If the gravitational constant $G$ , Plank's constant $h$ and speed of light $c$ be chosen as basic units, find the dimension of mass.
Dimensional formulae: $G : M^{- 1} L^{3} T^{- 2}, h M L^{2} T^{- 1}$

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$m \propto G^{a} h^{b} c^{c}$
$M \propto {[M^{- 1} L^{3} T^{- 2}]}^{a} {[M L^{2} T^{- 1}]}^{b} {[L T^{- 1}]}^{c}$
$M^{1} L^{0} T^{0} \propto M^{a + 2 B + c} T^{2 a - B - c}$
Comparing power of $M, L$ and $T$
$- a + B = 1$
$3 a + 2 b + c = 0$
$- 2 a - b - c = 0$
Adding (ii) and (iii) $\Rightarrow a + b = 0$
Adding (i) and (iv) $\Rightarrow b = \frac{1}{2}, a = - \frac{1}{2}, c = \frac{1}{2}$

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Updated on:21/07/2023

Class 11 PHYSICS DIMENSION AND MEASUREMENT

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