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 was last held in the 2015 season. It consists of both a build and test portion involving the fundamental concepts of simple machines, including the types of simple machines, their uses, input and output forces, mechanical advantage, and more.

Contents

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.

Force

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

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

Efficiency describes the effect 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

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

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}

where

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

where

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

History

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.

Lever

Discovered by Archimedes in 3rd century BC along with pulley and screw. Archimedes also discovered the idea of mechanical advantage in a lever.

The first use of a lever by people was opening and breaking into shells and fruits to eat the food inside.

By around 200 BC, scientists like Archimedes were figuring out why levers worked.

Pulley

It is not recorded anywhere when or by whom the first pulley was made. However, it is believed that Archimedes was the first person to have a documented block and tackle pulley system, as recorded by Plutarch.

Inclined Plane

Inclined planes have been used by people since prehistoric times, to lift heavy objects.

The inclined plane was the last simple machine to be recognized as a machine. This is because it is motionless and can be found in nature in the form of hills and slopes. The ancient Greek philosophers that stated the other five simple machines never decided that an inclined plane should be a machine.

However, the inclined plane was finally recognized during the Renaissance along with other simple machines.

The first elementary rules of sliding friction on an inclined plane were discovered by Leonardo da Vinci, written in his notebooks in between the years of 1452 and 1519, but they still remain unpublished.

Wheel and Axle

The wheel was first patented in Ur, Mesopotamia 3500 BC by a potter named Sumerian Sam.

Wedge

The origin of the wedge is still unknown today. One of the first examples of a wedge is a hand axe.

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.

Pulleys

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.

A pulley system with an ideal mechanical advantage of 2.
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 multiple pulleys orchestrated into a system. Both fixed and movable pulleys may be integrated in a system, however, two movable pulleys may not be placed adjacent to each other. A common method for determining the ideal mechanical advantage of a system of pulleys is counting the number of lengths of rope directly supporting the load. Another method is counting the number of pulley wheels in the system. For example, in the picture to the left, there is two lengths of rope directly supporting the load, along with a separate length of rope that is used to apply the input force. Because there is two lengths of rope, the ideal mechanical advantage is 2.

In the pulley system to the left, imagine a force is applied the string with the little arrow. If the string is pulled 2 meters downwards, the hook will rise 1 meter. This is because there are two strings that lift the hook and only one string that is being pulled. This means that the distance over which the input force is applied is twice as large as the distance over which the output force is applied. Because of this, the ideal mechanical advantage (which is the ratio of input distance to output distance) of this pulley system is 2:1, commonly stated as just 2. This means that the input force required to lift the load is only half as much as the force of gravity on the load. For example, if a load of 50 newtons is attached to the hook in the pulley system to the left, only 25 newtons of force would be required to lift the load. Pulleys can be arranged into even larger systems with even more pulleys, increasing the ideal mechanical advantage even further.

Note that in this event, pulley systems are limited to two double pulleys in a single system.

Pulleys can often be purchased at hardware stores. They can also be constructed out of a wide variety of household materials.


Inclined Planes

An inclined plane is a flat surface (a plane) that is on an angle (an incline). Inclined planes are used to raise masses to a higher elevation by extending the distance over which the force used to raise the mass is applied. The ideal mechanical advantage of a inclined plane is the ratio of the diagonal length (the length of the inclined surface) to the vertical length of which the surface rises to. This is illustrated in the formula IMA = {d_i \over d_v}, in which "IMA" represents the ideal mechanical advantage of the inclined plane, "di" represents the distance of the inclined surface, and "dv" represents the vertical distance to which the inclined surface is raised.

An inclined plane with an ideal mechanical advantage of 4.

For example, in the diagram to the right, the inclined surface (the red line) is 4 units long, and the vertical distance to which that inclined surface is raised to (the green line) is 1 unit long, yielding a ideal mechanical advantage of 4. This means that moving the load up the inclined plane theoretically only requires one-fourth of the force required to lift the load up directly. For example, if a 60 newton load is placed at the bottom of the inclined plane, it would theoretically only require 15 newtons of force to move the load up the inclined plane. However, there would generally be significant amounts of friction, resulting in an actual mechanical advantage much less than the ideal mechanical advantage of 4.

Wheel and Axles

An example of a wheel and axle system depicting the formula for finding a wheel and axle system's ideal mechanical advantage

A wheel and axle consists of two parts, a wheel and an axle, in which both parts rotate with each other as a force is transferred from one to another. A wheel and axle system can be used in many ways, including to transport something, to turn something else on the axle, or to turn another wheel and axle.

As depicted in the diagram to the right, the formula for finding the ideal mechanical advantage of a wheel and axle system is IMA = {R \over r}, in which "IMA" represents the ideal mechanical advantage of the wheel and axle, "R" represents the radius of the wheel, and "r" represents the radius of the axle.

Levers

A lever is a rigid bar resting on a pivot point, known as the fulcrum. There are three types of levers, characterized by the position of the input force, output force, and fulcrum in relation to each other:

  • 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. Because the effort force is always a greater distance from the fulcrum than the resistance force (the force exerted on the load), a second class lever always has an ideal mechanical advantage greater than one (refer to the formula described below).
As shown from left to right: 1st class, 2nd class, and 3rd class levers.
  • 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 an elbow. Because the resistance force is always a greater distance from the fulcrum than the effort force, a third class lever always has an ideal mechanical advantage of less than 1 (refer to the formula described below).
The locations of the load and effort arms of all three classes of levers.

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. This can be represented by the formula IMA = {d_i \over d_o}, in which "IMA" represents the ideal mechanical advantage of the lever, "di represents the distance from the input force to the fulcrum (known as the input or effort arm), and "do represents the distance from the output force to the fulcrum (known as the load, output, or resistance arm). These two distances are depicted in each of the three classes of levers in the diagram to the lower right.

"FRE 123" is a commonly used mnemonic to remember the different classes of levers.
One way to remember the different classes of levers is with the mnemonic "FRE 123", in which "FRE" stands for fulcrum, resistance, and effort, respectively. These correspond in order to the class of lever (the "123" in the mnemonic). Depending on which component of a lever (fulcrum, resistance force, or effort force) is in between the other two, the class of lever can be determined by matching that letter with its corresponding number. For example, knowing that the effort force is in the middle between the fulcrum and resistance force in a pair of tweezers, it can be determined that tweezers are a 3rd class lever (because 3 corresponds the "E" in "FRE"; "E" representing the effort force, which is in the middle for tweezers).

Wedges

A wedge is a triangular shaped compound inclined plane. A wedge converts a force applied to its blunt end (the side opposite of where the two inclined surfaces meet) to forces perpendicular to the inclined surfaces.
The IMA of a wedge is the ratio of the depth of penetration to the width of the wedge.
Uses of a wedge include separating two objects, splitting an object, lifting an object, or holding an object in place. Common everyday examples of wedges include knives and axes. The ideal mechanical advantage of a wedge is the ratio of the length of the wedge (often referred to as the "depth of penetration") to the width of the blunt end. This is shown in the formula IMA = {L \over w}, in which "IMA" represents the ideal mechanical advantage of the wedge, "L" represents the length of the wedge, and "w" represents the width of the blunt end. This is depicted in the diagram to the left.

Screws

Note that screws are intentionally excluded from the rules

A screw is essentially inclined plane wrapped around a central axis. Screws convert rotational force to vertical force. 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.

Materials

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

Work, Energy, and Power

Hyperphysics - Simple Machines

Simple Machine Sample Exercises

Trial Event Rules

Franklin Institute page on simple machines

Wisc-Oline Mechanical Advantage Presentation