Dimensions of universal gravitational constan | vedclass

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Question

(B) The dimensional formula for the universal gravitational constant $G$ is $[M^{-1}L^3T^{-2}]$.
The dimensional formula for Planck's constant $h

Vedclass · Accepted Answer

$[hT^{-1}LM^{-2}]$

Vedclass · Answer

(B) The dimensional formula for the universal gravitational constant $G$ is $[M^{-1}L^3T^{-2}]$.
The dimensional formula for Planck's constant $h$ is $[ML^2T^{-1}]$.
We need to express $G$ in terms of $h, L, M, T$. Let $G = h^a L^b M^c T^d$.
Substituting the dimensions: $[M^{-1}L^3T^{-2}] = [ML^2T^{-1}]^a [L]^b [M]^c [T]^d$.
$[M^{-1}L^3T^{-2}] = M^a L^{2a} T^{-a} \cdot L^b \cdot M^c \cdot T^d = M^{a+c} L^{2a+b} T^{-a+d}$.
Comparing the powers of $M, L, T$ on both sides:
$a + c = -1$ $(1)$
$2a + b = 3$ $(2)$
$-a + d = -2$ $(3)$
Testing option $(B)$: $a = 1, b = 1, c = -2, d = -1$.
From $(1)$: $1 + (-2) = -1$ (Correct).
From $(2)$: $2(1) + 1 = 3$ (Correct).
From $(3)$: $-1 + (-1) = -2$ (Correct).
Therefore, $G = h^1 L^1 M^{-2} T^{-1}$, which is $[hT^{-1}LM^{-2}]$.

Dimensions of universal gravitational constant $(G)$ in terms of Planck's constant $(h)$, distance $(L)$, mass $(M)$ and time $(T)$ are . . . . . . .

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