Gravitational metric system

In the gravitational metric system(s) the base unit of force is not normalised to one mass unit (gram or kilogram) times one length unit (metre or centimetre) per time unit squared (second) as in the SI, but it depends on a selected or locally measured gravitational constant g_n. This constant is usually set to an acceleration of 9.80665 m/s² on Earth. Other derived units inherit this factor. Sometimes alternately the base unit of mass is the one carrying the constant.

Where the difference between weight and mass is not important or not understood, gravitational units are still common despite adoption of the SI, an absolute metric system.

Units

Force

In English contexts the unit of force is usually formed by simply appending “force” to the unit of mass, thus gram-force (g_f) or kilogram-force (kg_f), which follows the tradition of pound-force (lb_f). In other, international contexts the special name pond (p) or kilopond (kp) respectively is more frequent.

1 p = 1 g_f 

= 1 g · g_n = 9.80665 g·m/s² = 980.665 g·cm/s² = 980.665 dyn

1 kp = 1 kg_f 

= 1 kg · g_n = 9.80665 kg·m/s² = 980 665 g·cm/s²

Three approaches to mass and force units^[1] v · d · e

Base	force, length, time		weight, length, time		mass, length, time
Designations	gravitational, technical		colloquial, pre-newtonian		absolute, scientific
Type	coherent		incoherent		coherent
Force (F)	$F = m\cdot {a} = w\cdot\tfrac{a}{g}$		$F = m\cdot\tfrac{a}{g_c} = w\cdot\tfrac{a}{g}$		$F = m\cdot {a} = w\cdot\tfrac{a}{g}$
Weight (w)	$w = m\cdot g$		$w = m\cdot\tfrac{g}{g_c} \approx m$		$w = m\cdot g$
System	BG	GM	EE	EM	AE	CGS	MTS	SI
Acceleration (a)	ft/s2	m/s2	ft/s2	m/s2	ft/s2	gal	m/s2	m/s2
Mass (m)	slug	hyl	lbm	kg	lb	g	t	kg
Force (F)	lb	kp	lbF	kp	pdl	dyn	sn	N
Pressure (p)	lb/in2 (PSI)	at	lbF/in2 (PSI)	atm	pdl/in2	Ba	pz	Pa

Mass

The hyl, metric slug (mug), or TME (German: technische Masseneinheit, technical mass unit), is the mass that accelerates at 1 m/s² under a force of 1 kg_f.^[2] The hyl has also been used as the unit of mass in a metre-gram force-second (mgfs) system.^[3]

1 TME 

= 1 kp / 1 m/s² = 1 kp·s²/m = 9.806 65 kg

1 hyl 

= 1 kp·s²/m = 9.806 65 kg or

1 hyl (alternate definition - mgfs) 

= 1 p·s²/m = 9.806 65 g

Pressure

The (only) gravitational unit of pressure is the technical atmosphere (at). It is the gravitational force of one kilogram, i.e. 1 kg_f, exerted on an area of one square centimetre.

1 at 

= 1 kp/cm² = 10 000 × g_n kg/m² = 98 066.5 kg/(m·s²) = 98.066 5 kPa

Energy

There is no dedicated name for the unit of energy, “metre” is simply appended to “kilopond”, but usually the symbol of the kilopond-metre is written without the middle dot.

1 kpm 

= 1 kp·m = g_n kg·m = 9.806 65 kg·m²/s² = 9.806 65 J

Power

In 19th-century France there was as a unit of power, the poncelet, which was defined as the power required to raise a mass of 1 quintal (1 q = 100 kg) at a velocity of 1 m/s. The German or metric horsepower (PS, Pferdestärke) is arbitrarily selected to be three quarters thereof.

1 p_q 

= 1 q_f·m/s = 100 kp·m/s = 100 × g_n kg·m/s = 980.665 kg·m²/s³ = 0.980 665 kW

1 PS 

= ³⁄₄ p_q = 75 kp·m/s = 75 × g_n kg·m/s = 735.498 75 kg·m²/s³ = 0.735 498 75 kW

References

1. ^ Lindeburg, Michael, Civil Engineering Reference Manual for the PE Exam 
2. ^ Measurements, Units of Measurement, Weights and Measures - Numericana
3. ^ metre-kilogram-force-second systems of units