Question

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 makes two full turns per minute. There is a red marker on Cog1 to help track this motion.

(a) Consider only the placement of the red marker on Cog1. It is defined by two variables, X and Y . X is the horizontal displacement from the vertical line through the center of Cog1 and Y is the vertical displacement from the horizontal line through the center of Cog1. Let θ be the angle from the positive horizontal axis 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 marker as functions of θ, as the marker rotates. Clearly label the axes, the maximums and minimums of the functions, and any points where the graphs cross the axes.

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

Answer 1