Answer:
CM = (0, 0)
Explanation:
We can apply
me = σ(x, y)*Ae
where Ae = π*(7)² = 49*π
then
me = (σ₀*x² + y²)*49*π
cm_e = (cm_ex, cm_ey) = (0, 0)
mi = σ(x, y)*Ai
where Ai = π*(2)² = 4*π
then
mi = (σ₀*x² + y²)*4*π
cm_i = (cm_ix, cm_iy) = (0, 0)
We can apply the equation
mt = me -mi
where is the total mass of the region
then
mt = me - mi = (σ₀*x² + y²)*49*π - (σ₀*x² + y²)*4*π
⇒ mt = 45*π*(σ₀*x² + y²)
then we apply the equation
x_cm = (me*cm_ex - mi*cm_ix) / mt
x_cm = ((σ₀*x² + y²)*49*π*(0) - (σ₀*x² + y²)*4*π*(0)) / (45*π*(σ₀*x² + y²))
x_cm = 0
y_cm = (me*cm_ey - mi*cm_iy) / mt
y_cm = ((σ₀*x² + y²)*49*π*(0) - (σ₀*x² + y²)*4*π*(0)) / (45*π*(σ₀*x² + y²))
y_cm = 0
finally, we get
CM = (x_cm, y_cm) = (0, 0)