Consider the following mass distribution where the x and y coordinates are given in meters: 5.0 kg at (0.0, 0.0) m, 3.6 kg at (0.0, 4.1) m, and 4.0 kg at (2.9, 0.0) m. Where should a fourth object of 7.7 kg be placed so the center of gravity of the four-object arrangement will be at (0.0, 0.0) m?

The fourth mass should be located at (-1.506 m, -1.917 m).

Explanation:

Given that each mass can be treated as puntual objects, the location of center of gravity can be determined by using weighted averages. That is:

Where:

, - Horizontal and vertical component of the location of the center of gravity, measured in meters.

- Horizontal components of the location of first, second, third and fourth masses, measured in meters.  

- Vertical components of the location of first, second, third and fourth masses, measured in meters.

- Masses of first, second, third and fourth masses, measured in kilograms.  

If , , , , , , , , , , and , then:

Both expression are simplified hereafter:

The solution of this system of equation is () and ().

The fourth mass should be located at (-1.506 m, -1.917 m).