Physic C............systems of particles?

Question

a cubical box, open at the top, with edge length 40 cm, is constructed from metal plate of uniform density and negligible thickness. Find the coordinates of the center of mass of the box

Answer 1

The centre of mass will be on a vertical line passing through the centre of the base.

Imagining the box with edge length a balanced with one of its sides on a knife edge at distance x from the base. take moments about the knife edge:

2(a - x)(a - x) / 2 = 2x(x / 2) + ax

(a - x)^2 = x^2 + ax

a^2 - 2ax + x^2 = x^2 + ax

a^2 - 3ax = 0

a(a - 3x) = 0

x = a / 3.

With x and y axes along the edges of the base, the co-ordinates of the centre of mass are (a / 2, a / 2, a / 3).

Putting a = 40 cm, this gives (20, 20, 40 / 3) cm.