Bicycle Wheel Math

Consider a car moving at 30 mph. Now, consider one of the cars wheels:

How fast is each of the following points on the wheel moving: at the top, bottom, right, left, and center?

First, lets look at the center. The center movies in a straight line with the car, at the same speed of the

car, so the speed of the center is also 60 mph:

Now, lets consider the point on the very top of the wheel. It is turning (because the wheel is turning)

and it also moving forward (because the car is moving forward). Since both of these actions move the

point to the right, we can simply add their speeds together. So, the overall speed of the point on the top

is 30+30 mph, or 60 mph.

Now lets look at the bottom. It also both spinning and moving forward. However, it is spinning to the

left, and moving forward to the right. Since they are moving the point in opposite directions, we must

subtract their speeds, giving us 30-30 mph, or 0 mph.

30 mph

60 mph

Next, consider the point on the right side of the tire. It is being spun upwards, and moving to the right.

So, we can add these vectors together to get its overall velocity. This is the same for the point on the

right, expect its spinning downwards.

Since both of the triangles shown above are isosceles right triangles with side lengths of 30 (because

they are moving at 30 mph) the actual speed at that point is 30 mph.

It is interesting to note that if you extend any velocity vector along the rim of the tire, it will intersect the

tire at the very top: (the blue lines show the extensions of the vectors)

Of course, this makes sense because any point makes a right triangle with the top and bottom:

0 mph

So the two legs have to point to the top and bottom of the tire.

Find the area of the shaded region:

The analytic way of solving this is by making b and r into a right triangle, solving for the hypotenuse, and

subtracted the areas of the two circles:

sin

cos

Area of the shaded region=(b2-r2)-r2=b2

This answer makes sense, because essentially we are just pivoting b around the whole r circle, so we

are basically making a circle of radius b, because the movement up, down, and sideways does not affect

the area:

Suppose we have ramps of different slopes, all connected by a point at the top, and we release marbles

from the top onto all the different slopes. Where will the marbles be on each slope at some time t?

What shape do these points make?

b2+r2

This problem can be solved by using the kinematic equation y=.5at2, where a is acceleration and t is

time. Since all the marbles are under the influence of gravity, all of their accelerations are 9.8 m/s2. Also,

we arent really interested in y, were interested in the hypotenuse (r) of the triangle, so the expression

becomes r=.5acos()t2. Since a and t are both constants, this is really just the equation of a circle in polar

coordinates. So, the shape made by all the marbles is a circle!

Now try to solve the homework problems

Start

Some time t