How To Calculate Levers & Leverage

Virtually everyone knows what a ​lever​ is, although most people might be surprised to learn just how wide a range of ​simple machines​ qualify as such.

Loosely speaking, a lever is a tool that is used to "pry" something loose in a way that no other non-motorized apparatus can manage; in everyday language, someone who has managed to gain a unique form of power over a situation is said to possess "leverage."

Learning about levers and how to apply the equations pertaining to their use is one of the more rewarding processes introductory physics offer. It includes a little bit about force and torque, introduces the counter-intuitive but crucial concept of ​multiplication of forces​, and dials you in to core concepts such as ​work​ and forms of energy in the bargain.

One of the main advantages of levers is that they can be easily "stacked" in such a way as to create a significant ​mechanical advantage​. Compound lever calculations help illustrate just how powerful yet humble a well-designed "chain" of simple machines can be.

​Isaac Newton​ (1642–1726), in addition to being credited with co-inventing the mathematical discipline of calculus, expanded on the work of Galileo Galilei to develop formal relationships between energy and motion. Specifically, he proposed, among other things, that:

Objects resist changes to their velocity in a manner proportional to their mass (the law of inertia, Newton's first law);

A quantity called ​force​ acts on masses to change velocity, a process called ​acceleration​ (​F = ma​, Newton's second law);

A quantity called ​momentum​, the product of mass and velocity, is very useful in calculations in that it is conserved (i.e., its total amount doesn't change) in closed physical systems. Total ​energy​ is also conserved.

Combining a number of the elements of these relationships results in the concept of ​work​, which is ​force multiplied through a distance​:

\(W=Fx\)

It is through this lens that the study of levers begins.

Levers belong to a class of devices known as ​simple machines​, which also includes ​gears, pulleys, inclined planes, wedges​ and ​screws​. (The word "machine" itself comes from a Greek word that means "help make easier.")

All simple machines share one trait: They multiply force at the expense of distance (and the added distance is often cleverly hidden). The law of conservation of energy affirms that no system can "create" work out of nothing, but even if the value of W is constrained, the other two variables in the equation are not.

The variable of interest in a simple machine is its ​mechanical advantage​, which is just the ratio of the output force to the input force:

\(MA=\frac{F_o}{F_i}\)

Often, this quantity is expressed as ​ideal mechanical advantage​, or IMA, which is the mechanical advantage the machine would enjoy if not frictional forces were present.

A simple lever is a solid rod of some sort that is free to pivot about a fixed point called a ​fulcrum​ if forces are applied to the lever. The fulcrum can be located at any distance along the length of the lever. If the lever is experiencing forces in the form of torques, which are forces acting about an axis of rotation, the lever will not move provided the sum of the forces (torques) acting on the rod is zero.

Torque is the product of an applied force plus the distance from the fulcrum. Thus a system consisting of a single lever subject to two forces ​**F₁​ and ​F₂**​ at distances x₁ and x₂ from the fulcrum is in equilibrium when ​F​_​1​x₁ = ​F​_​2​x₂.

•The product of F and x is called a ​moment​, which is any force that compels an object to begin rotating in some way.

Among other valid interpretations, this relationship means that a strong force acting over a short distance can be precisely counterbalanced (assuming no energy losses due to friction) by a weaker force acting over a longer distance, and in a proportional manner.

The distance from the fulcrum to the point at which a force is applied to a lever is known as the ​lever arm,​ or ​moment arm​. (In these equations, it has been expressed using "x" for visual simplicity; other sources may use a lowercase "l.")

Torques do not have to act at right angles to levers, though for any given applied force, a right (that is, 90°) angle yields the maximum amount of force because, to simply the matter somewhat, sin 90° = 1.

For an object to be in equilibrium, the sums of the forces and the torques acting on that object must both be zero. This means that all clockwise torques must be balanced exactly by counterclockwise torques.

Usually, the idea of applying a force to a lever is to move something by "leveraging" the assured two-way compromise between force and lever arm. The force you are trying to oppose is called the ​resistance force​, and your own input force is known as the ​effort force​. You can thus think of the output force as reaching the value of the resistance force at the instant the object starts to rotate (i.e., when equilibrium conditions are no longer met.

Thanks to the relationships between work, force and distance, MA can this be expressed as

\(MA+\frac{F_r}{F_e}=\frac{d_e}{d_r}\)

Where d_e is the distance the effort arm moves (rotationally speaking) and d_r is the distance the resistance lever arm moves.

Levers come in ​three types​.

•​First-order:​ The fulcrum is between the effort and resistance (example: a "see-saw").
•​Second-order​: The effort and the resistance are on the same side of the fulcrum, but point in opposite directions, with the effort farther from the fulcrum (example: a wheelbarrow).
•​Third-order:​ The effort and the resistance are on the same side of the fulcrum, but point in opposite directions, with the load farther from the fulcrum (example: a classic catapult).

A ​compound lever​ is a series of levers acting in concert, such that the output force of one lever becomes the input force of the next lever, thus allowing ultimately for a tremendous degree of force multiplication.

Piano keys represent one example of the splendid results that can arise from building machines that feature compound levers. An easier example to visualize is a typical set of nail clippers. With these, you apply force to a handle that draws two pieces of metal together thanks to a screw. The handle is joined to the top piece of metal by this screw, creating one fulcrum, and the two pieces are joined by a second fulcrum at the opposite end.

Note that when you apply force to the handle, it moves much farther (if only an inch or so) than the two sharp clipper ends, which only need to move a couple of millimeters to close together and do their job. The force you apply is easily multiplied thanks to d_r being so small.

A force of 50 newtons (N) is applied clockwise at a distance 4 meters (m) from a fulcrum. What force must be applied at a distance 100 m on the other side of the fulcrum to balance this load?

Here, assign variables and set up a simple proportion. F₁= 50 N, x₁ = 4 m and x₂ = 100 m.

You know that F₁x₁ = F₂x₂, so

\(x_2=\frac{f_1x_1}{F_2}=\frac{50\times 4}{100}=2\text{ N}\)

Thus only a tiny force is needed to offset the resistance load, as long as you are willing to stand the length of a football field away to get it done!