1. Pebble in the tire problem: as you stop your cat at a trafficlight, a pebble

Question

1. Pebble in the tire problem: as you stop your cat at a trafficlight, a pebble becomes wedged between the tire treads. When youstart moving again, the distance betweeen the pebble and thepavement varies sinusoidally with the distance you have gone. Theperiod is the circumferernce of the wheel. Assume that the diameterof the wheel is 24m.

a)      sketch the graph of thissinusoidal function.

b)      Find a particular equation forthe function. (it is possible to get an equation with zero phasedisplacement)

c)      What is the pebble’sdistance from the pavement when you have gone 15m?

d)      What are the first twodistances you have goen when the pebble is 11m from thepavement?

Explanation / Answer

The pebble will trace out a curve called a cycloid. Go here toread about it and see how to plot it: http://mathworld.wolfram.com/Cycloid.html In your case, the parametric equations of the pebble's pathwill be:    {x, y} = {12(t - t sin[t]), 12(1-cos[t])} Each trip around the wheel corresponds to0<t<2. To answer part (c) solve the following for t.    15 = 12(t - t sin[t]) Now plut this value of t into 12(1- cos[t]) to obtain thedistance of 18.15m. For part (d) solve:    12(1- cos[t]) = 11 to obtain the first twodistances of 1.487m and 4.796m

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