Simple Machines · MachineLab

Law of the lever

With one crowbar a worker can lift a stone of several hundred kilograms. The bar turns about the wooden block, and the block is the fulcrum O. Drag the force gauge on the handle down to pry the stone up.

The force that turns the lever is the effort F₁; the force that resists is the load F₂ (the stone pressing on the tip — about half its weight, because the stone tips about its far bottom edge). An arm is the distance from the fulcrum to the line of action of a force, not to the point where it acts.

F₁ × L₁ = F₂ × L₂

Move the fulcrum closer to the stone and grip farther out: the effort arm grows, the load arm shrinks, and it gets easier. Try “At a slant”: the force is the same, but its arm is shorter, so more force is needed. One person can exert about 600 N at most; with too little force the bar will not budge.

Classes of levers

By the lengths of the effort arm and the load arm, levers fall into three classes:

Force-saving levers (L₁ > L₂) save force but cost distance. On a wheelbarrow the wheel is the fulcrum, the weight of the sand is the load, and your hands lift the handles. The closer the sand sits to the wheel, the shorter the load arm and the easier it is.

Force-costing levers (L₁ < L₂) cost force but save distance. The boom of a lorry crane pivots at its root, and the hydraulic ram pushes very close to that root — it must push far harder than the pipes weigh, but a tiny extension of the ram lifts the hook a long way.

Equal-arm levers (L₁ = L₂), such as a balance, neither save nor cost force. A steelyard changes its arm by moving the counterpoise: drag it until the beam is level, then read the mark.

Fixed and movable pulleys

A pulley whose axle stays put is a fixed pulley; one whose axle moves with the load is a movable pulley. Drag the force gauge on the rope end to hoist a bucket of bricks: the crane scale shows the mass of the load and the gauge shows the pull on the rope.

Fixed pulley: F = G, and the rope end moves s = h. It saves no force but changes the direction of the force — pulling down lifts the bricks. Changing the rope direction leaves the pull unchanged.

Movable pulley: two strands pull the pulley up together,

F = (G + Gp) / 2 s = 2h

Half the force, but twice as much rope. A fixed pulley is really an equal-arm lever; a movable pulley is a force-saving lever whose effort arm is twice its load arm.

Block and tackle

Combining fixed and movable pulleys both saves force and changes its direction. Count the strands attached directly to the movable block (they are numbered on screen): that is n.

F = (G + Gp) / n s = n h

When n is even the rope is tied to the fixed block; when n is odd it is tied to the movable block. The last fixed pulley the rope passes over only changes direction and carries none of the load — try “Pull up / Pull down”: n stays the same, and so does the pull.

The bigger n, the less force, but the more rope to haul: saving force always costs distance.

Inclined plane

Lifting a 100 kg crate straight onto a truck bed 1.2 m high takes 1000 N. Lay a plank and pull it up along the plank, and it takes far less. Drag the force gauge to pull the crate up the plank.

Without friction, the work done by the pull equals the work of lifting straight up:

F L = G h → F = G h / L

The longer the plank and the gentler the slope, the less force, but the longer the path. In reality friction must also be overcome, so the pull is larger:

F = G h / L + f

Change the surface: steel is slicker than rough wood, and rollers turn sliding into rolling, so friction drops further. Mountain roads, stairs and the threads of a screw are all inclined planes.

Wheel and axle

The windlass is how people in ancient China drew water: a wooden axle is mounted over the well, the rope winds onto the axle, and one end carries a crank. Drag the crank around the axle: the rope wraps on turn by turn and the bucket rises.

The circle traced by the crank is the “wheel”. Wheel and axle turn about the same line, so this is a lever that keeps turning: the fulcrum is at the centre, the effort arm is the wheel radius R and the load arm is the axle radius r.

F × R = G × r

The larger R and the smaller r, the less force, but the hand travels farther: when the bucket rises h, the hand moves h × R / r. Door handles, steering wheels, spanners and screwdrivers are all wheels and axles.

Mechanical efficiency

What we want is to raise the load: the useful work Wuseful = G h. What the pull actually does is the total work Wtotal = F s. The difference, the extra work, goes into lifting the movable pulleys and overcoming friction.

η = Wuseful / Wtotal × 100%

There is always some extra work, so η is always below 100%. Pull the load steadily to the top, press Record data, change a condition and repeat, then look at the table and the η–G curve:

· With the same block and tackle, the heavier the load, the higher η (the extra work hardly changes, so the useful share grows).· Heavier movable pulleys and more friction lower η.· A smoother slope gives a higher η, and so does a steeper one.

Machines in daily life

Tower crane: the jib and counter-jib sit on top of the tower like one giant lever, and the counterweight balances the moment of the load. The farther the trolley runs out, the longer the load’s arm and the less it can lift — the load moment limiter in the cab works with “load × radius”, sounds the alarm on overload and refuses to hoist or travel outward. The hook is a block and tackle: with 4 falls each wire rope carries only 1/4 of the load’s weight, and the maximum load is twice that with 2 falls. Try dragging the trolley.

Well sweep: a water-drawing tool more than 2000 years old. A pole rests on a post, with a bucket at one end and a stone tied to the other. Pull the rope down to lower the empty bucket; the stone’s weight helps lift the full one. Drag the rope to draw a bucket of water, compare the force for lowering and for lifting, and find the best counterweight.

In-class testing