Final answer:
To find the center of mass of a metal plate shaped by the curve y=sqrt(x) and the x-axis over [0,1], you would calculate the moments about both axes and the total mass using integrals, then divide the moments by the total mass to obtain the coordinates of the center of mass.
Step-by-step explanation:
To find the center of mass of a metal plate that is bounded by the curve y=sqrt(x) and the x-axis over the interval [0,1], and assuming the plate has a constant density, we can use calculus. We need to calculate the moments about the x-axis (Mx) and the y-axis (My), as well as the total mass (m) of the plate.
The moment about the x-axis for a differential element is given by dMx = y ∙ dm, where dm is the mass element and y is the distance from the x-axis. Similarly, the moment about the y-axis is given by dMy = x ∙ dm. The total mass is the integral of dm over the whole plate.
To find dm, we multiply the density (which is constant and therefore can be called ρ) by the differential area element, dA = y dx. The moments and the mass are then calculated as follows:
- For total mass (m): m = ∫ dA = ∫01 sqrt(x) dx
- For Mx: Mx = ∫ y dA = ∫01 y ∙ sqrt(x) dx
- For My: My = ∫ x dA = ∫01 x ∙ sqrt(x) dx
The x-coordinate of the center of mass (Øx) is found by dividing My by the total mass (m), and the y-coordinate of the center of mass (Øy) is found by dividing Mx by the total mass (m).
Integrating the above expressions, we would find the values for Mx, My, and m. Then the center of mass is found by these formulas: Øx = My/m and Øy = Mx/m.