Dimension of Universal Gravitational Constant
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Theorem
The dimension of the universal gravitational constant $G$ is $M^{-1} L^3 T^{-2}$.
Proof
From Newton's Law of Universal Gravitation:
- $\mathbf F = \dfrac {G m_1 m_2 \mathbf r} {r^3}$
We have that:
- The dimension of force is $M L T^{-2}$
- The dimension of displacement is $L$
- The dimension of mass is $M$.
Let $x$ be the dimension of $G$.
Then we have:
- $M L T^{-2} = x \dfrac {M^2 L}{L^3}$
Hence, after algebra:
- $x = M^{-1} L^3 T^{-2}$
$\blacksquare$