Gc (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 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.

Motivations

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

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

Specific systems of units

International System	English System 1	English System 2
gc = 1 (kg·m)/(N·s2)	gc = 32.174 (lb·ft)/(lbf·s2)	gc = 1 (slug·ft)/(lbf·s2)

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 (9th ed.). McGraw-Hill Education. p. 6. ISBN   9781259822674.