Simple Rotational Machines - Maple Programming Help

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Simple Rotational Machines

Concept

 Mechanical advantage Mechanical advantage  is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The mechanical advantage 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.

 Screw A screw is a simple machine that provides mechanical advantage by amplifying and converting a small rotational force into a large linear force. A common screw consists of a threaded cylindrical shaft with the shape of the thread being a  helix. The pitch of a screw is defined as the axial distance between two adjacent threads while the lead of a screw is defined as the axial distance traveled by the screw for one complete, 360°, rotation. For a common helical screw, the pitch is equal to the lead. The work required to turn the screw one complete rotation is:       where ${F}_{i}$ is the applied force and $r$ is the distance from the screw's axis to the point of application of the force. Taken together,  constitutes the applied torque. The work done is represented by:     where ${F}_{o}$ is the axial output force applied by the shaft of the screw on a load. Neglecting friction, the work done by the machine is equal to the work done on the machine, ${W}_{o}={W}_{i}$ so that the mechanical advantage of the screw is the ratio of the circumference of the circle along which the torque is applied , to the lead of the screw, $l$:   .
 Wheel and Axle The wheel and axle is a simple machine that provides mechanical advantage by reducing the amount of force required to move an object. Commonly, a wheel and axle construction requires two cylindrical objects of different diameters that are attached together so that rotation of one object will cause the rotation of the other about the same central axis. Assuming that energy is not stored or dissipated by the wheel and axle, the torque applied by the axle will equal the torque applied to the wheel: ${\mathrm{τ}}_{\mathrm{wheel}}={\mathrm{τ}}_{\mathrm{axle}}.$ Since the torque is the product of the force with the distance between the point of application and the axis of rotation,     the mechanical advantage of the wheel and axle is the ratio of the radius of the wheel to the radius of the axle:   .   The wheel and axle can also be used to increase the linear speed. If the input force is applied at the axle and the amount of time is the same, the wheel will cover a larger linear distance than the axle by a factor of the mechanical advantage.

In the activity below, explore how a screw converts rotational motion into linear motion. Click and rotate the screw head on the left hand side and watch the resulting motion of the screw shaft and threads on the right hand side. The blue ball is there to help you visualize the linear motion of the shaft.



 

A windlass is an application of the wheel and axle that is used for hoisting objects, such as an anchor or a water bucket. The activity below demonstrates the concept of a windlass. Try to use the windlass to bring water to the surface by rotating the wheel on the left hand side.

  

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