To find the position of the center of mass of the system, we need to first calculate the coordinates of the center of mass along both the x-axis and y-axis.
Let's begin by finding the coordinates of the center of mass along the x-axis:
We can use the formula:
x_cm = (m1x1 + m2x2 + m3x3 + m4x4) / (m1 + m2 + m3 + m4)
where m1, m2, m3, and m4 are the masses of the point masses at each corner, and x1, x2, x3, and x4 are the x-coordinates of each point mass.
In this case, we have:
m1 = 2kg, x1 = 0cm
m2 = 4kg, x2 = 2cm
m3 = 6kg, x3 = 2cm
m4 = 4kg, x4 = 0cm
Substituting these values into the formula, we get:
x_cm = (2kg x 0cm + 4kg x 2cm + 6kg x 2cm + 4kg x 0cm) / (2kg + 4kg + 6kg + 4kg)
x_cm = 24 / 16
x_cm = 1.5cm
Therefore, the x-coordinate of the center of mass is 1.5cm.
Now let's find the coordinates of the center of mass along the y-axis:
We can use the formula:
y_cm = (m1y1 + m2y2 + m3y3 + m4y4) / (m1 + m2 + m3 + m4)
where m1, m2, m3, and m4 are the masses of the point masses at each corner, and y1, y2, y3, and y4 are the y-coordinates of each point mass.
In this case, we have:
m1 = 2kg, y1 = 0cm
m2 = 4kg, y2 = 0cm
m3 = 6kg, y3 = 2cm
m4 = 4kg, y4 = 2cm
Substituting these values into the formula, we get:
y_cm = (2kg x 0cm + 4kg x 0cm + 6kg x 2cm + 4kg x 2cm) / (2kg + 4kg + 6kg + 4kg)
y_cm = 20 / 16
y_cm = 1.25cm
Therefore, the y-coordinate of the center of mass is 1.25cm.
So the center of mass of the system is located at the point (1.5cm, 1.25cm) from the corner.