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As part of a mechanical device there is a circular cog, Cog1, that has a diameter...

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Mechanical Engineering

As part of a mechanical device there is a circular cog, Cog1,that has a diameter of 50 cm. From the mechanic’s point of view,when the device is in action, Cog1 rotates anticlockwise and makestwo full turns per minute. There is a red marker on Cog1 to helptrack this motion.

(a) Consider only the placement of the red marker on Cog1. It isdefined by two variables, X and Y . X is the horizontaldisplacement from the vertical line through the center of Cog1 andY is the vertical displacement from the horizontal line through thecenter of Cog1. Let θ be the angle from the positive horizontalaxis to the line between the center of C og1 and the red marker.Assume θ is measured anticlockwise and in radians.

(i) Sketch a representation of this situation.

(ii) Let the red marker begin at θ = 0. Sketch the graphs X(θ)and Y (θ); the horizontal and vertical position of the red markeras functions of θ, as the marker rotates. Clearly label the axes,the maximums and minimums of the functions, and any points wherethe graphs cross the axes.

(iii) Give explicit expressions for X(θ) and Y (θ).

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