The Planck force, FP = c4/G, is the unit force defined in terms of the other Planck units. It is the force needed to give the unit mass the unit acceleration that goes with Planck length and time. Planck force is the basic tension in string theory and in that context is described as approximately 1040 tons. In general relativity this force is the conversion factor between spacetime curvature and the energy density giving rise to it. (Frank Wilczek, Physics Today June 2001, "Scaling Mount Planck") The Planck force constant also plays simpler more elementary roles: for instance, if multiplied by the fourth power of the (dimensionless) circular orbit speed some distance from a central body it tells what the body would weigh at that distance.
FP = 12.1×1043 newtons.
Some readers may want to write out the F=ma equation with the Planck mass as mass and unit acceleration c/tP in order to verify that the Planck force is indeed what was said, namely c4/G. This is easy to do:
tP = Planck time = SQRT (h-bar G/c5).
mP = Planck mass = SQRT (h-bar c/G).
FP = mPc /tP = (h-bar c/G)1/2c(h-bar G/c5)-1/2 = c4/G.
The Planck force is, along with the speed of light, the Planck quantity existing most plainly and simply in nature. As such, an intuitive development of the Planck units can be based on it. Quantities like the Planck length and mass can be derived from the Planck force rather than the other way around. The algebra just has to be run in reverse. Begining with speed c and force c4/G one defines Planck power c5/G. Then h-bar lets us define Planck frequency ZP as that (angular) frequency whose square connects with Planck power by h-bar:
c5/G = h-bar ZP2
Planck time is then defined as the reciprocal of the frequency:
tP = 1/ZP = (h-barG/c5)1/2
Once Planck time is defined the rest is fairly direct:
lP = ctP = c(h-barG/c5)1/2 = (h-barG/c3)1/2
EP = lPFP = (h-barG/c3)1/2c4/G = (h-bar c/G)1/2c2.
mP = EP/c2 = (h-bar c/G)1/2.
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