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Simple machines are devices we use to make work easier. Simple machines include such things as door knobs, can openers, clothespins, pencil sharpeners, wheelbarrows, and staplers. Simple machines are also used in heavy equipment (e.g. cranes, bulldozers, and backhoes), sporting goods (e.g. baseball bats, hockey sticks, and bicycles), and automobiles (e.g. steering wheels, door handles, transmissions, differentials, engines, and jacks). In short, it is very difficult to go through the day without using a simple machine. In fact, machines have become so commonplace that we often have difficulty recognizing one when we see it. This course is designed to help you understand the various types of simple machines, how they work, how to identify them, and hopefully, to appreciate just how valuable they are to our everyday world.
Definition:A simple machine is a device that either multiplies or redirects a force. They are called simple in order to separate them from complex machines. A complex machine is nothing more than two or more simple machines working together.
There are six types of simple machines:
Recall the laws of thermodynamics:
In essence what this says is that even though a simple machine may feel like it is making work easier, in actuality it is not. In fact, because we add additional friction when using a machine, it actually requires more work to use a machine than it would if you did not use the machine. Read on to find out why.
Climbing a Hill:If you wanted to get to point C on the illustration below you have two options: start at A and go straight up, or start at B and walk up the slope. Which is easier? Well, actually, they both take the same amount of work. The only thing you have control over is if you put forth a lot of effort over a short distance (situation A-C) or a little effort for a long distance (situation B-C). Lets exam what work really is (at least in scientific terms).
Work, as used in the sciences, is defined as a force (F) acting through a distance (d):
The SI unit of work is the Joule (J), and is the amount of energy required to lift a one kilogram object one meter.
How is it that the path from A-C takes the same amount of work as path B-C? In A-C we have to exert a lot of force to get up the vertical side, but we don't have to go very far. Let's suppose that the force required is 100 Newtons and the distance is 4 meters, the amount of work done is 400 Joules:
Getting from B-C does not require near as much force as A-C, but we do have to move a considerably longer distance. In fact, the actual force and distance are 25 Newtons and 16 meters. Calculating the work gives:
Notice that the work is the same in both cases. If we wanted to we could lengthen the ramp even more. This would result in using even less force, but we would have to move an even longer distance. The choice is yours; do you want to put out a lot of effort and get it over with, or do you want to take it easy (i.e. small effort) and take a long time to finish? Either way, the work done is the same (OK, in actuality it requires more work with the machine because we have to overcome some friction, but for purposes of this class we will ignore the effects of friction).
Now, there are lots of ways to multiply two different numbers to get 400. In this particular illustration we used 100 x 4 and 25 x 16. There are numerous other combinations (infinitely many to be precise) that will multiply to give 400 (e.g. 1 x 400, 2 x 200, 20 x 20, 800 x ½, etc.).
In every situation involving a machine there is a fundamental law that must be obeyed; the amount of work you get out of a machine is equal to the amount of work you put into it:
Now, since work in defined as a force acting through a distance (W=Fd), we can substitute into the above equation:
Where F is the force in Newtons (N), d is distance in meters (m), and the subscripts E and R represent effort (what you put in) and resistance (what you get out) respectively. In the preceding problem, going from point A to C did not require the use of a machine, only 100 N applied over 4 m. Going from B to C used a ramp (a type of simple machine) and required 25 N to be applied of 16 m. Substituting these values into the above equation results in a true statement:
Let's go back to our original definition of a simple machine, "a device that either multiplies or redirects a force." Note how we used 100 N in the first situation, but only 25 N in the second. The ramp effectively reduced the effort force by a factor of four. However, we had to pay for that ease by walking a greater distance, four times the original distance to be exact. This is the trade-off a machine requires of us: we may not have to apply as much force, but we must apply it for a greater distance.
Mechanical AdvantageWhen trading effort for distance, we gain the advantage of multiplying our effort force using what is called a mechanical advantage. The mechanical advantage is the ratio of the resistance force to the effort force and may be calculated using either force or distance:
The mechanical advantage tells you how much you can multiply the effort force, and at the same time, how much distance it will cost you. A machine with a MA of 4 will increase the effort force by a factor of four, and at the same time inform you that such a force multiplier will require you to move the effort distance four times farther than the resistance distance.
DETAILED ANALYSIS OF THE SIX SIMPLE MACHINESInclined plane
An inclined plane is an even sloping surface. The inclined plane may slope at any angle between the horizontal ( ---------- ) and the vertical ( | ). The inclined plane makes it easier to move a weight from a lower to higher elevation. An inclined plane is illustrated below:
The mechanical advantage of an inclined plane is equal to the length of the slope divided by the height (aka rise) of the inclined plane. (This assumes that the effort force is applied parallel to the slope.) As an example, for the inclined plane illustrated above, assume that the length of the slope (S) is 15 feet and the height (H) is 3 feet. The mechanical advantage would be:
While the inclined plane produces a mechanical advantage, making it feel like it is easier to "lift" the object, it does so by increasing the distance through which the force must move. In the above situation the mechanical advantage of 5 multiplies our effort force five fold, but we have to pay for it by moving five times as far. In other words, if you can lift a 50 kg drum from the ground up onto a one meter tall platform, using the inclined plane above would allow you roll a 250 kg drum up the ramp to a height of one meter, but you would have to roll it for 5 meters..
Most schools have a wheelchair ramp for handicapped access. As an application activity, see if your students can correctly calculate the MA of your school's wheelchair ramp. Does it meet the architectural standard of no more than a one inch rise in twelve inches of length? What is the minimum mechanical advantage of a wheelchair ramp?
Wedge
The wedge is a modification of the inclined plane. Wedges are used as either separating or holding devices.
There are two major differences between inclined planes and wedges. First, in use, an inclined plane remains stationary while the wedge moves. Second, the effort force is applied parallel to the slope of an inclined plane, while the effort force is applied perpendicular to the direction of travel when using a wedge.
A wedge can either be composed of one or two inclined planes. A double wedge can be thought of as two inclined planes joined together with their sloping surfaces outward. Single and double wedges are illustrated below:
A common example of single wedge is a simple door stop or a wheel block. Though the "person holding the door" below is unique and entertaining, its forward foot is nothing more than a wedge.
Single and double wedges are often used in the timber industry where they aid in the felling of trees and splitting of wood. Note in the first picture below that when the wedge (an ax is nothing more than a double wedge fixed to the end of a handle) moves downward, the work it does is in splitting the log sideways (i.e. perpendicular to the direction of force). The railroad engine on the right uses a rather large wedge to plow snow off of railroad tracks, and in this case works both forward and backward.
The mechanical advantage of a wedge can be found by dividing the length of the slope (S) by the thickness (T) of the big end. As an example, assume that the length of the slope is 20 cm and the thickness is 5 cm. The mechanical advantage is equal to 20/5 = 4. As with the inclined plane, the mechanical advantage gained by using a wedge requires a corresponding increase in effort distance. In this case for every 20 cm you drive the wedge you spread the log 5 cm.
Which of these wedges has the greatest mechanical advantage?
The wedge on the right has the best of both worlds due to its wedge-within-a-wedge design. It starts out with a long thin slope then, once the tip gets things started, the broader top wedge helps to push things apart. (note the same property on the ax in the photo above.)
The photo below of a plane breaking the sound barrier (as shown by the water vapor condensing around the sound barrier) would not be possible without the aid of the wedge shaped nose of the plane. You can find this same aerodynamic feature on most modern automobiles (although I doubt you will see an automobile break the sound barrier). Some older automobiles used wedges in an effort to improve aerodynamics, hold the car to the track, and increase traction and cornering stability. Because of the newness and unfamiliarity of the nose-wedge feature, the cars below had a hard time selling for $3000 in 1970. Today their value is around $200,000. For more details on the airplane picture click here.
If you have a wedge doorstop, have your students determine its mechanical advantage. If you really want to get tricky, try to calculate the mechanical advantage of a butter-knife blade.
Screw
| The screw is also a modified version of the inclined plane. While this can be
somewhat difficult to visualize, it may help to think of the threads of the screw
as a type of circular ramp (or inclined plane). The activity from the workshop
should help you understand how this works and illustrate the property to your
students (i.e. wrapping a paper inclined plane around a pencil). The vertical distance between two adjacent screw threads is called the pitch of a screw. One complete revolution of the screw will move it into an object a distance equal to the pitch of the screw. As an example, assume that you place a ruler parallel to a screw and count 10 threads in a distance of one inch. The pitch of the screw would be 1/10. Since there are 10 threads per inch of screw, the distance between two adjacent screw threads is 1/10 of an inch. Also, remember that one complete revolution of a screw will move the screw into an object a distance equal to the pitch of the screw. Therefore, one complete revolution will move a screw with 1/10 pitch a distance of 1/10 of an inch into an object. |
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In the United States, the convention for describing threads is to give the number of threads per inch. So, for example, in the United States one might ask for a "2-inch quarter-twenty bolt," which would be
Metric sizes are described in a different way, for example "M3.5 × 1.2". The number following "M" is the diameter in millimeters; the number following "×" is the pitch (also in millimeters), which is the distance from one thread to the corresponding point on the next thread.
The mechanical advantage of a screw can be found by dividing the circumference of the screw by the pitch of the screw. The gentler the pitch (i.e. finer the thread), the easier it moves, but you have to make a lot of turns. Below are examples of several screws:
Not all screws are typical of what most people think of when they hear the term. Screws appear in cider presses, ships, irrigation head gates, torture devices (the device shown is a thumbscrew), airplane propellers, and household fans.
Screws have special applications when it comes to gearing. Using a "worm" gear (i.e. a screw) has the unique property that only allows the round gear to be driven by the worm gear (i.e. a screw), and not the opposite. If you try to use the round gear to move the worm gear the worm gear will simply bind the round gear and hold it stationary (in this case the worm gear acts like a wedge to hold the round gear stationary). Such a situation is very useful in many instances, such as when hoisting heavy objects or making the steering gear box of an automobile.
Lever
A lever is a rigid bar that rotates around a fixed point called a fulcrum. The bar may be either straight or curved.
The mechanical advantage of a lever is the ratio of the length of the arc on the effort side to the length of the arc on the resistance side (whew, that sounds technical J). Fortunately there is an easier approximation for the mechanical advantage of a lever: the length of the effort arm divided by the length of the resistance arm. The arm is measured from the fulcrum to the point of effort or resistance, as the case may be.
There are three different classes of levers. The class of a lever is determined by the location of the applied and resistance forces relative to the fulcrum. Each of the three classes of levers will be discussed next.
First-Class Lever
In a first-class lever the fulcrum is located at some point between the effort and resistance forces. Common examples of first-class levers include crowbars, scissors, pliers, tin snips and playground seesaws.
A first-class lever always changes the direction of force (i.e. a downward effort force on the lever results in an upward movement of the resistance force) and may or may not change the force itself. A first-class lever is illustrated below:
With a first-class lever, when the length of the resistance and effort arms are the same the mechanical advantage is 1. In such a situation there is no increase in effort, nor do you have to move a greater distance. The advantage is that the effort side gets pushed downward, thus benefiting from gravity, while the resistance moves upward. When the fulcrum is closer to the resistance force (resistance arm less than effort arm), the effort force is increased, but there is a corresponding decrease in both the distance the resistance force moves and the speed with which it moves. Conversely, when the fulcrum is closer to the effort force (resistance arm > effort arm), the resistance force is decreased and there is a corresponding increase in both the distance the resistance moves and the speed with which it moves.
Second-Class Lever
With a second-class lever, the resistance is located between the fulcrum and the effort force. Common examples of second-class levers include nut crackers, wheel barrows, and certain types of bottle openers.
In a second-class lever the effort and the resistance move in the same direction. When the fulcrum is located closer to the resistance than to the effort, an increase in force results. The mechanical advantage is still the ratio of the effort arm to the resistance arm. A second-class lever is illustrated below:
Third-Class Lever
With a third-class lever, the effort force is applied between the fulcrum and the resistance force. Examples of third-class levers include tweezers, ice tongs, baseball bats, and hockey sticks. In a third-class lever the effort and resistance both move in the same direction. Third-class levers always decrease the output force, but gain in terms of the distance and speed with which the resistance moves. A third-class lever is illustrated below.
The human bicep muscle is a classic example of a third class lever. Similar analogies can be drawn from almost all joints in the human body.
Pulley
A pulley consists of a grooved wheel that turns freely in a frame called a block. A pulley can be used to simply change the direction of a force or to gain a mechanical advantage, depending on how the pulley is arranged.
A pulley is said to be a fixed pulley if it does not rise or fall with the load being moved. A fixed pulley changes the direction of a force; however, it does not create a mechanical advantage. It does however, allow one to take advantage of gravity instead of working against it when lifting an object. A fixed pulley is illustrated below.
A moveable pulley rises and falls with the load that is being moved. A single moveable pulley creates a mechanical advantage; however, it does not change the direction of force.
The mechanical advantage of a moveable pulley is roughly equal to the number of ropes that support the moveable pulley. (When calculating the mechanical advantage of a moveable pulley, count each end of the rope as a separate rope). As shown in the following illustration, two rope ends support the moveable pulley, thus the MA is two.. Therefore, an effort force of 50 Newtons will lift a resistance force of 100 Newtons. One way to think of this is in terms of carrying an ambulance gurney, a stretcher, or a casket. If the load is 100 kg and one person has to lift it, that person lifts all 100 kg. If two people lift the load each person only needs to lift 50 kg. If four people are lifting each one only has to lift 25 kg, and so on. The load is distributed equally among each person lifting. In a movable pulley system you can think of each supporting rope as a person helping to lift the load.
The illustration below has four support ropes, and thus a mechanical advantage of 4. We only need to apply a force of ¼ the resistance (i.e. 25 kg), but we must move the effort four times as far as we want the resistance to move.
Perhaps the greatest example of the use of pulleys is the story of Archimedes and the launching of King Hieron's ship.
Wheel and axle
The wheel and axle is a simple machine consisting of a large wheel rigidly secured to a smaller wheel or shaft, called an axle. When either the wheel or axle turns, the other part also turns. One full revolution of either part causes one full revolution of the other part.
Occasionally people confuse things that look like a wheel and axle but really are not. Consider the pulley, it has a wheel and an axis on which it rotates. Many would think that a pulley is a wheel and axle, but it is not. Note that in a pulley the wheel rotates on an axis, not an axle. Since the wheel is not rigidly bound to the axis it is not a wheel and axle situation, but rather a wheel on a spindle because the wheel spins on the axis. On a typical rear-wheel drive car the rear wheels are a wheel and axle set-up, with the axle being driven by the driveline, which is subsequently driven by the engine. The front wheels are on a spindle and simply spin around as the rear wheels power the car. In a front-wheel drive car the situation is reversed.
The work of a wheel and axle is the result of the larger wheel being used to turn the relatively smaller axle. Such a situation multiplies the effort force, but as always requires one to pay for it by moving the effort a greater distance. A common example would be the steering wheel and shaft of a car or a screwdriver. In some cases the axle is used to spin the wheel, such as in the rear axle and wheel of a car or the twirling of an umbrella. The mechanical advantage of a wheel and axle is the ratio of the radius of the wheel to the radius of the axle as illustrated below. The MA of the illustration is
The MA of 5 in this case means that the in one full rotation the wheel and axle both turn 360 degrees, but the wheel travels 5 times farther than the axle during that one rotation. The pay-out is that the torque on the axle is 5 times greater than the effort applied to the wheel. In the days before power steering it was not uncommon to have a steering wheel with a MA of 20, most modern car steering wheels have an MA of 13.
See if you can recognize the wheel and axle assemblies below. For a classroom application activity, have your students determine the MA of the wheel and axle assemblies in your classroom (e.g. pencil sharpener, door knob, hands on the clock, and the volume dial on the radio).
Though we have tried to use pure examples of simple machines in this tutorial, the fact is that many of the examples are complex machines. Take another look at the thumbscrew in the screw section. The screw is rather obvious, but did you also recognize the wheel and axle (the big wing nut that you use to tighten the clamp)? For some further knowledge, especially with respect to using gears in a machine (gears can be thought of as a modified wheel and axle assembly), take a look at the Inventor's Toolbox web page. If you would like to assess your understanding of machines try taking their Gadget Anatomy assessment. The gadget anatomy page will amaze you with how many simple machines can be found in one common gadget, and it is great practice for your upcoming exam in this course.
We hope you have enjoyed learning about simple machines and will be better qualified and prepared to teach these concepts to your students.
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