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## Simple Machines: Force Transformers

### Definition

A simple machine is a device that acts to increase or decrease the amount of force output by the machine, Fo, compared to the force input to the machine, Fi.

· Energy is still conserved

Although a machine like a lever can increase the force, it does not create any new energy, because energy is a conserved quantity. The amount of work input into the machine must equal the amount of work output by the machine (minus some unavoidable energy loss due to friction).

Wo = Wi (ignoring friction losses)

Since work is force multiplied by the distance that the force is exerted,

Fodo = Fidi

In order for work out to equal the work in, the product of force times distance must remain the same. This means that if the force increases, then the distance must decrease, and vice-versa. In fact the change in force is inversely proportional to the change in distance.

(ignoring friction)

Thus, these machines represent a trade-off:

· You can amplify your force, but only by paying the price of having to exert the weaker force over a longer distance.

The mechanical advantage of a machine is the factor by which the force is increased.

A machine that doubles the force has a mechanical advantage of 2. The Ideal Mechanical Advantage (IMA) is the theoretical result achieved when there is no friction. In that case we can use the formula derived above to get

The Actual Mechanical Advantage (AMA) will always be less than the IMA because of friction losses:

AMA < IMA

Because the exact amount of frictional losses cannot be predicted, the AMA can only be precisely determined by actually measuring the input and output forces for an actual machine.

We can calculate the IMA exactly, but we do so knowing that the actual result will be less. Even though the IMA cannot be achieved in practice, it is still a useful calculation because it tells us the minimum amount of force that we will need in a given situation, and often the frictional losses are small enough that the IMA is a pretty good approximation.

### The Lever

Levers are divided into three classes, depending on the relative arrangement of

1. The fulcrum (the fixed pivot point)

2. The load (the output force).

3. The effort (the input force)

#### Class 3

Notice that a class 3 lever is the only one that does not amplify the input force. Rather, it is used to increase the distance that the load is moved, but of course this means that the input force must be larger than the output force.

#### IMA of Levers

For levers, the distances that the forces act through are proportional to the lengths of the two lever arms, or

Proof: In the figure below, the angles are the same on both sides of the fulcrum, and so the ratio of arc length s to radius r is the same, because . Thus

### Wheel and Axle, Pulleys, and Gears

A wheel and axle is really just a lever, because at any instant there is a tangential force on the wheel and axle, acting on a lever arm equal to the radius of the wheel or axle. The long lever arm is the radius of the wheel, and the short lever arm is the radius of the axle. Note that both the wheel and the axle are cylinders.

Just as with levers, the mechanical advantage is the ratio of the lever arms, which is this case are the radii:

· The ratio of diameters is the same as the ratio of radii, so you can use diameters if it is more convenient.

This relation holds true if the two cylinders are on the same shaft, as pictured above, or connect by a belt (as with pulleys), or in tangential contact (as with gears). The key difference between pulleys and gears is that pulleys turn in the same direction, but gears in contact rotate in opposite directions.

#### Systems of Pulleys: The Block and Tackle

Lifting a heavy object by running a rope over a fixed pulley does not offer any mechanical advantage. A fixed pulley only changes the direction of the force, allowing you to stand in a convenient place. By adding a movable pulley we can gain mechanical advantage.

In the example above, two strands now support the weight, which means that the tension in the rope is half the weight. Of course, the gentleman in the picture now has to move his end of the rope twice as far to move the mass the same distance as before. It is possible to make a mechanical advantage of many different (integer) values by arranging different combinations of moving and fixed pulleys. The thing to look for is to see how many strands are supporting the weight (think of a free-body diagram).

### Inclined Plane

An inclined plane is a common method of trading force for distance. The amount of work needed to raise a heavy object to a given height does not depend on the path taken, but by sing a ramp w can do the work with less force. Suppose we want to lift a mass m to a height h. We know that the work required is mgh. Indeed, if we lifted it straight up we would need an average force of mg exerted over a distance h , so by W = Fd we get W = mgh. Now suppose that we use a ramp at an angle q from the horizontal.

We will need to exert a force that overcomes the component of the weight that is in the direction of the ramp. This turns out to be mg sin q, which is less that mg. So far so good, but now instead of just lifting it a height h we have to push it a distance d along the hypotenuse of the triangle. We can find d in terms of h and q as follows:

As expected, the distance over which we need to apply the force is increased by exactly the same factor of sin q by which our force is reduced. Notice that the work done remains the same:

#### IMA of an inclined plane

Because the input force is reduced by a factor of sin q (see above), we can conclude that for inclined planes

### The Screw

A screw is a variation of an inclined plane (or a wedge) in which the incline is wrapped around a shaft. The pitch of a screw is the distance between its threads, and this is the distance that the screw will advance with one complete rotation. To calculate to IMA you need to know the pitch, but you also need to know the length of the lever arm being used to turn the screw. This lever arm could be the radius of a screwdriver handle, of the length of the handle on a vise or a C-clamp. This will allow you to calculate how far the screw will advance for a given lever movement.

Suppose the lever arm is L and the pitch of the screw threads is p.The trick is to consider one full revolution. For one revolution the end of the lever travels a distance equal to the circumference of the circle:

di = 2pL

For one full revolution the screw will advance a distance p, so

do = p

Then the IMA is

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Last Updated 07/08/04 by JWB