If the velocity of light c, gravitational constant G and Planck's constant h are chosen h as fundamental units, what are the dimensions of mass, length and time in the new system?

c = [L] [T]^(-1) G = [M]^(-1) [L]^(3) [T]^(-2) h = [M]^(1) [L]^(2)[T]^(-1) Thanks!

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Nov 2, 2016

$\left[m\right] = {c}^{\frac{1}{2}} {G}^{- \frac{1}{2}} {h}^{\frac{1}{2}}$

Explanation:

Here, $\left[c\right] = L {T}^{-} 1$
$\left[G\right] = {M}^{-} 1 {L}^{3} {T}^{-} 2$
$\left[h\right] = {M}^{1} {L}^{2} {T}^{- 1}$
Let $m \alpha {c}^{x} {G}^{y} {h}^{z}$

By substituting the dimensions of each quantity in both the sides,

$M = {\left(L {T}^{-} 1\right)}^{x} {\left({M}^{-} 1 {L}^{3} {T}^{-} 2\right)}^{y} {\left(M {L}^{2} {T}^{-} 1\right)}^{z}$
$M = \left[{M}^{y + z} {L}^{x + 3 y + 2 z} {T}^{x - 2 y - z}\right]$
By equating the power of M, L, T in both the sides:

$- y + z = 1 , x + 3 y + 2 z = 0 , - x - 2 y - z = 0$

By soling the above equation ,
We get, $x = \frac{1}{2} , y = - \frac{1}{2} , \mathmr{and} z = \frac{1}{2}$

So, $\left[m\right] = {c}^{\frac{1}{2}} {G}^{- \frac{1}{2}} {h}^{\frac{1}{2}}$

In above Manner, solve for Length And Time.

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