Mechanical systems applied

Lever: effort arm ÷ load arm. Gears: driven teeth ÷ driver teeth. Belt: driven diameter ÷ driver diameter. Pulley block: the number of rope falls supporting the moving block.

The ratios multiply. Stages of 15 and 6 give an overall MA of 90, not 21.

When speed or reach matters more than force — tweezers, a fishing rod, a broom, the human forearm.

The calculation is pure geometry; friction at pivots, bearings and gear teeth takes a share of the effort before it reaches the load.

The distance the effort moves ÷ the distance the load moves, or input speed ÷ output speed for a rotating drive. It comes from geometry alone.

One mesh at a time. Gears sharing a shaft turn at the same speed, so each stage output is the next stage input. The overall ratio is the stage ratios multiplied.

Nothing — its tooth count cancels. It reverses the output direction and bridges a gap; the ratio is set by the first and last gears only.

VR is a ratio of distances or speeds, fixed by geometry. MA is a ratio of forces, reduced by friction. In a perfect machine they would be equal.

Efficiency = MA ÷ VR × 100%, or useful energy out ÷ energy in. VR is what the geometry promised; MA is what was delivered.

Gear teeth roll against each other; a worm thread slides across the wheel teeth, and sliding friction is far larger. A worm can be at 50% where a spur pair is at 97%.

It multiplies. Three stages at 95% deliver about 86%, so removing a stage is often worth more than improving two.

In a screw jack and a worm hoist, where friction stops the load running back when the operator lets go. Making them efficient would need a separate brake.

Input speed ÷ required output gives the overall ratio; split it into stages of about 5:1 or less; choose tooth counts that multiply to it; then check direction and physical size.

A single 30:1 pair needs a gear thirty times the driver's diameter. Two stages of 5.5:1 multiply to the same ratio inside a hand-sized housing.

Use an even number of external meshes, or add an idler. The idler restores the direction and leaves the ratio unchanged.

The same pair of teeth then meet on every revolution, so any imperfection wears a pattern into that pair. A ratio like 5.5:1 spreads the contact.

Largest radius minus smallest radius. That is how far the follower rises, and it sets the stroke of whatever the follower drives.

A stretch of constant radius where the follower holds still while the shaft turns. It is how a valve stays open for part of a cycle and how a feed waits for a needle to clear.

Backwards: draw the displacement graph of follower height against shaft angle, then wrap it round the shaft so each height becomes a radius at that angle.

A knife edge traces fine detail and wears fast; a roller wears well and is usual; a flat follower cannot enter a concave curve, so the profile must suit it.

Effort × effort arm = load × load arm, with both arms measured from the fulcrum.

A longer effort arm must give a smaller effort. On a first- or second-class lever the effort should be less than the load.

It shortens the load arm and lengthens the effort arm at the same time, so the ratio improves twice over.

The user. A large MA needs a long effort arm and therefore a long hand movement, and handles that must open beyond an arm's reach are useless.