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?

1 Answer
A08
May 19, 2018

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#

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