A plane lamina occupies the region in the xy-plane between the two circles x2 + y2 = 4 and x2 + y2 = 49 and above the x-axis. Find the center of mass of the lamina if its mass density is σ(x, y) = σ0 x2 + y2 kg/m2

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)

the bal will not change its velocity.