#=>y=1/2# ........(5)

Inserting this value in (4)

#x+1/2=1#
#=>x=1/2# .......(6)

Inserting value of #xandy# in (2)

#3xx1/2+2xx1/2+z=1#
#=>z=-3/2# ......(7)

Dimension becomes

#[L]=[G]^(1/2) [h]^(1/2) [ c]^(-3/2)#