Center of gravity

The center of gravity of a body denotes the mean of all positions, weighted according to the attacking gravitational force at the respective point.

• For a homogeneous gravitational field (e.g. near the earth's surface ) the center of gravity coincides with the center of mass of the body. Therefore, both terms are often undifferentiated as a focus .
• In the general case of inhomogeneous gravitational fields (third case below) the center of gravity and center of mass are different. Which of the two points is designated as the “focus” then depends on the author.

overview

On closer inspection, the concept of the center of gravity as the center of gravity shows a more complex structure than one expects from the intuitive view - under simplified conditions such as constant acceleration of gravity and homogeneous density .

• With homogeneous density and homogeneous gravity ( acceleration due to gravity ), the total center of gravity of a cluster can be determined from the weighted sum of the centers of gravity of all subsystems: ${\displaystyle \left(\rho =\mathrm {konst.} \neq \rho ({\vec {x}})\right)}$ ${\displaystyle g=\mathrm {konst.} \neq g({\vec {x}})}$

${\displaystyle {\vec {x}}_{s}={\frac {g\cdot \rho }{G}}\int {\vec {x}}\;\mathrm {dV} }$

With ${\displaystyle G=g\cdot \rho \int \mathrm {dV} \;\;\Rightarrow \;\;{\vec {x}}_{s}={\frac {\int {\vec {x}}\,\mathrm {d} V}{V}}.}$

the subsystems are chosen so that their focus is easy to determine.

Are there

• x - position vector
• V - volume
• G - total weight

• For bodies with inhomogeneous density , the z. B. are irregularly shaped, and with a constant gravitational field, the total center of gravity is calculated as the first moment of the distribution function of the density of an accumulation in space, normalized to the total weight: ${\displaystyle \rho =\rho ({\vec {x}})\neq \mathrm {konst.} }$

${\displaystyle {\vec {x}}_{s}={\frac {g}{G}}\int {\vec {x}}\cdot \rho ({\vec {x}})\,\mathrm {dV} }$

With ${\displaystyle G=g\int \rho ({\vec {x}})\,\mathrm {dV} \;\;\Rightarrow \;\;{\vec {x}}_{s}={\frac {\int {\vec {x}}\cdot \rho ({\vec {x}})\,\mathrm {d} V}{\int \rho ({\vec {x}})\,\mathrm {dV} }}.}$

In this case, e.g. B. approximately on the earth's surface or with objects that are so small that the force of gravity does not noticeably change in the area of ​​their volume, the center of gravity of the system coincides with its center of mass.

• If the gravitational field is also inhomogeneous ( ), one does not integrate via the density (masses), but via the specific gravity : ${\displaystyle g=g({\vec {x}})\neq \mathrm {konst.} }$ ${\displaystyle \rho ({\vec {x}})\cdot g({\vec {x}})}$

${\displaystyle {\vec {x}}_{s}={\frac {1}{G}}\int {\vec {x}}\cdot \rho ({\vec {x}})\cdot g({\vec {x}})\,\mathrm {d} V}$

With ${\displaystyle \,G=\int \rho ({\vec {x}})\cdot g({\vec {x}})\,\mathrm {d} V\ .}$

The distribution function is the product of an external and an internal component: the external is formed by the location-dependent gravitational acceleration , the internal, defined by the accumulation, is the density. This density indicates where how much of “what is assigned to the system under consideration” is located; outside it is zero. The density function describes the shape of the objects. ${\displaystyle g({\vec {x}})}$

Effect of the deviation from the center of mass and center of gravity

A barbell of length d apparently falls “weightlessly” in a low orbit around the earth. It is oriented at an angle to the vertical, so that the two weights have a height difference of one meter. The gravitational acceleration g decreases by about 3 · 10 ⁻⁷ g per meter of height . In relation to the center of mass (barycenter), the center of gravity is 1.5 · 10 ⁻⁷ d closer to the lower end of the dumbbell. Gravity acts in the center of gravity, while the force of inertia ( centrifugal force ) acts in the barycentre. There is a small torque in the direction of vertical alignment, see gravitational stabilization of satellites .

Torque is created in the same way during tidal friction .

See also

• Geometric focus
• Gravity field
• Mass distribution

Individual evidence

1. ↑ James H. Allen: Statics for mechanical engineers for dummies . John Wiley & Sons , 2012, ISBN 978-3-527-70761-4 , pp. 158 ( Google Books ).