Final answer:
To find the center of mass of a thin plate of constant density covering the region bounded by the parabola y = x and the line y = ? , we can use the concept of calculus and integration. The x-coordinate of the center of mass, X, can be found by integrating the product of the x-coordinate of the elemental area and the mass of the elemental area over the region. The y-coordinate of the center of mass, Y, can be found in a similar way.
Step-by-step explanation:
To find the center of mass of a thin plate of constant density covering the region bounded by the parabola y = x and the line y = ? , we can use the concept of calculus and integration. Let's consider a small elemental area dA on the plate. The mass of this elemental area is given by dm = ρdA, where ρ is the density of the plate. The x-coordinate of the center of mass, denoted by X, can be found by integrating the product of the x-coordinate of the elemental area and the mass of the elemental area over the region. Similarly, the y-coordinate of the center of mass, denoted by Y, can be found by integrating the product of the y-coordinate of the elemental area and the mass of the elemental area over the region.
To find the x-coordinate of the center of mass, X, we integrate x·dm from y = x to y = ? . Since the parabola and the line intersect at (0, 0) and (1, 1), we can set up the integral as follows:
- Find the mass of the plate by integrating the density over the region:
Find the x-coordinate of the center of mass, X, by integrating x·dm over the region:
- X = (1/m)∫∫xρdA = (1/m)∫(∫xρ dy) dx
Solve the integrals to get the values of m and X.
Similarly, to find the y-coordinate of the center of mass, Y, we integrate y·dm from y = x to y = ? . The steps are the same as above, just replace x with y in the integrals. Solve the integral to find the value of Y.