The Science of Friction and the Cart Wheel
Friction as a fourth of the weight, lever-and-axle calculations, and a classification of rubbing surfaces
This sheet is one of Leonardo's systematic studies of friction, illustrated with cart and artillery wheels turning on their axles (poles). He states the law that the friction of dense bodies equals a fourth of the weight rubbed, then works the lever-versus-counter-lever of a wheel and axle numerically by the rule of three, finding that 200 pounds of power move the friction of 800 pounds dragged on the ground. The lower half sets out an ordered classification of rubbing surfaces—hard or soft, rough or polished—and notes that interposing round grains (like sand) makes the motion easy, closer to motion over a liquid.
On this page
Friction is a fourth of the weight
In the simple draggings of dense bodies one upon another, the friction is always equal to the fourth part of the weights rubbed together. Composite draggings, done with levers, are of various natures—simple when one lever acts, composite when many levers move one another—but here Leonardo treats only the simple.
Artillery cart wheel on its axle
An artillery cart 2 1/2 braccia high with an axle 1/4 thick gives 20/8 of lever against one eighth of counter-lever. Multiplying the lever by 4 as many times as it contains its counter-lever, subtracting 1, and dividing the axle's friction by the remainder yields the power the lever must have; of eight thousand, 2000 is the friction, divided by 79.
Friction by the rule of three
With 10 of lever against 1 of counter, one in the lever stands against 10 in the counter-lever. By the rule of three: if one stands against 10, what against 8000 that has 2000 of friction? The answer is 200 pounds of power, which is the friction of 800 pounds dragged along the ground.
A classification of rubbing surfaces
Leonardo lists the pairings that govern difficulty of motion: hard on hard, soft on soft, hard on soft, rough on polished, and so on. Rough hard on rough hard is the motion of utmost difficulty; rough soft on rough soft is of middling difficulty.
Round grains ease the motion
When sand is interposed between the movable body and the plane, round granules make the motion easy, and larger round grains make it easier still. Leonardo notes this belongs less to frictional motion than to motion over liquids—an anticipation of rolling anti-friction elements.