This time the surface is the thing that tilts. Put an object on a board and raise the board slowly. At some point it must slide, and the question is at what angle.
The forces at work are the same as on level ground. Gravity still points straight down. What has changed is the direction the block is free to move in: only along the surface.
So gravity is split in two, along the surface and perpendicular to it. As the figure shows, the component along the surface is and the perpendicular component is .
As the tilt increases these two move in opposite directions. grows and shrinks. The force trying to slide the block increases while the force pressing it against the surface decreases. That disagreement is what settles the slipping.
The last lecture gave the limit of friction as . So what is on a slope?
What presses on the surface is not gravity itself but only its perpendicular component. The component along the slope presses on nothing; it merely drags the block downhill. So .
That on level ground was only because made equal to 1. What we had learned was a special case.
Raise the tilt in the figure and the pressing component shrinks together with the normal force pushing back. The marker for the limit closes in with them. The limit of friction is , so a smaller means a smaller limit. The more the board is tilted, the weaker the force holding the block in place.
Everything needed is now on the table. Trying to slide the block is ; holding it back, at most, is .
Raise the tilt and the first grows while the second shrinks. Moving in opposite directions, they must meet somewhere. Until they do, friction absorbs everything, and the two arrows in the figure face each other at equal length. The instant it is overtaken, the block starts down.
The condition for slipping is . Look at the two sides. appears in both. And is positive, so both sides may be divided by it.
Dividing gives , that is, . Both and have gone. What remains is the tilt and , which stands for the pair of surfaces.
That dropped out means the weight has nothing to do with the angle of slipping. The two blocks in the figure differ greatly in size, yet they begin to move at the same instant and run down side by side.
The reason it drops out is plain. Make the block heavier and , which tries to slide it, grows. But , which presses it onto the surface, grows by just as much, and so does the limit . Both sides rise in the same proportion, so the angle at which the contest is decided does not move.
The angle at the boundary is called the angle of friction, and it is fixed by . Tilt a board and measure the angle at which the object starts to move, and that alone gives you . No ruler and no spring balance required.
After the slip the friction drops to kinetic friction, so the block accelerates. That acceleration is , and again it contains no mass. This is why the two blocks in the figure descend without parting.