Weight value

The weighing value (previously also weight in air or immersion weight ) of a body (goods to be weighed) is equal to the mass of the weights that keep the balance in equilibrium or provide the same display on the scale as the goods to be weighed. The weight is given in the unit of mass, i.e. in kilograms . When weighing in air , the weight of a body deviates from its mass due to the buoyancy if it has a different density than the weight.

Derivation

Two forces act on a body to be weighed: the downward weight force and - when weighing in air - the upward lift . In the following it should be assumed that a body of unknown mass is compared with weights of known mass with the aid of a beam balance . ${\displaystyle F_{\mathrm {G} }}$ ${\displaystyle F_{\mathrm {A} }}$ ${\displaystyle m}$ ${\displaystyle m_{\mathrm {G} }}$

The weight acting on the body is given by, where the gravitational acceleration denotes. When the body, the volume has in it and the air density is weighed, then its buoyancy corresponding to the weight of the displaced air from it: . If the density of the body is known, the volume in turn results as . Hence the force acting on the scales is: ${\displaystyle F_{\mathrm {G} }=mg}$ ${\displaystyle g}$ ${\displaystyle V}$ ${\displaystyle \rho _{\mathrm {L} }}$ ${\displaystyle F_{\mathrm {A} }=\rho _{\mathrm {L} }Vg}$ ${\displaystyle \rho }$ ${\displaystyle V=m/\rho }$

${\displaystyle F=F_{\mathrm {G} }-F_{\mathrm {A} }=mg-\rho _{\mathrm {L} }Vg=mg\left(1-{\frac {\rho _{\mathrm {L} }}{\rho }}\right)}$

A corresponding consideration also applies to the weights (mass , density ). The balance is in equilibrium when the force on both sides of the balance beam is equal: ${\displaystyle m_{\mathrm {G} }}$ ${\displaystyle \rho _{\mathrm {G} }}$

${\displaystyle m\left(1-{\frac {\rho _{\mathrm {L} }}{\rho }}\right)=m_{\mathrm {G} }\left(1-{\frac {\rho _{\mathrm {L} }}{\rho _{\mathrm {G} }}}\right)}$

According to the definition, the weight of the body is equal to the mass of the weights that keep it in balance. By weighing a value that deviates from the actual mass of the body is determined, which is related to this as follows: ${\displaystyle W}$ ${\displaystyle m_{\mathrm {G} }}$ ${\displaystyle m}$

${\displaystyle W=m\cdot {\frac {1-{\frac {\rho _{\mathrm {L} }}{\rho }}}{1-{\frac {\rho _{\mathrm {L} }}{\rho _{\mathrm {G} }}}}}}$

The mass and weight of a body therefore differ slightly. Therefore sometimes the term is for a better linguistic distinction true mass (en: true mass ) used for mass. If the two densities and are equal, the weight is equal to the mass of the body. ${\displaystyle m}$ ${\displaystyle W}$ ${\displaystyle \rho }$ ${\displaystyle \rho _{\mathrm {G} }}$

Conversely, with precise measurements, this systematic deviation can be mathematically taken into account as a lift correction. For a measured weight value , the mass is obtained from the equation: ${\displaystyle W}$ ${\displaystyle m}$

${\displaystyle m=W\cdot {\frac {1-{\frac {\rho _{\mathrm {L} }}{\rho _{\mathrm {G} }}}}{1-{\frac {\rho _{\mathrm {L} }}{\rho }}}}}$

Conventional weight

The conventional mass is in accordance with a recommendation of the International Organization for Legal Metrology an operand that is associated with a weight piece, when at a temperature of 20 ° C to a reference weight at the density 8000 kg / m ³ in air density 1.2 kg / m ³ keeps the balance.

If you insert these reference densities into the above equation, you get for the conventional weight of a body: ${\displaystyle W_{\mathrm {k} }}$

${\displaystyle W_{\mathrm {k} }=m\cdot {\frac {1-{\frac {1,2\,\mathrm {kg\,m^{-3}} }{\rho }}}{1-{\frac {1,2\,\mathrm {kg\,m^{-3}} }{8000\,\mathrm {kg\,m^{-3}} }}}}=m\cdot {\frac {1-{\frac {1,2\,\mathrm {kg\,m^{-3}} }{\rho }}}{0{,}99985}}}$

When comparing weights that are made of steel and whose density deviates only slightly from 8000 kg / m ³ , the conventional weight value defined in this way has the advantage that the remaining air density correction (i.e. the difference between mass and conventional weight value) is very small and mostly can be neglected. On the one hand, this simplifies the method and, on the other hand, small calibration uncertainties can be implemented.

Examples

A piece of cork (density 0.15 g / cm ³ ) weighing 1 kg has a density of 1.2 kg / m ³ when weighed with steel weights (density 8.0 g / cm ³ = 8000 kg / m ³ ) in air ^{3 has} a weight value of 0.99215 kg.

literature

• Paul Profos and Tilo Pfeifer (eds.): Handbook of industrial measurement technology. 6th edition, Oldenbourg, Munich / Vienna 1993, ISBN 3-486-22592-8 .
• Roland Nater et al.: Weighing dictionary. Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-75907-2 .

Individual evidence

1. ↑ DIN 1305: mass, weight value, force, weight force, weight, load; Terms.
2. ↑ International Organization of Legal Metrology: OIML D 28 Conventional value of the result of weighing in air. ( PDF file; 0.16 MB ).