A cubical box consists of 4 square sides and a square base but has no top.
The sides and the base are all made of thin sheet metal of uniform thickness.
If the edges of the box are all 200 mm in length, how far above the base is the
box’s centre of mass?

Question

Layla Ahmed · Accepted Answer

From the symmetry we see that the center of mass of the box is on a vertical line directly above the center of the base. We also know that weight  #w="mass"xxg# 
#=>w=("Volume"xx "Density")xxg# 
#=>w=(("Area" xx "Thickness")xx "Density")xxg#  It is given that thickness and density is same for base and all four sides of the box. Let edge of each side and base #=l#. Area of the base #= l^2#
Area of each side #=l^2# As such #l^2# represents the mass of the base and also of each side. Let the box be so placed on its base so that origin coincides with the center of mass of base, #i#.#e#., center of base. 
Height of base #=0.# We also see that center of each side is located at a height #=l/2# Height of CoM is given by #h_"CoM"=(m_1h_1+m_2h_2...)/(m_1+m_2....)#
Inserting various values we get  #h_"CoM"=(l^2xx0+l^2xxl/2+l^2xxl/2+l^2xxl/2+l^2xxl/2)/(5l^2)#
#=>h_"CoM"=(2l)/5#
#=>h_"CoM"=(2xx200)/5=80\ mm#

Sebastian Acosta · Answer

undefined The center of mass of the cubical box is located one-third of the distance from the base to the top along the vertical axis. Therefore, the center of mass is 133.33 mm above the base of the box.

Axel Acevedo · Answer

undefined The center of mass of a uniform density object is located at the geometric center. For a cubical box without a top, the center of mass is at the center of the cube. Since all edges of the cube are equal in length, the center of mass is equidistant from each face.

To find the distance from the base to the center of mass, we divide the length of one edge by 2. Therefore, for a cube with edges of 200 mm in length, the distance from the base to the center of mass is half of the edge length, which is 100 mm.

A cubical box consists of 4 square sides and a square base but has no top. The sides and the base are all made of thin sheet metal of uniform thickness. If the edges of the box are all 200 mm in length, how far above the base is the box’s centre of mass?