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18 August, 23:07

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

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  1. 19 August, 00:54
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    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)
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