Three of the fundamental constants of physics are the speed of light, c= 3.0 x 1

Question

Three of the fundamental constants of physics are the speed of light, c= 3.0 x 10^8 m/s, the universal gravitational constant, G=6.7 x 10^-11 m^3*kg^-1*s^-2, and Plancks constant, h=6.6 x 10^-34 kg*m^2*s^-1
(a) Find a combination of these three constants that has the dimensions of time. This time is called the Planck time and represents the age of the universe before which the laws of physics as presently understood cannot be applied.
(b) Using the formula for the Planck time derived in part (a), what is the time in seconds?

Explanation / Answer

If you call the combination of the three constants C^a * G^b * h^c and solve for the powers a,b,c which leaves the units in s: power of meter unit: 1a+3b+2c = 0 power of kg unit: 0a-1b+1c=0 power of seconds unit: -1a-2b-1c=1 0a-1b+1c=0 => b=c 1a+3b+2c=0 with b=c => 1a=-5c -1a-2b-1c=1 with b=c and 1a=-5c => 2c=1 => c=1/2 b=c=1/2 a=-5c = -5/2 So its c^(-5/2) * G^(1/2) * h^(1/2) =134.9 * 10^-45 seconds

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