g_c (engineering)

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 English units, g_c is not equal to 1, and is required to obtain correct results.[2]

Motivations

According to Newton's law of motion, 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 = 32.2 ft/s², then

${\displaystyle 1~{\text{lbf}}=K\cdot 1~{\text{lb}}\cdot 32.2~{\frac {\text{ft}}{{\text{s}}^{2}}}}$

Leading to

${\displaystyle K={\frac {1~{\text{lbf}}}{1~{\text{lb}}\cdot 32.2~{\frac {\text{ft}}{{\text{s}}^{2}}}}}=0.03106~{\frac {{\text{lbf}}\cdot {\text{s}}^{2}}{{\text{lb}}\cdot {\text{ft}}}}}$

g_c is defined as the reciprocal of the constant K

${\displaystyle g_{\text{c}}={\frac {1}{K}}=32.2~{\frac {{\text{lb}}\cdot {\text{ft}}}{{\text{lbf}}\cdot {\text{s}}^{2}}}}$

or equivalently, as

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

Specific systems of units

International System	English System	British Customary System
gc = 1 (kg·m)/(N·s^2)	gc = 32.2 (lb·ft)/(lbf·s^2)	gc = 1 (slug·ft)/(lbf·s^2)

References

1. Janna, William (2015). Introduction to Fluid Mechanics (5th ed.). CRC Press. p. 5. ISBN 9781482211610.
2. Cengel, Yunus; Boles, Michael (2014). Thermodynamics: An Engineering Approach (8th ed.). McGraw-Hill Education. p. 6. ISBN 9780073398174.