 Inclined Planes and Wedges - Maple Help

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Inclined Planes and Wedges

Main Concept Mechanical advantage Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. It is defined as the ratio between the equivalent output force ${F}_{o}$ achieved by the machine, and the input force ${F}_{i}$ supplied to the machine:   . Simple Machine

A simple machine is one of the six basic non-motorized devices identified by Renaissance scientists drawing from Greek texts on technology:

 • Lever
 • Pulley
 • Wheel and axle
 • Screw
 • Inclined plane
 • Wedge

These simple machines provide a mechanical advantage by changing the direction or magnitude of a force. Simple machines can be combined with other devices to form more complicated machines and can thus be considered the building blocks of engineering. Inclined Planes An inclined plane is a simple machine that provides mechanical advantage by reducing the force required to lift an object. However, the reduced force must be applied over a longer distance. The machine consists of a flat surface that is tilted by an angle such that one end of the surface is higher than the other.   For a frictionless inclined plane the amount of effective work done equals the potential energy gained, which is proportional to the vertical displacement $h$, that is, . On the other hand, the actual work done can be calculated as the product of the force applied and the distance, $l$, traveled along the inclined plane:   .   Since the effective force overcome is the object's weight, , the mechanical advantage works out to .   Given that   ,   where, $\mathrm{θ}$ is the angle of inclination, the mechanical advantage can also be represented by:   $\frac{1}{\mathrm{sin}\left(\mathrm{θ}\right)}$,   With the addition of friction opposing the movement of the mass, the mechanical advantage is reduced as some of the input energy is converted to heat. As a result, more effort is required in order to push the object up the inclined plane. Wedges A wedge is a simple machine that is made up of two inclined planes forming a triangular shape. It provides mechanical advantage by converting the force applied at its end into forces perpendicular to the inclined surfaces. As such, a wedge is a good tool to cut objects (for example, an axe), hold objects in place (for example, a doorstopper), and lift objects in order to separate them from the surface they are resting on.   Similar to an inclined plane, the mechanical advantage for a wedge is the ratio of the length of its slope to its height:   .

In the following example, for the sake of simplicity, the coefficient of friction is assumed to be independent of whether or not the object is in motion. Use the sliders to adjust the appropriate parameters. The magnitude and directions of the forces acting on the box are represented by vectors in the diagram.

 Angle $\mathbf{θ}$:  Weight ${\mathbit{F}}_{\mathbit{g}}$:  Coefficient of friction $\mathbf{μ}$:  Applied force ${\mathbit{F}}_{\mathbit{a}}$ (as a fraction of ${\mathbit{F}}_{\mathbit{g}}$):  More MathApps