Static friction is really important when we look at how things stay still. It's especially useful when figuring out problems with forces acting on objects that aren't moving. To get a clear picture of these situations, we need to know the basic rules about static friction.

Here’s the main idea:

• The basic formula for static friction is:

  $f_s \leq \mu_s N$

  Let’s break that down:

  • $f_s$ is the force of static friction.
  • $\mu_s$ is the friction coefficient, which tells us how "sticky" the surfaces are.
  • $N$ is the normal force, or how hard the object is being pushed against a surface.

Static friction can be any number between zero and a maximum limit. This limit is found by multiplying the friction coefficient ($\mu_s$) with the normal force ($N$). So, static friction can change to balance out other forces pushing the object, but only up to that maximum amount.

For an object to stay still, all the forces and torques acting on it need to add up to zero:

• The force balance equations are:

  $\sum F_x = 0$

  $\sum F_y = 0$

• The torque equation, which often relates to a point like a pivot or the center of the object, is:

  $\sum \tau = 0$

When friction is part of the mix, we need to include the frictional force in both the horizontal and vertical force equations. For example, if something is about to slide because of a push, we can write:

$F_{applied} = f_s$

Here, $F_{applied}$ is the external force trying to move the object.

To wrap it up, understanding static equilibrium is all about knowing how forces work together, what static friction is, and the factors that come into play. By figuring out the maximum static friction, we can better predict how different structures and systems will behave when they face various forces.