Levers and pulleys

Try opening a door by pushing near its hinges, then near its handle. The same force has different effects. What gives the advantage: door length, where you push, or the push's direction?

A seesaw can balance a heavy person on a short arm against a light person on a long arm. Comparing forces alone is not enough: their ability to turn the board about the pivot matters. A rigid object that can turn about a support is called a lever. One force may tend to turn it clockwise, another anticlockwise. First identify the axis and possible turning directions, then compare their effects.

Extend a force arrow in both directions to obtain its line of action. Draw a perpendicular from the axis to that line: its length l is the moment arm. Why this distance? A push along the door towards its hinges barely turns it, even with your hand far from the axis: the line of action passes through the axis and the moment arm is zero. Distance to the application point equals the moment arm only for the appropriate force direction, not always.

The turning effect increases with both force and moment arm. Its measure is the torque magnitude M = Fl. Doubling one factor while keeping the other fixed doubles torque; a line through the axis gives l = 0 and zero torque. Force in N times arm in m gives N·m. Torque and work have the same dimensions but different meanings: no displacement is specified here, so the torque unit is called a newton-metre, not a joule.

For a stationary lever not to start rotating, opposite turning effects must balance. For two such forces, F₁l₁ = F₂l₂. Twice the moment arm requires only half the force: the products stay equal. Include every torque, including the lever's gravity; ignore it only if instructed, or if its line of action passes through the axis and its torque is zero. The formula describes rotational equilibrium about a fixed axis.

Pass a rope over a wheel with a fixed axle: pull down and the load rises. This is a fixed pulley. In the ideal model, a massless inextensible rope with no friction has equal tension on both sides. During slow lifting, the pull therefore equals the load's gravitational force: there is no force gain. The pulley makes pulling more convenient by changing its direction. Each metre the load rises requires a metre of movement of the free end.

Now attach the load to a pulley that can move, supported by two vertical rope strands. Each ideal strand has the same tension T, giving 2T together. For slow lifting and negligible pulley weight, 2T = mg, so the free-end pull is T = mg/2. The movable pulley thus gives a twofold force gain. Inclined strands or a heavy pulley require us to revise this equation.

Have we obtained work for free? Raise a movable pulley by 1 m: each supporting strand shortens by 1 m. The free end must therefore take up 2 m of rope. Half the force over twice the distance leaves Fs unchanged. This is the golden rule of mechanics: an ideal simple machine trades force for distance without reducing work. Tracking rope length reveals the reason, not just the rule.

A real pulley has axle friction, bending rope and deforming parts. The same lift therefore requires more input work than in the ideal model. A smaller effort may still make a task possible, but it does not mean lower total energy expenditure. The ideal model gives a clear limit: it shows the maximum force gain and helps us account separately for extra energy needed by a real device.

A force of 50 N acts with a moment arm of 0.4 m. Find the force producing the opposite torque with a moment arm of 0.1 m. Neglect the lever's weight.

The first force's torque is 50·0,4 = 20 Н·м.

For equilibrium, the second torque must also be 20 N·m. Thus F₂ = 20/0,1 = 200 Н.

The moment arm is four times shorter, so the required force is four times greater.

The force is 200 N: a moment arm four times shorter needs four times the force for the same torque.

Torque M = Fl depends on distance to the force's line of action; even a force applied far from the axis can have zero moment arm.

Equal opposite torques explain why a long moment arm lets a smaller force balance a larger one.

A fixed pulley changes pull direction; a movable pulley gains force at the cost of distance. A real machine requires extra input work.

For a real machine, we will compare useful work with total input and follow energy transformations.