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
Dimension of the Gravitational constant
Dimension of Planck's constant
Dimension of the velocity of light
Radius of Gyration will have units of length. Let the dimension of length in new system be
Using first three dimensions we get
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
#3xx1/2+2xx1/2+z=1#
#=>z=-3/2# ......(7)
Dimension becomes
#[L]=[G]^(1/2) [h]^(1/2) [ c]^(-3/2)#