The distance from the rounded tip of the plate to the center of mass is 1.87 m.
What is center of mass?
The center of mass is a point inside or outside the mass where all of the mass is concentrated.
The y-coordinate of the centroid is given by the ratio of two definite integrals;
Yc = ∫ydm/∫dm,
where dm is a density function
For the uniform plate, δ does not change with position in the plate.
Yc = ∫yδdA/∫δdA
Yc = ∫ydA/∫dA.
dA is a horizontal slice of the plate with dimensions xdy.
Solving the parabola for x,
y = 0.5x²
x = ± √(y/0.50), where the negative value corresponds to the left half of the parabola and the positive to the right half.
dA = (√(y/0.50)
= √(y/0.50))dy
= 2(√(y/0.50))dy
The limits of integration are from zero to 1.870, the top of the plate.
∫ydA = ∫2y√(y/0.50)dy = 7.232 m³
∫dA = ∫2√(y/0.50)dy = 3.868 m²
∫ydA/∫dA = 7.232 m³/3.868 m²
∫ydA/∫dA = 1.869700 m
∫ydA/∫dA = 1.87 m
Thus, the distance from the rounded tip of the plate to the center of mass is 1.87 m.
Learn more about center of mass.
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