Final answer:
To keep the center of gravity at (0, 0), a 9.00-kg object should be placed at (0, -1.733) m, counterbalancing the other masses' distribution.
Step-by-step explanation:
The question is asking where to place a fourth object of mass 9.00 kg so that the center of gravity of the four-object system will be at (0, 0). The center of gravity (CoG) or center of mass (CoM) refers to the point at which all the weight of a body or system is considered to act. To solve this, we need to ensure that the net moments about both axes are zero when including the fourth mass.
Let's denote the position of the fourth mass as (x, y). Since the overall CoG must be at the origin and we have a symmetrical distribution on the x-axis, the fourth mass will contribute to the y-axis balance. Given that we already have a system with a net mass of 14.60 kg (4.00 + 5.20 + 5.40) and a center of gravity at (0, 0), placing a 9.00-kg object at (0, -y) should maintain the balance.
We can calculate the exact position using the formula:
(m1*y1 + m2*y2 + m3*y3 + m4*y4) / (m1 + m2 + m3 + m4) = CoG_y
Substituting the known values:
((4.00 * 0) + (5.20 * 3.00) + (5.40 * 0) + (9.00 * -y)) / (4.00 + 5.20 + 5.40 + 9.00) = 0
Solving for y gives us:
9.00 * y = (5.20 * 3.00)
y = (5.20 * 3.00) / 9.00
y = 1.733 m
Thus, to balance the system, the 9.00-kg object must be placed at (0, -1.733) m.