** PULLEYS**

**BACKGROUND INFORMATION:**

Archimedes was a great mathematician and engineer who was born in 287 BC in Syracuse, Sicily. He is credited with the development of many of our modern day mathematical and mechanical principles (such as Archimedes' principle, the concept of pi, and geometric proofs) and machines like the lever, a pump, and pulleys. According to Plutarch, Archimedes had stated in a letter to King Hieron that he could move any weight with pulleys; he boasted that given enough pulleys he could move the world! The king challenged him to move a large ship in his arsenal, a ship that would take many men and great labor to move to the sea. On the appointed day, the ship was loaded with many passengers and a full cargo, and all watched to see if Archimedes could do what he said. He sat a distance away from the ship, pulled on the cord in his hand by degrees, and drew the ship along "as smoothly and evenly as if she had been in the sea."

Archimedes understood the concept of mechanical advantage and how to use it to move or lift heavy objects with less force. The mechanical advantage of a machine is the ratio of the output and input forces that are used within the machine. A good mechanical advantage is a number that is greater than 1. The output force generated should be larger than the input force used to start the machine. For a simple machine like a pulley or a lever, these forces are easy to determine. For a pulley, the output force is the weight of the object and the input force is the force applied on the end of the rope.

A force is a push or a pull on an object or
machine that may cause an action. Forces are measured in
units of pounds-force (lbf) or newtons (N).
A newton is a kilogram times a meter divided by seconds
squared (N = kg m/s^{2}). A force
is a vector; it has both a magnitude (numerical
value) and a direction. If an object is held
up by a rope, for example, it has a
force called the weight (the mass times the gravitational acceleration)
acting downward, and it causes a tension in the rope,
which acts upward. If the object is in equilibrium,
the downwards weight of the object will be equal to the
upwards tension. When something is in equilibrium, it means
that it is not moving; all the forces are balanced.
A book sitting on a table is in equilibrium. The
weight of the book is balanced by the reaction force of
the table on the book. The study of objects with
forces in equilibrium is called Statics.

Archimedes knew that he could improve his mechanical advantage for lifting or moving an object by using pulleys. A pulley is an object that is usually round with a smooth groove around its outside edge. A pulley transfers a force along a rope without changing its magnitude. When engineers work with pulleys, they often assume that the rope through the groove of a pulley moves smoothly and evenly, without catching. They say it moves without friction. When two rough surfaces are rubbed together (like two wooden blocks), they become warm; the heat is caused by friction. If the two surfaces were slicked with oil and then rubbed together, they would move much more smoothly and very little heat would be generated. There is much less friction. Engineers also assume that the pulley and rope weigh very little compared to the weight on the end of the rope, so they can ignore these two weights and make their calculations with only the heavy weight on the end of the rope.

The first figure shows a single pulley with a weight on one end of the rope. The other end is held by a person who must apply a force to keep the weight hanging in the air (in equilibrium). There is a force (tension) on the rope that is equal to the weight of the object. This force or tension is the same all along the rope. In order for the weight and pulley (the system) to remain in equilibrium, the person holding the end of the rope must pull down with a force that is equal in magnitude to the tension in the rope. For this simple pulley system, the force is equal to the weight, as shown in the picture. The mechanical advantage of this system is 1! The output force is the weight to be held in equilibrium and the input force is the applied force.

Figure 1 and Figure 2

The pulley in the first figure is a fixed pulley; it doesn't move when the rope is pulled. It is fixed to the upper bar. In the second figure, the pulley is moveable. As the rope is pulled up, it can also move up. The weight is attached to this moveable pulley. Now the weight is supported by both the rope end attached to the upper bar and the end held by the person! Each side of the rope is supporting the weight, so each side carries only half the weight (2 upward tensions are equal and opposite to the downward weight, so each tension is equal to 1/2 the weight). So the force needed to hold up the pulley in this example is 1/2 the weight! Now the mechanical advantage of this system is 2; it is the weight (output force) divided by 1/2 the weight (input force).

Each additional figure shows different possible pulley combinations with both fixed and moveable pulleys. The mechanical advantage of each system is easy to determine. Count the number of rope segments on each side of the pulleys, including the free end. If the free end is to be pulled down, subtract 1 from this number. This number is the mechanical advantage of the system! To compute the amount of force necessary to hold the weight in equilibrium, divide the weight by the mechanical advantage! In the third figure, for example, there are 3 sections of rope. Since the applied force is downward, we subtract 1 for a mechanical advantage of 2. It will take a force equal to 1/2 the weight to hold the weight steady. The fourth figure has the same two pulleys, but the rope is applied differently and it is pulled upwards. The mechanical advantage is 3, and the force to hold the weight in equilibrium is 1/3 the weight. Each additional figure shows another possible pulley configuration and lists the force necessary to lift and hold the weight still. The mechanical advantage for the system will be the number in the denominator of the force. Check out the pulley problems in the interactive section to test your knowledge of the mechanical advantage of pulleys!

Figure 3 and Figure 4

Figure 5 and Figure 6

Figure 7 and Figure 8

These systems are known as simple pulley systems because they use the same rope throughout the system. If the pulleys were attached with several different ropes (not one continuous rope), the system would be a complex pulley system. The force necessary to hold a complex pulley system in equilibrium would have to be computed using other Statics methods. Once it was known, however, the mechanical advantage of the system would still be computed by dividing the weight to be held by the force applied to hold it!

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