### Belts and Circles

Posted:

**Tue Nov 08, 2011 2:49 am**2 circles of radius r

Now if we wrap a belt around the 2 circles so its perfectly tight, then L(D

Its easy to see if D

Now it gets trickier if D1 not equal to D2. My instinct is that if D1 < D2, then the belt will be tangent to Circle

Any ideas about where to go to get started?

_{1}and r_{2}are placed such their centers are D_{1}+D_{2}apart where D_{1}and D_{2}are the diameters of the circles. For ease of explanation lets assume circle one is centered at the origin and circle 2 is centered at (D_{1}+D_{2},0).Now if we wrap a belt around the 2 circles so its perfectly tight, then L(D

_{1},D_{2}) = length of the belt.Its easy to see if D

_{1}=D_{2}then the L(D,D) = pi*D+2*(D1+D2). The belt is tangent to each circle for 180 degrees, then the top and bottom lengths are lines going from (0,D) to (D1+D2,D) and (0,-D) to (D1+D2,-D).Now it gets trickier if D1 not equal to D2. My instinct is that if D1 < D2, then the belt will be tangent to Circle

_{1}for 180 - [theta_{1}] degrees; and will be tangent to Circle_{2}for 180+[theta_{2}] degrees. I also think that [theta_{1}] = k*[theta_{2}] where k is some function of D_{1}and D_{2}. But I have no idea about where to start to evaluate the thetas.Any ideas about where to go to get started?