Find the x- and y-coordinates of the center of mass of the flat triangular plate of height H = 87.8 cm and base B = 50.7 cm shown in the figure.

x =
y =

I have tried this problem a multitude of times and I cannot seem to get this right, Please help!

I have used the equation Xcm = 1/m (integral sign) xdm

Without the figure or a good description of it, we can't help you.

Coordinates of the center of mass depend how the triangle is oriented in the coordinate system. They also depend upon whether the triangle is isosceles or scalene

To find the x- and y-coordinates of the center of mass of a flat triangular plate, you need to divide the plate into infinitesimal small portions and find the coordinates of their center of mass. Then, integrate these coordinates over the entire plate to get the coordinates of the overall center of mass.

Let's break down the steps to solve this problem:

Step 1: Set up the coordinate system
In this problem, let's set up the coordinate system such that the origin (0, 0) is at the bottom left corner of the triangular plate. The x-axis will extend horizontally to the right, and the y-axis will extend vertically upwards.

Step 2: Divide the plate into small portions
To calculate the center of mass, divide the triangular plate into an infinite number of small rectangles or strips. Each of these small elements will have an area of dx*dy.

Step 3: Express the x-coordinate of each element's center of mass as a function of x and y
Considering the symmetry of the triangle, let's assume that the vertex apex of the triangle is located at (0, H) in the coordinate system. The base of the triangle will lie along the x-axis, with its length being B. For any small element with coordinates (x, y), the x-coordinate of its center of mass is simply the position of the element, i.e., x.

Step 4: Express the y-coordinate of each element's center of mass as a function of x and y
The y-coordinate of each element's center of mass can be expressed as (H-y)/2, as the distance from the bottom vertex to any point on the triangle is H-y, and we divide it by 2 to obtain the distance from the origin (0, 0) to the center of mass of each element.

Step 5: Convert the Cartesian coordinates (x, y) to their corresponding polar coordinates (r, θ)
To simplify integration, let's convert the Cartesian coordinates (x, y) into polar coordinates (r, θ). Here, r represents the distance from the origin to a point on the plate, and θ represents the angle between the x-axis and the radial line connecting the origin to that point.

Given that r = sqrt(x^2 + y^2) and x = r*cos(θ), y = r*sin(θ), we can substitute the values of x and y in terms of r and θ in steps 3 and 4 above.

Step 6: Calculate the differential element of mass dm
The area element, dm, of each small rectangular strip, can be expressed using its height (dy) and its width (dx). Since the height of the triangle is H, and the base is B, we can use similar triangles to find that: dx = (B/H)*dy.

Step 7: Integrate to find the x-coordinate of the center of mass
Using the equation Xcm = (1/m) * ∫x * dm, where m represents the total mass of the plate, and integrating over the entire plate, we can solve for the x-coordinate of the center of mass.

Step 8: Integrate to find the y-coordinate of the center of mass
Similarly, we can use the equation Ycm = (1/m) * ∫y * dm, where m represents the total mass of the plate, and integrate over the entire plate to solve for the y-coordinate of the center of mass.

While these steps provide the theoretical framework to solve the problem, actually carrying out the calculations may be quite involved. It might be helpful to break down the calculations into smaller parts and evaluate them separately using numerical methods or software such as MATLAB or Mathematica.