Find the center of mass and the moment of inertia about origin of the thin plate bounded by the x-axis, the lines x=±1 and y=x^2 with a density function ρ(x,y) = y+1 .

Answer 1

Plate Mass is M = ∫_{(-1 to 1)}∫_{(0 to x-squared)}(y + 1)dydx which goes to ∫_{(-1 to 1)}[0.5y² + y|_{(0 to x-squared)}]dx equal to

∫_{(-1 to 1)}[0.5x⁴ + x²]dx or [0.1x⁵ + 3^-1x³|_{(-1 to 1)}] or 1/5 + 2/3 or 13/15.

First Moment about the x-axis is M_x equal to ∫_{(-1 to 1)}∫_{(0 to x-squared)}(y² + y)dydx or 

∫_{(-1 to 1)}[3^-1y³ + 0.5y²|_{(0 to x-squared)}]dx

or ∫_{(-1 to 1)}[3^-1x⁶ + 0.5x⁴]dx or [x⁷/21 + x⁵/10|_{(-1 to 1)}] equal to 31/105.

First Moment about the y-axis is M_y equal to ∫_{(-1 to 1)}∫_{(0 to x-squared)}(xy + x)dydx which goes to

∫_{(-1 to 1)}[0.5xy² + xy|_{(0 to x-squared)}]dx or ∫_{(-1 to 1)}[0.5x⁵ + x³]dx or [12^-1x⁶ + 4^-1x⁴|_{(-1 to 1)}] or 0.

Coordinates of the Center Of Mass are given by (x_bar, y_bar) or (M_y/M, M_x/M) equal to

[0/(13/15), (31/105)/(13/15)] or [0, 31/91].

Second Moment (or Moment Of Inertia) about the x-axis is I_x equal to ∫_{(-1 to 1)}∫_{(0 to x-squared)}[y³ + y²]dydx

or ∫_{(-1 to 1)}[4^-1y⁴ + 3^-1y³|_{(0 to x-squared)}]dx or ∫_{(-1 to 1)}[4^-1x⁸ + 3^-1x⁶]dx or [x⁹/36 + x⁷/21|_{(-1 to 1)}] or 19/126.

Second Moment (or Moment Of Inertia) about the y-axis is I_y equal to ∫_{(-1 to 1)}∫_{(0 to x-squared)}(x²y + x²)dydx

or ∫_{(-1 to 1)}[0.5x²y² + x²y|_{(0 to x-squared)}]dx or ∫_{(-1 to 1)}[0.5x⁶ + x⁴])dx or [14^-1x⁷ + 5^-1x⁵|_{(-1 to 1)}] or 19/35.

Moment Of Inertia about the Origin is I_o equal to ∫_{(-1 to 1)}∫_{(0 to x-squared)}(x² + y²)(y + 1)dydx, rewritten as

∫_{(-1 to 1)}[0.5x²y² +x²y + 0.25y⁴ + 3^-1y³|_{(0 to x-squared)}]dx or ∫_{(-1 to 1)}[0.5x⁶ + x⁴ + 0.25x⁸ + 3^-1x⁶]dx.

Integrate this last to [14^-1x⁷ + 0.2x⁵ + 36^-1x⁹ +21^-1x⁷|_{(-1 to 1)}] which amounts to (1/7 + 2/5 + 1/18 + 2/21)

or 437/630. Note that 437/630 is also obtained by the sum of I_x and I_y or 19/126 +19/35.