A square sheet with a uniform mass per unit area has a circular hole cut out of it. The square extends from x=1 to x=5, and from y=0 to y=4. The circle has a radius of 1 and is centered at x=2.5, y=2.5. Where is the center of mass of the sheet with the hole?

The center-of mass equation can be written:

X_cm = ( x₁m₁ + x₂m₂ ) / M

X_cm M = x₁m₁ + x₂m₂

Here X_cm and M represent the x-coordinate of the center-of-mass position, and the mass, of the sheet without the hole. The subscript 1 represents the sheet with the hole, and the 2 represents the piece cut out to make the hole. We're looking for x₁.

x₁ = ( X_cm M - x₂m₂ ) / m₁

X_cm = 3, and x₂ = 2.5.

How do we handle the masses? Because the sheet is uniform, the mass of a piece of sheet is proportional to the area. In fact:

m = σ A, where σ is the mass/unit area of the sheet.

M = 16σ , m₂ = πσ , and m₁ = (16-π)σ

Substituting these values into the expression for x₁ gives:

x₁ = σ ( 48 - 2.5π ) / [(16-π )σ]

Note that the factors of σ cancel.

x₁ = 3.12

A similar method can be used to find that y₁ = 1.88.