A system consists of the following masses loacted in the xy- plane: 3.0 kg at the point (0,5), 7.5 kg at the point (3,8) and 7.0 at the point (-3, -6). Find the distance from the origin to the center of mass.
x cm: (3*0+7.5*3+7*-3)/17.5
7 cm:(3*5+7.5*8+7*(-6))/17.5
distance to cm; sqrt( (xcm)^2+(ycm)^2 )
To find the distance from the origin to the center of mass, we need to calculate the center of mass first. The center of mass for a system of particles is given by the weighted average position of all the particles, where the weights are the masses of the particles.
Let's denote the mass of the first particle as m1 = 3.0 kg located at the point (0,5).
Let's denote the mass of the second particle as m2 = 7.5 kg located at the point (3,8).
Let's denote the mass of the third particle as m3 = 7.0 kg located at the point (-3,-6).
The x-coordinate of the center of mass (Cx) is given by the formula:
Cx = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3),
where x1, x2, x3 are the x-coordinates of the respective particles.
Substituting the given values, we get:
Cx = (3.0 kg * 0 + 7.5 kg * 3 + 7.0 kg * (-3)) / (3.0 kg + 7.5 kg + 7.0 kg),
= (0 + 22.5 + (-21)) / (17.5 + 7.0 + 7.0),
= 1.5 / 31.5,
= 0.0476.
Using the same logic, the y-coordinate of the center of mass (Cy) is given by the formula:
Cy = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3),
where y1, y2, y3 are the y-coordinates of the respective particles.
Substituting the given values, we get:
Cy = (3.0 kg * 5 + 7.5 kg * 8 + 7.0 kg * (-6)) / (3.0 kg + 7.5 kg + 7.0 kg),
= (15 + 60 + (-42)) / (17.5 + 7.0 + 7.0),
= 33 / 31.5,
= 1.0476.
The center of mass is located at the point (Cx, Cy) = (0.0476, 1.0476).
Finally, to find the distance from the origin to the center of mass, we can use the distance formula:
Distance = sqrt((Cx - 0)^2 + (Cy - 0)^2),
= sqrt((0.0476 - 0)^2 + (1.0476 - 0)^2),
= sqrt(0.0023 + 1.0972),
= sqrt(1.0995),
= 1.048.
Therefore, the distance from the origin to the center of mass is approximately 1.048 units.