The last lecture found the limit of friction to be . is a number set by the pair of surfaces, and it changes with what is in contact: wood on wood, ice on metal. The other quantity, , is the one that moves.
is the normal force, the force with which the surface pushes back. It may equally be read as how hard the object presses on the surface.
In the figure, press down on the block from above. The surface pushes back by as much as it is pressed, so and it grows. The marker for the limit then moves away in the same proportion. Ease the hand off and both come back.
has not been touched once. Only is moving. Holding a sheet of paper down makes it harder to rub out a pencil mark for the same reason: the hand holding it down is increasing .
can be reduced as well. When pulling an object, try aiming a little above the horizontal rather than straight sideways.
Split the pull into horizontal and vertical parts. The horizontal component is the part trying to move the object, and the vertical component is the part taking over from gravity. The surface carries only what is left, so .
Raise the angle in the figure and the arrow shrinks while the marker for the limit draws nearer. The size of the pull has not been changed at any point. Only its direction has.
Dragging a heavy load feels lighter when the pull is aimed slightly upward rather than straight along the floor. The horizontal component loses a factor of , but the limit drops by more than that. Raise the angle too far, though, and the horizontal component is gone, and now nothing moves.
Here is something that runs against intuition. Friction does not depend on the area of contact.
The two objects in the figure are identical. The left one lies flat, the right one stands on end. The widths touching the floor are very different, yet the markers for the limit sit at the same distance. Push both with the same force and they begin to slip at the same instant, and they run side by side afterward.
The one touching over a wider area looks as though it ought to have more friction. But the wider the contact, the less pressing each part of it carries. The two effects cancel, and the area drops out of the balance.
Not one letter in stands for an area. The only things in it are , for the pair of surfaces, and , for how hard they are pressed together. That the equation says so is the surest answer there is.
Gather the three together. is how hard the surface is pressed: a force pushing down is added, the vertical component of a pull aimed upward is subtracted. With both at work, as in the figure, .
The limit of friction is nothing more than that multiplied by . Find first, then build . Keep to that order and no arrangement of forces will confuse you.
In the figure the downward push and the angle of the pull are strengthened in turn. The marker is farthest away while the block is being pressed down and nearest while it is being pulled up. The same , the same weight, and this much variation.
The next lecture tilts the surface itself. Once we know what becomes on a slope, the angle at which things start to slide is settled.