Simple Machines

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Simple Machines
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Question Marathons 2014
Division B Champion Piedmont Middle School
This event was not held last year in Division C

Simple Machines is an event that is currently being held in the 2015 season.


Event Overview

Simple Machines is an event in which competitors take a written test and use a homemade device (it must be a first class lever) to determine the ratios of unknown masses. The included simple machines are levers, pulleys, wheels and axles, inclined planes, and wedges.

A simple machine is a mechanical device for applying force. They are useful because they can make physical jobs easier, by changing the magnitude or direction of the force, or the distance that the force is applied over.

2008 version

The Written Test

The written test will include topics such as IMA, AMA, efficiency, work, torque, power, and history. A free response answer will be marked as wrong if significant figures are not taken into account, although some graders may give partial credit. Units should always be included.


A force is any action that tends to change the motion of an object. A force has the potential to accelerate any object with mass. The SI unit of force is the newton (N). One newton is equivalent to the force required to accelerate a mass of 1 kilogram by 1 meter per second every second.


Work is the application of a force over a distance. Work represents how much mechanical energy is being transferred from one object to another. The SI unit of work (and energy) is the joule (J). A joule is equal to the energy required to apply one newton of force applied over a distance of one meter.

Work can be negative. For example, if object 2 is transferring mechanical energy to object 1, then the work done by object 1 is negative. Emphasis should be put on the difference between work done on and work done by. The work done on an object refers to the mechanical energy transferred to that object, whereas work done by an object refers to the mechanical energy transferred from that object to another.

The amount of work performed by a force can be represented by the formula W = F \cdot d, in which "W" represents the work applied, "F" represents the amount of force, and "d" represents the distance over which the force is applied.

Conservation of Energy

The law of conservation of energy states that in a closed system with no outside influences, energy is neither lost nor gained. Despite of this, energy can change forms. Work can be converted into heat through friction. Work can be converted into sound. Heat can be converted into work, through engines. But energy is never lost nor gained. This is an important concept to keep in mind.

Mechanical Advantage

Mechanical advantage is the factor by which a machine multiplies force. It is described as the ratio of the output force to the input force. Because of mechanical advantage, machines are able to multiply the input force, resulting in a greater output force, therefore decreasing the amount of input force required to move an object or perform a task. Although machines may have mechanical advantage and can multiply the force applied, due to the law of conservation of energy, they can never multiply the energy (or work) that is applied. They are able to decrease the amount of force required to perform a task by increasing the distance over which the force is applied. If the distance is increased, a smaller force is required to perform the same amount of work.

Mechanical advantage does not have any units.

Mechanical advantage is the ratio of the output force to the input force, as described in the formula MA = {F_o \over F_i}, where "MA" represents the mechanical advantage of the machine, "Fo" represents the output force, and "Fi" represents the input force.

Ideal Mechanical Advantage

Ideal Mechanical Advantage (IMA) is the number of times a machine would multiply an effort force if there were no friction or wear on the machine. For example, if a machine has an IMA of 2, that means that the force applied was doubled by the machine (once again assuming no friction). If the IMA of a machine is 1/2, that means that the force applied was halved by the machine. If the IMA is 1, that means the force applied stayed the same.

However, machines with a high IMA are not always desirable. The higher IMA a machine has, the less distance the load moves in comparison to the distance of the input force. If a machine has an IMA of greater than 1, then the load is being moved less of a distance than the distance of which the force is applied. A machine with an IMA less than one will move an object a further distance, at the sacrifice of force.

The IMA is equal to the ratio of the distance over which the input force is applied to the distance over which the output force is applied. Each type of simple machine has a formula for determining its IMA, as described later in this article. However, the general formula for determining the ideal mechanical advantage is IMA = {d_i \over d_o}, where "IMA" represents the ideal mechanical advantage of the machine, "di" represents the distance over which the input force is applied, and "do" represents the distance over which the output force is applied.

Actual Mechanical Advantage

Actual Mechanical Advantage (AMA) is experimentally determined mechanical advantage which takes friction and wear of the machine into account.

The AMA is experimentally determined and is equal to the ratio of the output force to the input force. The formula for determining actual mechanical advantage is very similar to the general equation for determining mechanical advantage, and is described as AMA = {F_o \over F_i}, where "AMA" represents the actual mechanical advantage of the machine, "Fo" represents the output force, and "Fi" represents the input force).


Efficiency describes the affect of friction and wear of the device on the output work. The law of conservation of energy states that the amount of energy in a closed system is constant. However, some work is always converted into other undesired forms of energy, such as heat.

Efficiency is the ratio of output work to input work, and is normally expressed as a percent. This is described by the formula \eta = {W_o \over W_i}, in which eta (η) represents the efficiency of the machine, "Wo" represents the output work, and "Wi" represents the input work. Efficiency is always less then 100%. Another way to determine efficiency is the ratio of the actual mechanical advantage to the ideal mechanical advantage (\eta = {AMA \over IMA}), which amounts to the same thing.


Torque is the rotational analog of force. It is equal to the force times the perpendicular distance between where the force is applied and the fulcrum (moment arm). The fulcrum is what the body rotates about. If the force is expressed in Newtons and the distance is expressed in meters, then the units of torque would appear to be Joules. However, in order to put emphasis on the fact that torque is not work, the units would actually be Newton-meters. The net torque on a body whose rotational velocity is not changing is 0.


Power represent how fast energy is being transferred or work is being done from one object to another. It is equal to the amount of energy transferred or the amount of work done over the time it took to transfer that energy or to do the work. The SI unit of power is a Watt, which is equal to one Joule being transferred per second.

As described previously, the formula for determining power is P = {W \over t}


P is the power developed
W is the amount of work performed or energy transferred
t is the time over which the work was performed or the energy was transferred

Another useful formula for determining the power a motor is outputting is P=\tau \omega


\tau is the torque the motor is exerting
\omega is the angular velocity the motor's shaft.


Archimedes was credited with discovering the idea of Simple Machines around 3rd century BCE. The original ones were lever, pulley, and screw.

Galileo Galilei was the first person to recognize that simple machines do not change the energy, but only transform it.



Inclined Plane

Wheel and Axle


Significant Figures

See Significant Figures for information about significant figures. When performing calculations, answers must contain the appropriate number of significant figures when requested.

Types of Simple Machines

There are six types of simple machines; pulleys, inclined planes, wheel and axles, levers, wedges, and screws. All except screws are necessary knowledge for the event.


A diagram illustrating the difference between a fixed pulley and a movable pulley.

A pulley is a wheel on an axle that is designed to support movement and change of direction of a cable or belt along its circumference. There are two types of pulleys, fixed pulleys and movable pulleys. A fixed pulley is a stationary pulley that that doesn't move with the load. A movable pulley is a pulley that is freely suspended and moves with the load.

In a single fixed pulley, if a load is attached to one end of the string, the theoretical amount of input force required to raise the load is the same as the force of gravity on the load; it has an ideal mechanical advantage of 1. However, due to frictional losses and wear of the pulley, the actual mechanical advantage of a single fixed pulley is always less than 1. Although using a machine that reduces the force applied may seem ineffective, a single fixed pulley is useful for changing the direction of a force.

A single fixed pulley has an IMA of 1, and therefore can be balanced by placing two identical loads on each side.

Pulleys can be more useful than that when there is more than one pulley wheel to a pulley system. If the load's weight is spread between two strings, there is an IMA of 2. Here is an example of a pulley system using two pulley wheels, with an IMA of 2:


Imagine that you are pulling on the string with the little arrow. If you pulled it 2 feet downwards, the hook will rise 1 foot. This is because there are two strings that lift the hook and only one string that is being pulled. This means that you can lift a heavy load as if it were only half as heavy. Now, imagine if three strings were attached to the hook. The IMA would now be 3. But remember only count the supporting lines, or the lines that are being pulled up. In the diagram, the line with an arrow is being pulled down, so it is not counted in the IMA.

In conclusion, the easy way to tell what the IMA of your pulley system is to count the number of pulley wheels.

Pulleys are easily acquirable at your local hardware store. To make the circular part of the pulley, you could also use wheels from Lego sets (without the tires, of course).

Inclined Planes

"Inclined plane" is just a fancy word for a ramp. To find the IMA of an inclined plane, divide the diagonal length of the ramp by the vertical length of the ramp.


The IMA of this ramp is 60/5=12. Since the circular weight (the one being lifted) is less than 12 times heavier than the square weight, the square weight is able to lift the circular weight. (The picture is not to scale)

Wheel and Axles

A wheel and axle system can be used in many ways: to transport something, to turn something else on the axle, or to turn another wheel and axle.


This diagram shows how to find the IMA of a wheel and axle simple machine. Basically, the IMA of a wheel and axle is the radius of the wheel divided by the radius of the axle.


A lever is, in basic terms, a rigid bar resting on a pivot point, or the fulcrum. To work the lever, effort must be applied to move the load, which is usually opposite the effort, across the fulcrum. There are three types of levers.

  • First Class
    • The fulcrum is in the middle, the effort is on one side, and the load is on the other. An example of a first class lever would be a seesaw or a crowbar.
  • Second Class
    • The fulcrum is to one side, the load is in the middle, and the effort is on the other side. An example of a second class lever would be a wheelbarrow or a nut cracker
  • Third Class
    • The fulcrum is to one side, the load is on the other side, and the effort is in the middle. An example of a third class lever would be tweezers or your elbow.

1st, 2nd, and 3rd class levers respectively

To find the IMA of a lever, divide the distance between the fulcrum and the effort by the distance between the fulcrum and the load. Therefore, if the fulcrum is moved closer to the load, the easier it is going to be to lift a load, and the higher the IMA is.


A wedge is a simple machine that separates two objects by converting downward force to sideways force. Imagine a triangular block of wood and two adjacent bouncy balls. If you put the triangular block between the two bouncy balls and pushed down, it would separate the two bouncy balls. The IMA of a wedge is how far the wedge went down divided by the distance it separated the two objects.


Note that screws are intentionally excluded from the rules

Screws convert rotational force to vertical force. A screw is essentially inclined plane wrapped around a central axis. An example would be a scissor jack. IMA = \frac{2\pi L}{p}, where L is the length of the handle and p is the distance between adjacent screw threads.

The Device

In this part of the test you use a homemade first class lever to determine the ratios of three masses. The masses will be given to you at the event; you are not allowed to bring your own mass.

The goal is to determine the ratios of mass A to mass B and mass B to mass C as quickly and accurately as possible.


What is probably the strongest material to build your lever out of is metal. Unfortunately it is hard to work with, and you may end up with a more wobbly structure then if you had made it out of wood. It is also more expensive. So, if you have never worked with metal before, it may be a better idea to use wood.

If you use wood make sure not to use a wood that is too light (otherwise it might bend), and make sure that your lever is straight. You can always tape a ruler onto your lever, so it might be a good idea to cut the wood yourself just to make sure it's straight.

PVC pipe will bend and may break if the masses are too heavy. However, you might want to take your chances with this material so that you can use a sliding fulcrum design.

Designing your device

Practically all possible designs involve a stand to support your lever. Try and build a sturdy stand.

Here is a list of some possible designs:

1. Get a straight bar as your lever and put its center on an object acting as the fulcrum. This is far by the simplest, but it is not suggested. The lever may slide and get unbalanced, an it will probably be hard to work with. Plus, you will have to put the masses on top of the lever, meaning that it will be hard to tell what mark the objects are on.

2. Build a stand and then hang the lever off of it. This is a good design because it has minimal friction and thus is accurate. Because of it's low friction it may be time consuming, however, because you will spend a long time getting it adjusted just right so that it is balanced.

3. Build a stand and put a bar on top. Drill a hole through your lever and slide the bar through that hole. Effectively it is just like design 1, however it is easier to work with and it is higher off the ground. It has more friction and thus is not as accurate, but it is faster to use then design 2.

4. Build a stand and hang a ring from it that the lever can slide through. This design is the fastest. However, because you are moving the fulcrum and getting more torque on one side then the other, it is also the least accurate. If you use light wood or PVC pipe this error can be minimized.

Some tips

  • Keep in mind that a lever is not balanced when it is level, it is balanced when it is not rotating.
  • Remember that you don't know how large the masses are going to be. Its probably a good idea to keep your lever high off the ground so that you can be ready for any situation.
  • Practice.

Other Events that Involve Simple Machines

External Links

How Simple Machines Work

Calculating IMA for lever, wheel and axle, and pulley

Calculating IMA for incline, wedge, and screw

Simple Machine Sample Exercises

Trial Event Rules

Franklin Institute page on simple machines

Wisc-Oline Mechanical Advantage Presentation