Mathematics of pulleys
The mathematics of pulleys involves the study of simple machines designed to change the direction and magnitude of forces, as well as to transmit power. A pulley consists of a rotating cylinder, known as a drum or sheave, over which a rope or chain runs. This system can be utilized in various practical applications, such as construction and ship rigging, where it allows workers to lift heavy objects with less effort. The principles governing pulleys have been explored by mathematicians throughout history, including Archimedes, who famously demonstrated their utility in moving large weights.
Pulleys can change the direction of a force effectively when the input and output forces align along the rope's ends. They can also change the magnitude of forces; for example, a sailor can lift a weight with reduced effort by using a pulley system, although this requires pulling more rope, maintaining the relationship of work done. More complex systems, like block and tackle, employ multiple pulleys to provide greater mechanical advantages, though friction becomes a limiting factor. Additionally, pulleys are essential in power transmission, where a belt or chain mechanism connects multiple drums, enabling the efficient movement of machinery. Overall, the mathematical principles associated with pulleys continue to be relevant in both historical contexts and modern engineering applications.
Mathematics of pulleys
Summary: Pulleys provide mechanical advantage and help people do work.
A pulley is a simple machine consisting of a cylinder, called a “drum,” “wheel,” or “sheave,” rotating on an axle, and a rope, chain, or belt running over the cylinder without sliding. Pulley drums often have grooves and ribs that prevent their ropes from sliding over the edge. People use pulleys in three ways: to change directions of forces, to change magnitude of forces, and to transmit power. Pulleys are used in building and construction, ship rigging, and within belt-driven mechanisms.
Mathematicians have investigated many aspects of pulleys. There is evidence that Archimedes of Syracuse used a compound pulley to move a ship and studied the related theories. He famously expressed: “Give me a place to stand and I will move the Earth.” While his mechanical inventions brought him recognition among his contemporaries, he seems to have preferred pure mathematics. Guidobaldo del Monte reduced systems of pulleys to levers. Guillaume de l’Hôpital investigated the equilibrium of a pulley system, and mathematicians continue to explore his pulley problem using algebra, geometry, trigonometry, and calculus. A mechanical tide-predicting machine, which incorporated pulleys, is attributed to William Thomson, who later became Lord Kelvin.
Changing Directions of Forces
In an example of this use of pulleys, construction workers often attach pulleys to roofs of buildings. A builder standing on the ground can pull down on one end of the pulley’s rope and a weight on the other end will move up as the drum rotates.
The vectors of input and output forces always go along the two ends of the pulley’s rope. This means that a pulley can change the direction of a force within the plane that is perpendicular to the pulley’s axle but not sideways from that plane. The builder can also stand inside the building, pulling the rope through a window, or on the roof pulling horizontally, as long as the triangle formed by the worker, the weight, and the pulley’s drum is perpendicular to the pulley’s axle.
Changing Magnitudes of Forces
When a pulley is used to change the magnitude of a force, its axle is attached to the weight, and the pulley moves up together with the weight. For example, a sailor can attach one end of a line to a yardarm, string it around a pulley’s drum attached to a weight, and pull the other end up, standing on the yardarm. The sailor will only have to apply the force equal to one-half of the weight.
Does the other half of the force disappear, breaking the conservation of energy law and the work-energy theorem? No, it is distributed to the other, attached end of the rope. Moreover, the sailor will use half the force, but pull enough line to cover twice the distance the weight is lifted. The total work, which is equal to the product of the force and the distance, will be the same as in the fixed pulley case:
Changing Directions and Magnitudes of Forces: Blocks and Tackles
Because it is much easier to work for longer than to increase one’s force, movable pulleys are widely used. A block and tackle is a pulley system where the rope zigzags through movable and fixed pulleys. Depending on the way the tackle is rigged, it can provide a force advantage with the factor of two, as in the example above, or 3, 4, 5 and so on. At first sight, it would seem that a block and tackle can reduce the force required to lift weights by any factor. However, friction interferes increasingly with more pulleys used.
Marine cadets memorize rigging of common block and tackle systems, and the names of tackles corresponding to force advantage factors: factor 2: “gun”; factor 3: “luff”; factor 4: “double”; factor 5: “gyn.”
Drums for tackles may have multiple grooves to reduce rope friction. When tackles are combined, for example, a double tackle upon a luff tackle, their force advantage factors multiply, in this case, creating the force advantage of 3×4=12.
Transmitting Power
A belt or a chain going in a loop over two or more pulley drums makes all of them rotate when one is rotated. For example, a bicyclist rotates the special pulley drum, called a “crank,” to which pedals are attached. The rotation of this crank is transmitted to the rotation of the rear wheel’s crank, which makes the bicycle move. Using drums of different diameters, such as cranks on a sports bicycle drivetrain, can produce a force advantage.
Until the mid-twentieth century, factories typically used belts distributing power to individual machines from one central rotating drum, connected to a large steam, turbine, or animal-powered capstan engine. This power transmission system is called “line shaft.” Because most industries have switched to compact electric motors, one is currently more likely to meet this type of a pulley in a museum or a history book. A human-powered capstan is also a popular science or historical fiction trope, used to demonstrate oppression, for example, in Conan the Barbarian and Captain Blood.
Bibliography
Boute, Raymond. “Simple Geometric Solutions to De l’Hospital’s Pulley Problem.” College Mathematics Journal 30, no. 4 (1999).
Hahn, Alexander. Basic Calculus: From Archimedes to Newton to its Role in Science. Emeryville, CA: Key College Publishing, 1998.
Rau, Dana. Levers and Pulleys: Super Cool Science Experiments. Ann Arbor, MI: Cherry Lake Publishing, 2009.