Question

A thin uniform sheet of metal of uniform thickness is cut into shape bounded by the line x=a and y=+-kx2. find the coordinates of the center of mass

Answer 1

see diagram.  We use integration to solve this.

Obviously from the symmetry, the y - coordinate of the center of mass lies on the x - axis .  Hence it is 0.

We take a thin vertical strip at  x = x and of width dx.  Its area dA = (y₁- y₂) * dx.

X_com = [tex]\frac{1}{a} \int \limits_0^a {x\ (y_1 - y_2)} \, dx\\\\=\frac{1}{a} \int \limits_0^a {x\ (k x^2-(-kx^2))} \, dx\\\\=\frac{2k}{a} \int \limits_0^a {x^3} \, dx\\\\=\frac{2k}{a}*\frac{a^4}{4}\\\\=\frac{k\ a^3}{2}[/tex]

so answer is  ( k a³/2, 0) is the center of mass of the area enclosed by the two parabolas and the straight line x = a.