The Coolest Commercial, Check it out, it's awsome! Options

http://home.attbi.com/~bernhard36/honda-ad.html

yeah, yeah, yeah. It's a honda commercial, but you will enjoy it. trust me. (IMG:style_emoticons/default/mueba.gif)

Read this after:

http://www.dailytelegraph.co.uk/news/main..../13/ixhome.html

[Jeff Krieger]

My previous solution to the "tire rolling uphill" question was incorrect.
Here's the correct (I hope) solution. If you can find a mistake or know of a
simpler solution, let me know.

Let the tire sit just at the bottom of a ramp which has a positive slope. If
the tire's center of mass is a distance r from the tire's center and the
center of mass is located (initially) at the origin of the x-y coordinate plane (see my
diagram below), it's height (y) above ground is given by y = r(a-sina)sinb +
r(1-cosa)cosb. Here "a" is the angle formed between the radius of the center
of mass and the angle -pi/2 + b. "b" is the angle that the ramp makes
with the horizontal axis.

Since the tire's mass (m) and the acceleration due to gravity (g) are both
constant, the critical points of the tire's potential energy (mgh) equation
are determined completely by the the height of the tire's center of mass
above the ground h = y = r(a-sina)sinb + r(1-cosa)cosb. For fixed r and b,
dy/da = r(sinb)-r(sinb)(cosa) + r(cosb)(sina).

For the specific case where r = 4 and b = pi/6, dy/da = 2sqrt(3)(sina) -
2(cosa) + 2 which has critical numbers a = 0 + 2n(pi) and a = 4/3(pi) +
2n(pi) where n is an integer. a = 0 corresponds to the tire's center of mass
being at the origin and having 0 potential energy, a = 4/3(pi) corresponds
to the tire's center of mass being pi/6 radians after TDC (see my diagram
below) and is a local maximum. If at this point the tire is given a slight
uphill push, then it will roll uphill until it reaches a = 2(pi) radians
which is a local minimum for the potential energy equation. The rotation of the tire from a = 4/3(pi) to a =
2(pi) radians moves the center of the tire 4(2 - 4/3)pi units up the hill.

(IMG:http://persweb.direct.ca/aschwenk/diagram.jpg)

Here is a graph for y = r(a-sina)sinb +
r(1-cosa)cosb for the specific case when r
= 4 and b = pi/6. (In this graph x = a).

(IMG:http://persweb.direct.ca/aschwenk/PE2.jpg)