Question

Find the Center of mass of a thin plate of constant density 𝛿 covering the region bounded by the parabola 𝑦 = 𝑥² and the line 𝑦 = 4.

Answer 1

Solution

Points of intersection of given curves are solution of equation 𝑥² = 4 => x₁ = a = -2, x₂ = b = 2.

For density 𝛿 mass of the plate is

$M=\delta\int_{a}^{b}{\left(4-x^2\right)dx=}\delta\left(4x-\frac{1}{3}x^3\right)\left|\begin{matrix}2\\-2\\\end{matrix}\right.=\delta\left(16-\frac{16}{3}\right)=\delta\frac{32}{3}$

Equations of Moments

$M_x=\delta\int_{-2}^{2}{\frac{1}{2}\left[4^2-\left(x^2\right)^2\right]dx}=\delta\left[8x-\frac{1}{10}x^5\right]\left|\begin{matrix}2\\-2\\\end{matrix}\right.=\delta\left[32-\frac{64}{10}\right]=25.6\delta$

$M_y=\delta\int_{-2}^{2}x\left(4-x^2\right)dx=\delta\left[2x^2-\frac{1}{4}x^4\right]\left|\begin{matrix}2\\-2\\\end{matrix}\right.=0$

Center of Mass Coordinates

$x_C=\frac{M_y}{M}=0$

$y_C=\frac{M_x}{M}=\frac{256\bullet3}{320}=2.4$