Question

A uniform plate of height 1.480 m is cut in the form of a parabolic section. The lower boundary of the plate is defined by: y=1.050x2. Find the distance from the rounded tip of the plate to the center of mass.

Answer 1

The y-coordinate of the centroid is given by the ratio of two definite integrals;


y_c = ∫ydm/∫dm, where dm = a density function δ evaluated over dA.

For the uniform plate, δ does not change with position in the plate.

y_c = ∫yδdA/∫δdA = ∫ydA/∫dA.

dA is a horizontal slice of the plate with dimensions xdy.

Solving the parabola for x,

y = 1.050x²

x = ± √(y/1.050), where the negative value corresponds to the left half of the parabola and the positive to the right half.

dA = (√(y/1.050) - - √(y/1.050))dy = 2(√(y/1.050))dy

The limits of integration are from zero to 1.480, the top of the plate.

The integral in the numerator has an extra y.

Using a TI-83 Plus,

∫ydA = ∫2y√(y/1.050)dy = 2.080 m³

∫dA = ∫2√(y/1.050)dy = 2.343 m²

∫ydA/∫dA = 2.080/2.343 = .888 m up from rounded tip of plate.