Question

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?

Answer 1

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

Step-by-step 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:

`\bar x = (x_(1)\cdot m_(1) + x_(2)\cdot m_(2) + x_(3) \cdot m_(3) + x_(4)\cdot m_(4))/(m_(1) + m_(2) + m_(3) + m_(4))`

`\bar y = (y_(1)\cdot m_(1) + y_(2)\cdot m_(2) + y_(3) \cdot m_(3) + y_(4)\cdot m_(4))/(m_(1) + m_(2) + m_(3) + m_(4))`

Where:

`\bar x`,
`\bar y` - Horizontal and vertical component of the location of the center of gravity, measured in meters.

`x_(1), x_(2), x_(3), x_(4)` - Horizontal components of the location of first, second, third and fourth masses, measured in meters.

`y_(1), y_(2), y_(3), y_(4)` - Vertical components of the location of first, second, third and fourth masses, measured in meters.

`m_(1), m_(2), m_(3), m_(4)` - Masses of first, second, third and fourth masses, measured in kilograms.

If
`m_(1) = 5\,kg`,
`m_(2) = 3.6\,kg`,
`m_(3) = 4\,kg`,
`m_(4) = 7.7\,kg`,
`\bar x = 0\,m`,
`\bar y = 0\,m`,
`x_(1) = 0\,m`,
`x_(2) = 0\,m`,
`x_(3) = 2.9\,m`,
`y_(1) = 0\,m`,
`y_(2) = 4.1\,m` and
`y_(3) = 0.0\,m`, then:

`0\,m = ((0\,m)\cdot (5\,kg)+(0\,m)\cdot (3.6\,kg)+(2.9\,m)\cdot (4\,kg)+x_(4)\cdot (7.7\,kg))/(5\,kg + 3.6\,kg + 4\,kg + 7.7\,kg)`

`0\,m = ((0\,m)\cdot (5\,kg)+(4.1\,m)\cdot (3.6\,kg)+(0\,m)\cdot (4\,kg)+y_(4)\cdot (7.7\,kg))/(5\,kg + 3.6\,kg + 4\,kg + 7.7\,kg)`

Both expression are simplified hereafter:

`(4)/(7)\,m + (11)/(29)\cdot x_(4) = 0\,m`

`(738)/(1015)\,m + (11)/(29)\cdot y_(4) = 0\,m`

The solution of this system of equation is
`x_(4) = -(116)/(77)\,m` (
`x_(4) \approx -1.506\,m`) and
`y_(4) = - (738)/(385)\,m` (
`y_(4)\approx -1.917\,m`).

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