Units And Dimensions Question 12
Question 12 - 01 February - Shift 2
If the velocity of light $c$, universal gravitational constant $G$ and planck’s constant $h$ are chosen as fundamental quantities. The dimensions of mass in the new system is:
(1) $[h^{\frac{1}{2}} c^{-\frac{1}{2}} G^{1}].$
(2) $[h^{1} c^{1} G^{-1}]$
(3) $[h^{-\frac{1}{2}} c^{\frac{1}{2}} G^{\frac{1}{2}}]$
(4) $[h^{\frac{1}{2}} c^{\frac{1}{2}} G^{-\frac{1}{2}}]$
Show Answer
Answer: (4)
Solution:
Formula: Principle of homogeneity of dimensions
Say dimensional formale of mass is $H^{x} C^{y} G^{z}$
$M^{1}=(ML^{2} T^{-1})^{x}(LT^{-1})(M^{-1} L^{3} T^{-2})^{Z}$
$M^{1} L^{0} T^{0}=M^{x-z} L^{2 x+y+3 z} T^{-x-y-2 z}$
on comparing both side
$x-z=1$
$2 x+y+3 z=0$
$-x-y-2 z=0$
On solving above equations we get
$x=\frac{1}{2} \quad y=\frac{1}{2} \quad z=\frac{-1}{2}$
