Friction: Finding the angle at which an object slips

Imagine you have a ramp, made out of steel, and an object made out of wood. At some point, when raising that ramp, you'd expect the object to slide down the slope. How would you find the exact angle that an object's frictional force equals the pull of gravity? This is the point where if you raise the ramp any further it the object will slide.



Where to start? Well first lets look at some friction basics:

The image above is a free-body diagram of the forces acting on the wooden object. We know there is a gravitational force directed downward, and we need to isolate the components of the gravitational force. These are represented by Fg,parallel, and Fg,perpendicular.

At this point of equilibrium, we know that the maximum frictional force is applied by the ramp on the object, and any more force from gravity would be unbalanced; therefore moving the object. At this point there exists a ratio between the normal force(Fn) and the frictional force(Fs,max), it is called the coefficient of static friction. This value is higher for more friction, and less for less friction. For example, two blocks of sand paper will exhibit a high coefficient of static friction, while two ice blocks will have a very small value. But friction isn't the only factor we must consider, there is also the weight of the object acting perpendicular to the ramp. The weight is factored in as the normal force, because the force of the object pushing down is met by the ramp pushing back equally. Here is the equation for this ratio:

This value will be known, you can change it according to the friction between the surfaces of your two objects.

Now here's the tricky part. We know that Fn = Fg,perpendicular and that Fs,max = Fg,parallel because they have cancel out; they must cancel out for the object to remain in equilibrium. Equilibrium is simply the state of an object not moving. That pen on your desk is in equilibrium. This state implies there are no net forces acting on the object.

We also know:

As that is how you define those components of gravity relative to the ramp. Because of division, the forces of gravity(Fg) actually cancel out. This leaves Sine divided by Cosine. Which if you know enough about trigonometry is mathematically equivalent to the Tangent of that angle:

We're almost there. We have to use the inverse tangent of the coefficient of static friction to solve for theta, our angle:

There you go. Given a coefficient of static friction, we can find the exact angle that our object will slide if any infinitely small climb in the angle occurs. In other words, we found the limit of the static friction--the point where friction cannot hold the object against the forces of gravity anymore.