Question

If the velocity of light c, Planck's constant h and gravitational constant G are taken as fundamental quantities; then express mass, length and time in terms of dimensions of these fundamental quantities.

Answer 1

we know, $c=[LT^{-1}]$
$h=[ML^2T^{-1}]$
and $G=[M^{-1}L^3T^{-2}]$

we have to write mass in term of c, h and G

so, [M] = k $c^xh^yG^z$, k is proportionality constant

[M] = [LT^-1]^x[ML²T^-1]^y [M^-1L³T^-2]^z

[M] = [M^(y - z) L^(x + 2y + 3z) T^(-x -y - 2z)]

compare both sides,

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

after solving these equations we get,
x = 1/2 , y = 1/2 and z = -1/2

so, mass = $k\sqrt{\frac{ch}{G}}$

similarly, we have to write length in term of c , h and G

length = k $c^xh^yG^z$

[L] = k[M^(y - z) L^(x + 2y + 3z) T^(-x - y - 2z)]

compare both sides,

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

after solving theses equations ,
we get, x = -3/2 , y = 1/2 , z = 1/2

so, length = $k\sqrt{\frac{hG}{c^3}}$


again, we have to write time in term of c ,h and G.
Time = k$c^xh^yG^z$

[T] =k [M^(y-z) L^(x + 2y + 3z) T^(-x - y - 2z)]

compare both sides,

(y - z) = 0
x + 2y + 3z = 0
-x - y - 2z = 1

after solving these equations,
we get, y = z = 1/2 , x = -5/2

so, Time = $k\sqrt{\frac{hG}{c^5}}$

Answer 2

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