In engineering and physics, g_c is a unit conversion factor used to convert mass to force or vice versa.^[1] It is defined as

${\displaystyle g_{\text{c}}={\frac {ma}{F}}}$

In unit systems where force is a derived unit, like in SI units, g_c is equal to 1. In unit systems where force is a primary unit, like in imperial and US customary measurement systems, g_c may or may not equal 1 depending on the units used, and value other than 1 may be required to obtain correct results.^[2] For example, in the kinetic energy (KE) formula, if g_c = 1 is used, then KE is expressed in foot-poundals; but if g_c = 32.174 is used, then KE is expressed in foot-pounds.

According to Newton's second law, the force F is proportional to the product of mass m and acceleration a:

${\displaystyle F\propto ma}$

or

${\displaystyle F=Kma}$

If F = 1 lbf, m = 1 lb, and a = 32.174 ft/s², then

${\displaystyle 1~{\text{lbf}}=K\cdot 1~{\text{lb}}\cdot 32.174~{\frac {\text{ft}}{{\text{s}}^{2}}}}$

Leading to

${\displaystyle K={\frac {1~{\text{lbf}}}{1~{\text{lb}}\cdot 32.174~{\frac {\text{ft}}{{\text{s}}^{2}}}}}=0.03108~{\frac {{\text{lbf}}\cdot {\text{s}}^{2}}{{\text{lb}}\cdot {\text{ft}}}}}$

g_c is defined as the reciprocal of the constant K

${\displaystyle g_{\text{c}}={\frac {1}{K}}=32.174~{\frac {{\text{lb}}\cdot {\text{ft}}}{{\text{lbf}}\cdot {\text{s}}^{2}}}}$

or equivalently, as

${\displaystyle g_{\text{c}}={\frac {ma}{F}}}$