Question
If the velocity of light (c), the constant of gravitation (G) and Planck's constant (h) be chosen as the fundamental units, find the dimensions of mass, length and time in the new system.
✓
Answer
Let us write the dimensions of c, G and h in terms of M, L and T. $\text{Let}\text{M}=\text{Kc}^a\text{G}^b\text{h}^\text{y},$ where K is constant $\text{[M]}=\text{[LT}^{-1}]^\alpha\text{[M}^{-1}\text{[L}^3\text{T}^{-2}]^\beta\text{[ML}^2\text{T}^{-1}]^\gamma$ $=\text{[M}^{-\beta+\gamma}\text{L}^{\alpha+3\beta+2\gamma}\text{T}^{-\alpha-2\gamma}]$ Comparing the powers of M, L and T on both the sides, we have: $-\beta+\gamma=1$ $\alpha+3\beta+2\gamma=0$ $-\alpha-2\beta-\gamma=0$ Solving these equations, we get, $\alpha=\frac{1}{2},\beta=\frac{-1}{2},\gamma=\frac{1}{2}$ $\therefore \text{M}=\text{Kc}^\frac{1}{2}\text{G}^\frac{1}{2}\text{h}^\frac{1}{2}$ Taking$\text{K}=\text{I},$ we can write $\text{M}=_\text{c}^\frac{1}{2}\text{G}^{\frac{-1}{2}}\text{h}^\frac{1}{2}$ Similary,we can prove that. $\text{L}=_\text{c}^\frac{-3}{2}\text{G}^\frac{1}{2}\text{h}^\frac{1}{2}$ $\text{T}=_\text{c}^\frac{-5}{2}\text{h}^\frac{1}{2}\text{G}^\frac{1}{2}$
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