Moments and Centers of Mass

Moments of a plate

Symmetry and composite regions

Pappus's theorem

Variable density plates

What is the center of mass of a uniform thin plate, in terms of its moments and area?

A region is symmetric about the vertical line x = 2. What does symmetry tell us about its centroid?

The semicircular region of radius r has area pi r^2 over 2 and moment M_x equal the integral of one half (r^2 - x^2) dx from -r to r. What is its centroid's y coordinate?

Pappus's theorem states that the volume of a solid of revolution equals what?

The triangle with vertices (0, 0), (b, 0), and (0, h) has area bh/2. Its centroid lies at what point?

The rectangle [0, a] by [0, b] has centroid at half each side. What is x bar? Answer like a/2.

A square plate of side 2 with centroid (1, 1) has a unit square of centroid (1.5, 1.5) removed from its corner. The remaining area is 3. What is the x coordinate of the new centroid?

A vertical strip of height f(x) and width dx at position x has area f(x) dx. Its moment about the y axis is x times that area. As an expression, what is it? Answer like x f(x) dx.

The integral from -r to r of (r^2 - x^2) dx equals 4r^3 over 3. What is the value of M_x, one half of that? Answer like 2r^3/3.

Pappus's theorem for rotation about the y axis uses the centroid's x coordinate in the travel 2 pi x bar. As a single symbol, what is that coordinate? Answer like x with a bar described, using the letter x.