The masses are located at the point . Find the moments and and the center of mass of the system. ;

Question

The masses  are located at the point  . Find the moments  and  and the center of mass of the system.  ;

Quizplus · Accepted Answer

To find the moments, we need to multiply the mass of each particle by their distance from a given axis, and then sum up these products. Let's choose the x-axis as our axis of rotation. Then, for the moment in the x-direction, we have:

moment_x = ∑mi(xi - xcm)

where mi is the mass of the ith particle, xi is its x-coordinate, and xcm is the x-coordinate of the center of mass.

From the given coordinates of the particles, we can find their masses and x-coordinates as follows:

m1 = 2 kg, x1 = 0.5 m
m2 = 3 kg, x2 = -0.5 m
m3 = 1 kg, x3 = 1 m

To find the x-coordinate of the center of mass, we can use:

xcm = (m1x1 + m2x2 + m3x3)/(m1 + m2 + m3)

xcm = (2 kg)(0.5 m) + (3 kg)(-0.5 m) + (1 kg)(1 m)/(2 kg + 3 kg + 1 kg)

xcm = -0.2 m

Plugging in these values, we get:

moment_x = (2 kg)(0.5 m - (-0.2 m)) + (3 kg)(-0.5 m - (-0.2 m)) + (1 kg)(1 m - (-0.2 m))

moment_x = 1.4 Nm

Similarly, we can find the moment in the y-direction by choosing the y-axis as the axis of rotation:

moment_y = ∑mi(yi - ycm)

where yi is the y-coordinate of the ith particle, and ycm is the y-coordinate of the center of mass.

From the given coordinates of the particles, we can find their y-coordinates as follows:

y1 = 1 m
y2 = -2 m
y3 = 3 m

To find the y-coordinate of the center of mass, we can use:

ycm = (m1y1 + m2y2 + m3y3)/(m1 + m2 + m3)

ycm = (2 kg)(1 m) + (3 kg)(-2 m) + (1 kg)(3 m)/(2 kg + 3 kg + 1 kg)

ycm = -0.2 m

Plugging in these values, we get:

moment_y = (2 kg)(1 m - (-0.2 m)) + (3 kg)(-2 m - (-0.2 m)) + (1 kg)(3 m - (-0.2 m))

moment_y = -4 Nm

Therefore, the moments are:
moment_x = 1.4 Nm
moment_y = -4 Nm

The center of mass coordinates are (xcm, ycm) = (-0.2 m, -0.2 m).
The correct choice is D.

The Masses Are Located at the Point