Final answer:
To balance the center of gravity at the origin, the fourth object of 7.8 kg must be placed at (-1.79 m, -1.60 m), which counterbalances the moments created by the other given masses.
Step-by-step explanation:
To determine where the fourth object of 7.8 kg should be placed so that the center of gravity (CG) of the entire system is located at (0.0, 0.0) m, we must use the concept of the center of mass. The center of mass can be found using the weighted average positions of each mass, taking their quantities into account.
The x-coordinate of the center of mass, based on the distribution given, can be calculated as:
(5.0 kg * 0 m + 3.2 kg * 0 m + 4.0 kg * 3.5 m + 7.8 kg * x) / (5.0 kg + 3.2 kg + 4.0 kg + 7.8 kg) = 0
Similarly, the y-coordinate of the center of mass is:
(5.0 kg * 0 m + 3.2 kg * 3.9 m + 4.0 kg * 0 m + 7.8 kg * y) / (5.0 kg + 3.2 kg + 4.0 kg + 7.8 kg) = 0
By solving these equations, we can find the x and y coordinates where the fourth object should be placed. However, since the desired center of gravity is at the origin, this means that the sum of the moments about both the x and y axis must be zero. This directly implies that the 7.8 kg mass must be placed at an x and y position such that its distance times its mass cancels out the moments created by the other masses.
If we do the math, we find that for the x-coordinate:
(4.0 kg * 3.5 m) / 7.8 kg = x → x = -1.79 m (approximately)
And for the y-coordinate:
(3.2 kg * 3.9 m) / 7.8 kg = y → y = -1.60 m (approximately)
Thus, the fourth object should be placed at (-1.79 m, -1.60 m) to ensure the center of gravity of the system is at the origin (0.0, 0.0) m.