Suppose you design your system of dimensions and take velocity(v), Planck's constant (h) and gravitational constant (G) as the fundamental quantities. What will be the dimensions of length(L) in this system.

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$v^{3 / 2} h^{1 / 2} G^{1 / 2}$

$v^{3 / 2} h^{- 1 / 2} G^{1 / 2}$

$v^{- 3 / 2} h^{1 / 2} G^{- 1 / 2}$

$v^{- 3 / 2} h^{1 / 2} G^{1 / 2}$

answer is D.

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Detailed Solution

We Know

$[V] = [L T^{- 1}]; [h] = [M L^{2} T^{- 1}];$

$[G] = [M^{- 1} L^{3} T^{- 2}]$

$Let L = V^{a} h^{b} G^{C}$

$M^{0} L^{1} T^{0} = {(L T^{- 1})}^{a} {(M L^{2} T^{- 1})}^{b} {(M^{- 1} L^{3} T^{- 2})}^{c}$

Comparing the powers of M, L, T both sides

b-c=0 – (1), a+2b+3c=1 – (2) ;

-a-b-2c=0 - (3)

On solving (1), (2) & (3)

We get $a = - 3 / 2, b = 1 / 2, c = 1 / 2$

$L = V^{\frac{- 3}{2}} h^{\frac{1}{2}} G^{\frac{1}{2}}$

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