Simple Machines: Pulley
A pulley is a type of simple machine that utilizes one or more wheels and a rope to redirect or amplify an input force, making it easier to lift heavy objects. The simplest pulleys consist of a rope passing over a smooth surface, such as a hook, allowing the user to apply force in one direction to lift a load in another. More complex pulley systems, such as a block and tackle, can provide a mechanical advantage, allowing a smaller force to lift heavier loads, although this typically requires pulling a longer length of rope.
Mechanical advantage is a key concept related to pulleys, defined as the ratio of the output force to the input force. Simple machines like pulleys are foundational in engineering and play critical roles in various applications, including cranes, elevators, and even everyday items like glove straps. Despite their usefulness, real-world pulleys are not perfectly efficient; friction and other factors often reduce their effectiveness. Understanding the principles of work, force, and energy is essential for grasping how pulleys operate and their significance in both simple and complex machinery.
Simple Machines: Pulley
FIELDS OF STUDY: Classical Mechanics
ABSTRACT: A pulley is a simple machine consisting of a wheel with a rope, belt, or cord laid over it, typically resting in a groove. The simplest pulley is anchored to a surface, such as a ceiling beam, and is used to redirect a force without amplifying it. Systems with movable pulleys provide a mechanical advantage by amplifying the input force. The more movable pulleys there are in a system, the greater the advantage.
PRINCIPAL TERMS
- actual mechanical advantage: the ratio of the output force of a machine to the input force, taking into account friction and other factors that limit the efficiency of real-world machines.
- efficiency: the measure of how effective a machine is at transforming or transferring energy, quantified as the ratio of the actual performance of the machine to an idealized, theoretical version of it.
- ideal mechanical advantage: the ratio of the output force of a machine to the input force, ignoring friction and other factors that limit the efficiency of real-world machines.
- joule: the International System of Units (SI) unit of work and energy; one joule is equal to the work done by a force of one newton acting across a distance of one meter.
- net force: the sum of all of the forces acting on an object.
- newton: the International System of Units (SI) unit of force; one newton is equal to the force required to accelerate a one-kilogram mass at one meter per second per second.
- power: the rate of work or energy transfer over time.
- work: the use of force to successfully displace an object from its original position.
What Is a Pulley?
A pulley is a simple machine that uses one or more wheels and rope to redirect or amplify an input force. The simplest version consists of little more than a rope draped over a surface that it can slide easily over, such as a smooth metal hook, and serves only to redirect force. For instance, a rope thrown over a rafter in a barn will allow a person to pull down on one end of the rope, aided by gravity, to lift a load upward at the other. This is an impromptu example of a simple pulley system, with the rafter acting as an anchored pulley. More complex arrangements with moving pulleys produce a mechanical advantage, amplifying an input force. In exchange, the person applying the force has to move a greater amount of rope.
A simple machine is a device that makes a task—typically, moving some target object—easier by amplifying and directing an input force. A two-pulley system, with one anchored and one able to move freely, is the simplest possible way of arranging ropes over wheels to produce a mechanical advantage.
Simple Machines and Mechanical Advantage
Generally, simple machines differ from other, more complex machines by virtue of being the simplest ways to multiply a force. The multiplication of a force by a tool or machine is called mechanical advantage. Mechanical advantage (MA) is calculated as the ratio of output force (Fo) to input force (Fi):
A mechanical advantage of more than 1 indicates an amplification of force.
The six classic simple machines are the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. Gears are sometimes included in this category as well. Simple machines often act as the building blocks for more complex machines, sometimes called compound machines. An industrial crane, for example, might in fact be made up of several pulley systems and levers. An elevator operates using a system of pulleys connected to motors that can lift it via gear trains and wheels and axles.
Work Remains Constant
A force is said to do work if it displaces an object from its original position. If the net force acting on an object is positive in any direction, the object will move in that direction. A heavy bale of hay resting on the floor of a barn experiences a net force of zero. If somebody attaches a pulley system and starts lifting the bale upward, however, the net force is positive in the direction of movement. Work is being done.
In physics, work and energy are both measured in joules (J). One joule is equal to the work done or energy transferred when a force of one newton (N) moves something over a distance of one meter. Power is simply the rate of work performed over time. It is measured in watts (W), with one watt being equivalent to one joule of work done or energy transferred per second.
Mathematically, work (W) is equal to the product of the strength of the force (F) applied, the displacement of the object from its original position (s), and the cosine of the angle between the directions of force and displacement (θ):
W = F × s × cosθ
For the same reasons that energy can only be transformed, not created or destroyed, the total work done at either end of a simple machine must remain constant. In practical terms, this means that a simple machine that amplifies force via mechanical advantage must reduce the displacement caused by that force. The angle between the force and the displacement determines whether that force caused the displacement. A pulley system might enable a farmer to lift a heavy hay bale, but the bale will be displaced a shorter distance per unit of force applied.
Imperfect Machines
In the real world, there is no perfect machine. Even the most well-oiled pulley does not transmit forces perfectly; some energy is always lost to friction. In the real world, a distinction is made between ideal mechanical advantage (assuming a perfect machine) and actual mechanical advantage (taking into account friction and other forces).
The difference between a theoretically perfect machine and its real-world counterpart is measured in terms of its efficiency, which is the ratio of the actual, measured performance of a machine to its theoretically perfect performance. A perfect machine would have an efficiency value of 1. Many modern pulleys are designed with so much precision as to be near-perfect. However, impromptu pulley systems, like the hypothetical rope-over-rafter example above, are subject to large efficiency losses due to friction.
Sample Problem
Pulleys only offer mechanical advantage—i.e., an amplification of the input force—when they are able to move. The simplest pulley arrangement that both redirects and amplifies force consists of two pulleys. One pulley is anchored to the rafter, and the rope goes up from the person pulling and over this pulley, then descends to wrap up under a second pulley. The rope is then attached to the rafter and tied in place around a simple metal hook. In this arrangement, pulling the rope causes the second pulley to move up and down. Instead of being attached to the end of the rope, the load is attached to the movable pulley. This arrangement is often called a "block and tackle."
A farmer uses a block-and-tackle system to help lift a hay bale that is too heavy to move using a single anchored pulley. With a force of 90 newtons, he pulls his end of the rope 1 meter straight down. The hay bale attached to the movable pulley rises half a meter straight up. Assuming friction is not a factor, what is the mechanical advantage of this system?
Answer:
A simple machine generates mechanical advantage through the work-distance trade-off. The presence of the second pulley splits what would have been a single length of rope attached to the load into two lengths of rope. As a result, the total distance the rope needs to be moved to lift the load is doubled. Thus, with the block-and-tackle system, pulling one meter of rope straight down only lifts the bale half a meter.
Knowing that total work is constant at either end of a machine, use the equation for calculating work to compare input force and displacement to output force and displacement:
W = F × s × cosθ
Wi = Wo
Fi × si × cosθ = Fo × so × cosθ
Because the force is being applied straight down and the hay bale is being displaced straight up, the angle (θ) between the two is 180°. The cosine of 180 is −1. However, this value will remain the same on either side of the equation, so it is not necessary to include it in the calculations. Divide both sides by cosθ to simplify the equation:
Fi × si = Fo × so
Rearrange this to solve for the one missing variable, output force:
Plug in the known values from the description and calculate:
The output force is 180 N. To calculate the mechanical advantage (MA), simply calculate the ratio of the output force to the input force:
A mechanical advantage of 2 makes sense. The output displacement is one-half the input displacement, so the output force must be doubled to ensure the same total work at both ends of the machine.
Determining the mechanical advantage of any pulley system is even simpler than these calculations, however. In any system with multiple pulleys, the mechanical advantage will always be equal to twice the number of movable pulleys. In an ideal, frictionless world, each movable pulley would increase the mechanical advantage by a factor of 2.
Pulleys Are Everywhere
Simple machines permeate every aspect of daily life. Pulleys are common, especially in industries such as shipping, where large loads often need to be lifted. Some examples of pulleys are less obvious. Consider the hook-and-loop strap on a winter glove’s cuff. The strap goes through a slot and bends back over itself; pulling on the end of the strap makes the plastic slot act as a tiny movable pulley.
More importantly, the principles of power, work, force, and mechanical advantage, along with the basic structures of classic simple machines, provide the foundation for a deeper understanding of the more complex machinery encountered in everyday life.
Bibliography
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