Posted by Oisin on Monday, August 9, 2010 at 10:03am.
the change of height results in a change of energy:
changePE=changein KE
changePE= change in linear KE + change in rotational KE
that is the guiding energy relationship.
The key, is to relate angular rotation to linear speed, so you can combine the terms and solve. If the wheel does not slip, but rolls, you relate it this way:
condider a point on the wheel at the top, and the bottom, and the center of the wheel.
Assume the velocity down the hill at the center is v. THen the speed at the top must be wr+v (relative motion). The speed at the bottome is wr-v, but that has to be zero.
wr-v=0
or wr=v
Another way. Assume it went down the plane a length L in time t.
average velocity= L/t
maximum velocity=2L/t
rotations= L/2PIr
angular displaement= rotation*2PI=L/r
averge angular speed (rad/sec)=L/rt
final angular speed=2L/rt
so wfinal=2L/rt but t=2L/vfinal
final angular speed=2L/r*2L/vfinal=vfinal/r and again we get
wfinal*r=vfinal
You will have to make an assumption about the wheel for moment of inertia, that is, is it to be modeled as a disk,or a hoop. Most wheels a hoop works fine.