What is the moment of a force?

The moment of a force about a pivot is the force multiplied by the perpendicular distance from the pivot to the line along which the force acts. It is measured in newton metres. A lever balances when the clockwise moments equal the anticlockwise moments.

Why does a longer handle make a job easier?

Because the turning effect is force times distance. If you grip twice as far from the pivot, you need only half the force to produce the same turning effect, though your hand has to move twice as far.

What are the three classes of lever?

In a first-class lever the pivot sits between the effort and the load, as in a seesaw. In a second-class lever the load sits between the pivot and the effort, as in a wheelbarrow. In a third-class lever the effort sits between the pivot and the load, as in tweezers.

Do mechanical reasoning tests ignore the weight of the lever?

Often, when the question says to treat the bar as light. If a question says the plank is uniform and gives its weight, you treat that weight as a single force acting at the centre of the plank and include its moment.

How Levers Work: The Three Classes and the Moment Rule

A lever is a rigid bar that can turn about a fixed point, the pivot. Almost every lever question on a mechanical reasoning test reduces to one comparison: how big is the turning effect on one side of the pivot, and how big is it on the other?

The Moment Rule

The turning effect of a force is called its moment. It is the force multiplied by the perpendicular distance from the pivot to the line of the force: moment = force x distance. With force in newtons and distance in metres, the moment comes out in newton metres.

A lever balances, and stays level, when the total clockwise moment equals the total anticlockwise moment. That single statement answers most lever questions you will meet.

A worked example: the seesaw

A child weighing 300 N sits 1.5 m from the pivot of a seesaw. Her moment is 300 x 1.5 = 450 N m, clockwise. Where must an adult weighing 450 N sit, on the other side, to balance her?

The adult must supply 450 N m as well, so 450 x d = 450 and d = 1.0 m. Sense check: the heavier person sits nearer the pivot, which is what you expect.

The Three Classes of Lever

Levers are sorted by where the pivot, the effort (the force you apply) and the load (the force you are working against) sit along the bar.

First class: the pivot is between the effort and the load. A seesaw, a pair of scissors and a crowbar pivoting on a block are examples. It can multiply force or distance, depending on which side is longer.
Second class: the load is between the pivot and the effort. A wheelbarrow, a nutcracker and a bottle opener are examples. The effort arm is always longer than the load arm, so the force you need is always smaller than the load.
Third class: the effort is between the pivot and the load. Tweezers, a fishing rod and the human forearm lifting a weight in the hand are examples. The effort arm is shorter than the load arm, so you need more force than the load, in return for moving the load a larger distance and faster.

You rarely need to name the class in a test. You need to find the pivot and the two distances. The classes are useful because they tell you in advance whether the answer should be a force bigger or smaller than the load.

Mechanical Advantage: Force Traded for Distance

For an ideal lever, the mechanical advantage is the effort arm divided by the load arm: the length from the pivot to where you push, over the length from the pivot to the load. A value above 1 means the lever multiplies force.

Take a 2 m crowbar whose pivot is 0.25 m from the rock. The effort arm is the remaining 1.75 m, so the mechanical advantage is 1.75 / 0.25 = 7. A push of 100 N at the free end can lift 700 N. The price is distance: the end you push moves seven times as far as the rock does.

That trade, force gained and distance paid, shows up again in gears, pulleys and hydraulics. It is the single most useful idea in mechanical reasoning.

A Plank With Weight of Its Own

Some questions give the weight of the bar itself. Treat the weight of a uniform bar as one force acting at its centre.

A uniform 4 m plank weighing 120 N is pivoted 1 m from its left end. What load hung at the left end keeps it level? The plank's centre is 2 m from the left end, which is 1 m to the right of the pivot, so the plank's weight gives 120 x 1 = 120 N m clockwise. The load at the left end is 1 m from the pivot, so load x 1 = 120 and the load is 120 N.

Several Loads on One Side

With more than one load, work out each moment separately and add the ones that turn the same way. Take a light 2.4 m bar pivoted 0.8 m from its left end. A 150 N load hangs at the left end, and a 40 N load hangs 0.5 m to the right of the pivot. What load hung 1.0 m to the right of the pivot keeps the bar level?

Anticlockwise moment: 150 x 0.8 = 120 N m. Clockwise moments: 40 x 0.5 = 20 N m, plus W x 1.0. For balance, 120 = 20 + W, so W = 100 N. Sense check: the unknown load sits further out than the 40 N load, and the sum of the two right-hand moments has to reach 120 N m, so a load between 40 N and 150 N is plausible.

Try These Three

A seesaw has a 400 N child 2 m from the pivot on the left. Where must a 250 N child sit on the right to balance? Answer: 400 x 2 = 800 N m, so 250 x d = 800 and d = 3.2 m.
A bar lever has the pivot between a 600 N load, 0.2 m from the pivot, and your hand 1.0 m from the pivot on the other side. What effort balances the load? Answer: E x 1.0 = 600 x 0.2 = 120, so E = 120 N.
In a nutcracker, the nut sits 0.04 m from the hinge and your hand pushes 0.16 m from the hinge with 30 N. What force acts on the nut? Answer: F x 0.04 = 30 x 0.16 = 4.8, so F = 120 N, four times your push.

Three Traps to Watch For

Measuring from the wrong place. Distances are measured from the pivot, not from the end of the bar or from the other load.
Using a slanted distance. Only the perpendicular distance counts. A push along the bar, towards the pivot, has no turning effect at all.
Forgetting a load. With several loads on one side, add their moments together before you compare with the other side.

Try the idea on real questions with the JobCannon mechanical reasoning test, or work through solved examples on the mechanical reasoning practice page. Rope and gear versions of the same trade are in pulleys and mechanical advantage and gear ratios and gear trains.