Simple Machines: Mechanical Advantage

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Mechanical advantage is the amplification of force provided by many machines. It is a ratio of the output force to the input force. Mechanical advantage is generated by applying a smaller input force over a larger distance to one end of the machine in order to generate a larger force over a small distance at the output.

PRINCIPAL TERMS

• conservation of energy: the law that states energy can be neither created nor destroyed, only transformed and transferred. This is why there is a force-distance trade-off that governs mechanical advantage.
• directly related: a relationship between two parameters that exists if increasing one increases the other according to a set multiplier.
• force: a push, pull, or any other interaction that affects the motion of an object. Force is measured in newtons (N).
• fulcrum: the pivot point for a lever.
• inversely related: a relationship between two parameters that exists if increasing one decreases the other according to a set multiplier.
• law of the lever: the law that states if the fulcrum of the lever is closer to the output end than the input end, it will generate a mechanical advantage.
• power: the rate of work (energy transfer) over time. The International System of Units unit of power is the watt (W), which is equal to one joule per second (J/s).
• torque: twisting or rotational force.

What Is Mechanical Advantage?

When a machine amplifies the force put into it, it is said to provide a mechanical advantage. Perhaps the simplest way of generating mechanical advantage is with a lever—a simple machine consisting of a stiff plane balanced on a fulcrum. A playground seesaw is a classic lever. Archimedes (ca. 287–212 BCE), an ancient Greek mathematician and engineer, proved the law of the lever. This law states that the mechanical advantage of a lever is dependent on the relative position of the fulcrum. If it is closer to the output end (where the load is), then the lever will produce a mechanical advantage. If closer to the input, it will instead reduce the input force. The mechanical advantage offered by a lever is directly related to how close the fulcrum is to the output end.

In the real world, there is no such thing as a perfect machine. Even the simplest machines do not transmit forces perfectly; some is lost to friction, or resistance, between moving surfaces. The actual mechanical advantage of a machine is measured against the ideal mechanical advantage, or the theoretically perfect performance, of the same machine. The degree to which a machine achieves its ideal mechanical advantage is its efficiency.

Mechanical Advantage of Simple Machines

All simple machines generate mechanical advantage in one way or another. Generally, simple machines are distinguished from other, more complex machines by virtue of being the simplest ways of generating mechanical advantage. The six classical simple machines are the lever, wheel and axle, pulley, inclined plane, wedge, and screw. Below are descriptions of the different machines and the most useful formulas for determining their mechanical advantage (MA):

• Lever—a long, rigid surface balanced on a fulcrum. It redirects and amplifies force based on the relative position of the fulcrum. Pressing down on a lever with a fulcrum near the target force will result in an amplified upward force:

MA_lever= distance_{input-fulcrum} / distance_{fulcrum-output}

• Wheel and axle—a wheel fitted to a rod. Exerting torque (rotational force) on the wheel transfers the motion to the axle. If the wheel is bigger than the axle, then the force is reduced but the axle moves faster. If the wheel is smaller than the axle, then the force is amplified but the axle spins more slowly:

MA_wheel-axle = radius_wheel / radius_axle

• Pulley—rope hung over an anchored wheel (often grooved to keep the rope from slipping) can form the most basic pulley, which only redirects force. A system with movable pulleys can double the input force for every movable pulley in the system:

MA_pulley = 2 × number of movable pulleys

• Inclined Plane—a sloping ramp. It helps move objects vertically by making it easier to push or pull an object across its surface (not acting directly against gravity), but an object must be moved farther as a result:

MA_ramp= length_ramp / height_ramp

• Wedge—two inclined planes fused together along their bottoms to create a sharp point. Wedges can transform a force applied to their thick end into an amplified force out to either side perpendicular to the input. For instance, an axe swung downward splits a log outward to either side of the wedge:

MA_wedge = length_slope / width_base

• Screw—an inclined plane wrapped around a cylinder. Screws amplify force and direct a circular force into a linear force along the line of the central column. For example, turning a screw into wood with a screwdriver turns the circular motion of a wrist into linear motion into the wood:

MA_screw= circumference of screw / distance between threads

Power, Work, and Energy Remain Constant

Machines are able to produce a mechanical advantage, multiplying the force from input to output, but there is an important trade-off underpinning this multiplication.

A force is said to do work if it moves an object. An object will move if the net force on it—the sum of all forces acting upon it—results in a positive force in any direction. Therefore, a person standing absolutely still on the surface of the earth is experiencing a net force of zero. The force of gravity is performing no work on him or her. If a person is falling straight down, the net force is positive in the direction of gravity and the gravitational force of the earth is doing work.

Work and energy are both measured in joules (J). One joule is equal to the work performed (or energy transferred) when a force of one newton (N) moves something a distance of one meter. Since work is what happens when energy is transferred, the law of conservation of energy applies. The work input into a machine must equal the work output because the energy on either end of the machine must also remain constant. Power, measured in watts (W), is simply the rate of work over time, and it must also remain equal on either side of a machine. One watt is equivalent to one joule per second (J/s).

Work (w) is equal to the force (F) applied times the absolute distance the object acted upon moves from its original position (displacement, s) times the cosine of the angle between the force and displacement (θ):

w = F × s × cosθ

This formula is useful for understanding the force/distance trade-off inherent to the way machines work. For the same fundamental reasons that energy can only be transformed, not created or destroyed, the total work performed at either end of a simple machine must remain constant. In order to keep the work value constant, a simple machine that amplifies force via mechanical advantage must also reduce the displacement (total distance) caused by that force. Force and displacement are inversely related—increasing one by a certain factor will decrease the other by the same factor.

Sample Problem

A student has rented a moving truck to help her carry her things to her college apartment. One of the items she is trying to load into the truck is a massive dresser. The bed of the truck sits 1 meter off the ground, and she cannot lift the dresser even a centimeter. The truck has a built-in ramp, which the student pulls out to a length of 4 meters before it locks in place and she lowers her end to the pavement. What is the mechanical advantage of the ramp she just made?

Answer:

Recall that the mechanical advantage of an inclined plane is equal to the ratio of the length of the ramp (L) to the height it is raised at one end (h):

MA_ramp = L / h

Plug the given values in for length and height:

MA_ramp = 4 m / 1 m

MA_ramp = 4

She can quadruple her input force in exchange for moving the dresser one quarter of the distance. This illustrates the force-distance trade-off.

Mechanical Advantage Is Ubiquitous

Machines permeate every aspect of daily life. These machines constantly redirect, transmit, and amplify forces as needed to move heavy loads or generate additional torque. More importantly, the principles of work and force that govern mechanical advantage form the basis for a deeper understanding of even the most complex machinery one may need to interact with.

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