Simple Machines: Wheel and Axle

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: A wheel and axle consists of a broad disk mounted on a stiff rod so that when one rotates, the other does too. Typically, the rod is attached to the center of the disk. When the wheel spins, the force is amplified and transmitted to the axle. Spinning the axle rather than the wheel reduces the force but moves the wheel a greater distance.

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 effectively a machine transforms or transfers energy, quantified as the ratio of its actual performance to that of 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 Wheel and Axle?

A wheel and axle is a simple machine consisting of a disk or wheel mounted on a stiff rod so that when one spins, the other does too. A wheel and axle transfers rotational force (torque) from one part to the other and may also amplify this force. Alternatively, a wheel and axle may be used to amplify the distance the wheel rotates while reducing the force. A car provides examples of both uses. When a driver turns the steering wheel, the force is amplified as it is transferred to the steering column. Meanwhile, the car’s transmission, a system of axles, wheels, and gears, reduces the force output by the engine in exchange for increasing the distance covered by the tires.

A simple machine is a device that makes a task easier by amplifying and directing an input force. In general, simple machines are the simplest possible systems that generate mechanical advantage—that is, amplify an input force. The six classic simple machines are the lever, the inclined plane, the pulley, the wedge, the wheel and axle, and the screw. Gears are sometimes included as well. Simple machines often serve as the building blocks for more complex machines, sometimes called compound machines. Factory assembly lines, for instance, can be broken down into a series of ramps, gear trains, pulleys, and wheels and axles. Even the screw, typically considered a simple machine itself, acts much like a combination of an inclined plane and a wheel and axle.

Trading Force for Distance

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. 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

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 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 equal to one joule of energy transferred or work done per second.

This formula for work is useful for understanding the force-distance trade-off inherent in how simple machines function. 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. To keep the work value constant, a simple machine that amplifies force via mechanical advantage must also reduce the displacement caused by that force. Spinning the wheel of a wheel and axle amplifies torque on the axle but reduces the distance the axle rotates. Conversely, spinning the axle reduces the output torque but increases the rotational distance covered by the wheel.

Friction and Imperfect Machines

In the real world, there is no perfect machine. Even the simplest machines fail to transmit forces perfectly; some energy is always lost to friction. Thus, 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 an ideal machine and its real-world counterpart is measured in terms of its efficiency. Efficiency is the ratio of the actual, measured performance of a machine to its ideal performance.

Calculating Mechanical Advantage

The ideal mechanical advantage (MA) of a wheel and axle depends on the relative dimensions of the wheel and axle. The mechanical advantage provided by applying an input force to the wheel is equal to the ratio of the radius of the wheel (r_w) to that of the axle (r_a):

The mechanical advantage provided by applying an input force to the axle is calculated using the inverse:

Because what matters is the relative size of the two components, diameter or circumference can be used as well, as long as the dimension and unit of measurement are the same for both.

Sample Problem

Ships use a device called a capstan to lift heavy objects such as anchors. Before the widespread use of motors, most capstans were simply a vertical drum mounted on an axle to which several sailors would attach bars, forming the spokes of an impromptu wheel. The sailors would push on their rods in unison to spin the axle, wrapping a rope or chain around it and hauling the load attached to the other end. Calculate the mechanical advantage provided by a capstan if the axle is one foot in diameter and the "wheel" formed by the bars is four feet across.

Answer:

Calculate the mechanical advantage using the standard equation for the mechanical advantage of a wheel-and-axle system in which the force is applied to the wheel:

Because the diameter (d) of a circle is directly proportional to its radius (d = 2r), it is possible to use the values for the diameters without any conversion:

This capstan would provide a mechanical advantage of 4, meaning it would amplify the force applied by the sailors to the bars by four times. The larger the wheel is relative to the axle, the greater the mechanical advantage generated by spinning the wheel.

A Fundamental Machine

Due to the simple construction of the wheel and axle and its easy amplification of force, it has found countless applications throughout modern society. A mechanical winch used to lift a bucket from a well is a wheel-and-axle system, while a doorknob acts as a wheel and axle to unlatch a door. Numerous motor-powered devices feature wheels and axles connected to gear-and-pulley systems to transmit and redirect the force of the motor to its desired ends. Essentially a circular version of a lever, the wheel and axle is among the most fundamental of the simple machines.

Bibliography

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