The Cartesian coordinates of the point of contact of the rear wheel of a certain

Question

The Cartesian coordinates of the point of contact of the rear wheel of a certain bicycle are given by
xrear(t) = t
yrear(t)= sin(t)
where t is the number of seconds after the bicycle started to move, and both coordinates are
numbers of meters. Assume, further that the front and rear axles of the bike are one meter apart.
Find the coordinates of the front wheel as functions of time t .
Include a diagram that shows how the coordinates of the point of contact of the front wheel are
related to those of the rear wheel.
Graph the tracks of both wheels on the same coordinate system for t
between zero and nine
seconds.

Explanation / Answer

I realize there are a few assumptions I have to make in order to solve the problems. The contact point means the points on both wheels were in contact with the ground at the same time at some point in time. After that it would go around the wheel and come to contact with the ground once every cycle (no pun intended).

The front and rear axles of the bike are not one meter apart as stated. yrear(t) = sin(t) indicates radius of the wheel is one meter. If so the front and rear wheels would be in contact with each other which is not possible if it is to be a real bicycle. But, since one meter is given we assume somehow they are not in contact with each other by some magic.

We further assume that both wheels are the same size as they usually are.

1) Find the coordinates of the front wheel as function of time t. The front wheel is one meter in front of the rear wheel, then

xfront(t) = t + 1
yfront(t) = sin(t)

2) Include a diagram that shows how the coordinates of the point of contact of the front wheel are related to those of the rear wheel.

http://i42.tinypic.com/14xcutl.png
Here is a rough sketch of the bicycle. C.p. indicates contact points going around the wheels. As can be seen they are in sync with each other.

3) Graph the tracks of both wheels on the same coordinate system for t between zero and nine seconds.

y = sin(t) indicates that the x-axis is at the axle level not at the ground and at t=0 the contact point is [sin(0)=0] on the x-axis not on the ground.

Graph http://i44.tinypic.com/29kzyc5.png

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.