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CHAPTER 3 Simple Machines
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Fig. 3-7.
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Block on a wall.
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The relationship is described by the Pythagorean theorem, which states: a2 b2 c2 3-1
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where a and b are the lengths of the parallel and perpendicular sides, and c is the hypotenuse, or long side described by gravity in our example. Other ways to show this relationship are: p c a2 b2 3-2 a p c2 b2 3-3
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Note that this is the same relationship used to calculate the distance between two points, used in the section about velocity in 2, and equation (3-2) is the same as equation (2-1). Now, back to the ramp and how it changes our use of force. When you push an object across the oor, you only have to counteract its friction and momentum. Ignoring friction, we are only acting against the mass of the object. Lifting the object straight up, we act against the full force of gravity. When we push the object in the direction of the ramp, we are only pushing against the parallel force. Remember that we are ignoring friction, which is a function of the perpendicular force. For a very shallow ramp angle, there is almost no parallel force, and it is easy to push the object up the ramp against gravity. However, it doesn t get
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CHAPTER 3 Simple Machines
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very far o the ground unless we push it a long ways. Lifting straight up, all of the motion is up but it s much harder work. This is where we are trading side-to-side motion for force. The farther we push an object to raise it a certain distance, the less force we have to use pushing it but the longer we have to push.
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A wedge is essentially an inclined plane in portable form (Fig. 3-8). Instead of gravity pushing, we apply our own force. Looking at the wedge, if its sides are at 45 degrees to the towers, then for every meter we push it into the gap, it pushes outward by one meter. This just redirects the force and doesn t make the pushing any easier. If the wedge is long and skinny, however, so that for every two meters you push it down it pushes out one meter, it has a mechanical advantage of 2.
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Fig. 3-8.
A wedge for mechanical advantage.
CHAPTER 3 Simple Machines
You have to push it twice as far, but it generates twice as much force outwards. In the diagram in the corner of Fig. 3-8, you can see that the wedge is just a ramp on its side. The height of the ramp h is the width of the wedge and the length l is the length of the wedge. The mechanical advantage (MA) is de ned as: MA l h 3-4
In terms of energy, we haven t done anything except rearrange some terms. Remember, energy is just force in Newtons applied across distance. We have the energy we are putting into the wedge, Ein, and the energy the wedge is putting into the towers, Eout. The relationship is de ned as: MA Eout m Ein MA m 3-5
where m is distance. On one side, the mechanical advantage adds to the output force. On the other side, we have to move the wedge that much farther. Another way to look at it is as:   F 3-6 E m MA MA where E is the energy, or work being done, and m is still distance. The mechanical advantage MA just shifts the focus of our e orts from the force being applied to the distance we have to move to apply it. The wedge is a di erent way of looking at the inclined plane. A third look is the screw.
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