Static Equilibrium

We now want to look at bodies under the influence of gravity. For them to be in static equilibrium all forces acting on them have to add up to 0, and all torques have to add up to 0, too.

Let's look at this applet: Shown are 13 books, , resting on a support (dark gray rectangle). You can move each book (and with it the ones above it) left or right by simply dragging it with the mouse. You can also click on each book and then move it with the arrow keys on your keyboard.

Question: How far can you move the top book out without the pile crashing down? (We will not make the tower crash, but only display the word "Crash" - this way you can quickly correct any mistake without having to start again from the beginning ...) The numbers displayed on the right side give the location of the left edges of each book (in pixels) relative to the right edge of the support structure. Positive numbers mean that the entire book is outside the support.

The first condition for stability of the pile is that the center of mass of the top book is above book 2. Suppose all books each have a length L, and their c.m., xi, is at L/2 (i.e. exactly in the center of the book). Then the relationship between the c.m. of book 1 and book is maximally:

x1 = x2 + L/2

The next consition for stability is that the combined c.m. of the top two books, x1,2 = (x1+x2)/2, has to be supported by book 3. And so on. This leads to an equation for x:

x < (1 + 1/2 + 1/3 + 1/4 + ... + 1/12 + 1/13)L/2 - L = 0.59 L

where x is the location of the left edge of the top book relative to the right edge of the support.

Since we chose L as 100 pixels, x can maximally be 58 pixels.