Simple Machines: Lever

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Levers are simple machines consisting of a rigid plane balanced on a fulcrum, or pivot point. A playground seesaw is a classic lever. Levers redirect and, depending on the relative position of the fulcrum, amplify or reduce an input force. The closer the fulcrum is to the output end of the lever, the larger its mechanical advantage (force amplification) will be.

PRINCIPAL TERMS

• actual mechanical advantage: the ratio comparing the input force of a machine to the output force, taking into account friction and other factors that limit the efficiency of real-world machines. A mechanical advantage of more than one indicates an amplification of force.
• efficiency: the measure of how effective a machine is at transforming or transferring energy, quantified as the ratio of the actual performance of the machine to an idealized, theoretical version of the same machine. A perfect machine would have an efficiency value of one.
• fulcrum: the pivot point for a lever.
• ideal mechanical advantage: the ratio comparing the input force of a machine to the output force, ignoring friction and other factors that limit the efficiency of real-world machines. A mechanical advantage of more than one indicates an amplification of force.
• joule: abbreviated J, the International System of Units unit of work and energy. One joule is equal to the work done by a force of one newton acting across a distance of one meter.
• net force: the sum of all of the forces acting on an object. Forces with equal magnitude but opposite directions will cancel each other. An object moves in the direction of the net force acting on it.
• power: the rate of work done (energy transfer) over time. The International System of Units unit of power is the watt (W), which equals one joule per second (J/s).
• work: the force moving an object, or the successful transfer of energy. The International System of Units unit of work is the joule (J).

What Is a Lever?

A lever is one of the simplest of the simple machines. It is a tool that redirects and amplifies (or reduces) an input force. It consists of a stiff plank, bar, or rod—anything straight and rigid—balanced over a fulcrum. The fulcrum is usually wedge shaped or round, but anything that the lever can be pushed or pulled against can work. An impromptu lever can easily be made by laying a sturdy stick across a round rock. Even an action as simple as using a screwdriver to pry open a paint can lid is actually a form of lever use. The rim of the can serves as a fulcrum for the steel rod of the screwdriver.

Three types of levers exist and depend upon the location of the fulcrum with respect to the load and the effort. In a first-class lever, the fulcrum is between the load and the effort, as in a seesaw. The second-class lever places the load between the fulcrum and the effort. Lastly, in a third-class lever, the effort is between the load and the fulcrum.

What Is a Simple Machine?

Simple machines like levers are devices that make tasks—typically, moving some target object—easier by redirecting or amplifying an input force into a new output force. A lever is a very simple and effective machine, consisting of nothing more than a rigid plane and a fulcrum on which the lever is balanced. The ancient Greek mathematician Archimedes (ca. 287–212 BCE) came up with the concept of the simple machine.

Generally, simple machines are distinguished from other, more complex machines by virtue of being the simplest possible ways of generating mechanical advantage—that is, multiplying a force. The six classical simple machines are the lever, wheel and axle, pulley, inclined plane, wedge, and screw.

Power and Work Remain Constant

A force is said to do work if it moves an object. An object will move if the net force on it—the sum of all forces acting upon it—results in a positive force in any direction. Therefore, a person standing absolutely still on the surface of the earth is experiencing a net force of zero. The force of gravity is performing no work on him or her. If a person is falling straight down, the net force is positive in the direction of gravity and the gravitational force of the earth is doing work.

Work (w) is equal to the strength of the force (F) applied times the absolute distance the object moves from its original position (displacement, s) times the cosine of the angle between the force and displacement (θ):

w = F · s · cosθ

This formula is useful for understanding the force/distance trade-off inherent to the way machines work. For the same fundamental reasons that energy can only be transformed, not created or destroyed, the total work performed at either end of a simple machine must remain constant. In order to keep the work value constant, a simple machine that amplifies force via mechanical advantage must also reduce the displacement (total distance moved) caused by that force.

Work and energy are both measured in joules (J). One joule is equal to the work performed (or energy transferred) when a force of one newton (N) moves something a distance of one meter. Power, measured in watts (W), is simply the rate of work over time. One watt equals one joule per second (J/s).

Imperfect Machines

In the real world, there is no such thing as a perfect machine. Elements such as friction play a role in affecting the performance of a machine. In the real world, a distinction is made between ideal mechanical advantage and actual mechanical advantage. The former assumes a perfect machine, unimpeded by friction or design flaws, which transmits and transforms a given force perfectly. The latter is based on actual measurements of the ratio of input to output forces.

The difference between a theoretically perfect machine and its real-world counterpart is referred to as its efficiency. This is the ratio of the actual, measured performance of a machine to the theoretically perfect performance of the same machine, or the ratio of the actual to the ideal mechanical advantage. In the real world, levers lose efficiency due to friction between the fulcrum and the rigid bar.

The mechanical advantage (MA) of a lever depends on the ratio of two distances: the distance between the input and the fulcrum (d_in) and the distance between the fulcrum and the output (d_out):

MA_lever = d_in / d_out

Sample Problem

A hiker is climbing some rocks when he slips and falls. He drops his pack, and a particularly large boulder, disturbed by his fall, rolls onto the strap. Unfortunately, the boulder is very heavy and set in its new position—no amount of pushing or shoving budges it. Fortunately, there is a straight, sturdy branch of about 6 feet and an abundance of medium-sized rocks that could serve as a fulcrum. Assuming the hiker is at one end of the 6-foot branch and the boulder at the other, where should he place the fulcrum in order to obtain the maximum mechanical advantage?

Answer:

Recall that the mechanical advantage (MA) of a lever depends on the ratio of the distance between the input and fulcrum (between hiker and fulcrum, di_n) and that between the fulcrum and output (between fulcrum and boulder, d_out):

MA_lever = d_in / d_outIf the branch is 6 feet long and the fulcrum is placed at the midpoint, then d_in and d_out will both equal 3 feet. Note that mechanical advantage is a dimensionless quantity. Therefore, the units used for distance do not matter as long as they are the same for both distances. They will cancel each other out during the division.

MA_lever = 3 ft / 3 ft = 1

A mechanical advantage of exactly one means there is no mechanical advantage: the input force is exactly equal to the output force. If the fulcrum is only two feet away from the input, then d_in is 2 feet and d_out is 4 feet, giving

MA_lever = 2 ft / 4 ft = 0.5

A mechanical advantage of 0.5 means the input force is actually being reduced by 50 percent. Instead of amplifying force, this arrangement amplifies movement: moving the input end of the lever a small distance will result in larger motion at the output end.

Finally, what if the fulcrum is set four feet from the input? In this case, di_n is 4 feet and d_out is 2 feet:

MA_lever = 4 ft / 2 ft = 2

Setting the fulcrum two-thirds of the way toward the output gives a mechanical advantage of 2, meaning that any force applied to the input end of the lever will be doubled at the output end. If this is still not enough force to move the boulder, the hiker could try setting the fulcrum even closer to it.

Levers Are Ubiquitous

Simple machines are used every day to help with the effort put into a number of tasks. Examples of levers abound, with perhaps the most obvious being the classic playground seesaw. A wheelbarrow has the fulcrum at the far end to make lifting large weights easy. Levers can even work in combination. Scissors are made of two first-class levers acting against one another. In the human body, the forearm works as a pair of third-class bone levers attached to the fulcrum of the elbow that get pulled by muscle contractions.

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