Find the center of mass of a semi-circular plate

[Debelius]

Homework Statement

Find the center of mass of a semi-circular plate of radius r

Find the volume when the plate (above) is rotated around a line along its straight side

Homework Equations

2(pi) integral of r dr

The Attempt at a Solution

I honestly don't know how to do centroids. :-( I'd like to know how to actually solve this problem.

 

[Knissp]

The x coordinate of the center of mass should be zero (intuitively).
For the y coordinate, assuming that the mass is uniformly distributed, use this equation (which can generalize for x):
ycentroid = $$\frac{\int\int_R y dA}{A}$$

The volume is thus computed from the theorem of Pappus which states that
volume = (area of R) * (distance traveled by the centroid)
where the distance traveled by the centroid is 2*pi*ycentroid

Hope that helps, let me know if you need further clarification.
(Also, the volume when rotated about the straight side should come out to be that of a sphere, which is 4/3*pi*r^3, so you can use that to make sure you did it right.)

 

[Moetasim]

I understand your frustration with not knowing how to solve a problem. Centroids, also known as center of mass, can be calculated using the following formula:

x_bar = (1/A) * integral of x*dA
y_bar = (1/A) * integral of y*dA

Where A is the total area of the object and x and y represent the coordinates of each infinitesimal area element dA. In this case, we can use polar coordinates to simplify the integral.

First, let's determine the total area of the semi-circular plate. We know that the area of a circle is given by A = pi*r^2, so the area of the semi-circle will be half of that, or A = (1/2)*pi*r^2.

Next, we can express the coordinates of each infinitesimal area element dA in polar coordinates, where r represents the distance from the origin to the element and θ is the angle from the x-axis. We can then rewrite the formula for x_bar and y_bar as:

x_bar = (1/A) * integral of r*cos(θ)*r*dθ*dr
y_bar = (1/A) * integral of r*sin(θ)*r*dθ*dr

Now, we can solve these integrals using the limits of integration. For a semi-circle, the limits of integration for θ will be from 0 to π, and for r, it will be from 0 to r (the radius of the semi-circle). This gives us:

x_bar = (1/A) * integral from 0 to π of r*cos(θ)*r*dθ*dr
y_bar = (1/A) * integral from 0 to π of r*sin(θ)*r*dθ*dr

Solving these integrals, we get:

x_bar = (2*r)/(3*pi)
y_bar = 0

Therefore, the center of mass of the semi-circular plate will be at the coordinates (x_bar, y_bar) = ((2*r)/(3*pi), 0).

To find the volume when the plate is rotated around a line along its straight side, we can use the formula for volume of revolution:

V = integral from a to b of pi*r^2*dx

Where a and b are the limits of integration along the x-axis. In this case, a = -r and b = r, since the plate