Question

A cubical box, open at the top, with edge length 81.0 cm, is constructed from metal plate of uniform density and negligible thickness. One of the bottom corners of the box is at 0,0,0, the base of the box is in the x-y plane and the box is the positive quadrant. What is the x coordinate of the center of mass of the box (in cm.)?

What is the y coordinate of the center of mass of the box (in cm.)?

What is the z coordinate of the center of mass of the box (in cm.)?

Answer 1

here,

The center of mass of each square side is at its center.

Considering the four sides of the box (leaving top and bottom],

the center of mass is at [x/2, y/2, z/2] and the mass = 4m

the center of mass is at [40.5, 40.5, 40.5] and the mass = 4m

where m is the mass of each square.

The center of mass of the bottom side is at its center.
[X/2, y/2, 0] and its mass is m.

i.e (40.5 , 40.5 , 0)

Now we are having two mass points alone in the vertical line through the center of the base square.

One mass point [= m] is at the center of the base and another mass point [= 4m]

the x coordinate of the center of mass of the box (in cm.) , x = (40.5 * 4m + 40.5 * m) /(4m + m)

x = 40.5 cm

the x coordinate of the center of mass of the box is 40.5 cm

the y coordinate of the center of mass of the box , y = (40.5 * 4m + 40.5 * m) /(4m + m)

y = 40.5 cm

the y coordinate of the center of mass of the box is 40.5 cm

the z coordinate of the center of mass of the box z = (40.5 * 4m + 0 * m) /(4m + m)

z = 32.4 cm

the z coordinate of the center of mass of the box is 32.4 cm