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# Center of gravity, combined bodies

The center of gravity has to be calculated for the body shown. The body consists of two individual bodies of different materials.

Given values:

a = 30 mm, b = 25 mm, c = 60 mm, d = 30 mm, e = 50 mm

Body 1 consists of steel with a density of ρ = 7.850 kg/m³,

Body 2 is made of aluminium with a density of ρ = 2.660 kg/m³.

A note on the direction of projection: a circle with a center point indicates a coordinate axis that points at the viewer. A circle with a cross is a coordinate axis that points away from the viewer.

## Solution

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The equation for calculating the center of gravity coordinate is, noted here as an example for x S :

$\tag{1} x_S = \frac{\sum \rho_i \cdot V_i \cdot x_i}{\sum \rho_i \cdot V_i}$

The values ​​of the individual bodies are recorded in a table

$i$$V_i$$x_i$$y_i$$z_i$$\rho_i$$\rho_i \cdot V_i$$\rho_i \cdot V_i \cdot x_i$$\rho_i \cdot V_i \cdot y_i$$\rho_i \cdot V_i \cdot z_i$
${mm}^3$$mm$$mm$$mm$$10^{-6}kg/{mm}^3$$kg$$kgmm$$kgmm$$kgmm$
154.000-15-1507,850,424-6,359-6,3590
275.00012,5-502,660,22,494-0,9980
$\sum$0,624-3,865-7,3570

This results in the following center of gravity coordinates:

$\tag{2} x_S = \frac{-3.865 kg\,mm}{0.624kg} = -6.2\,mm$

$\tag{3} y_S = \frac{-7.357 kg\,mm}{0.624kg} = -11.8\,mm$

$\tag{4} z_S = \frac{0 kg\,mm}{0.624kg} = 0\,mm$

So the center of gravity of the combined body is determined.