A system consists of the following masses loacted in the xy- plane: 5.0 kg at the point (0,5), 7.5 kg at the point (3,8) and 6.5 at the point (-3, -6). Find the distance from the origin to the center of mass.
To find the distance from the origin to the center of mass, we need to calculate the coordinates of the center of mass first. The center of mass is the weighted average of the coordinates of the individual masses, where the weights are proportional to the masses.
Let's calculate the x-coordinate of the center of mass (Cox):
Cox = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3)
Here,
m1 = 5.0 kg
m2 = 7.5 kg
m3 = 6.5 kg
x1 = 0
x2 = 3
x3 = -3
Cox = (5.0 * 0 + 7.5 * 3 + 6.5 * (-3)) / (5.0 + 7.5 + 6.5)
= (-4.5) / 19
≈ -0.24
Now, let's calculate the y-coordinate of the center of mass (Coy):
Coy = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3)
Here,
y1 = 5
y2 = 8
y3 = -6
Coy = (5.0 * 5 + 7.5 * 8 + 6.5 * (-6)) / (5.0 + 7.5 + 6.5)
= 17 / 19
≈ 0.89
Therefore, the coordinates of the center of mass are approximately (Cox, Coy) = (-0.24, 0.89).
Finally, we can calculate the distance from the origin to the center of mass using the distance formula:
Distance = √((Cox - 0)^2 + (Coy - 0)^2)
= √((-0.24 - 0)^2 + (0.89 - 0)^2)
= √(0.0576 + 0.7921)
= √0.8497
≈ 0.92
So the distance from the origin to the center of mass is approximately 0.92 units.