## Belts and Circles

Shape and space, angle and circle properties, ...
drwhat
Posts: 41
Joined: Tue Sep 06, 2011 3:56 am

### Belts and Circles

2 circles of radius r1 and r2 are placed such their centers are D1+D2 apart where D1 and D2 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 (D1+D2,0).

Now if we wrap a belt around the 2 circles so its perfectly tight, then L(D1,D2) = length of the belt.
Its easy to see if D1=D2 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 Circle1 for 180 - [theta1] degrees; and will be tangent to Circle2 for 180+[theta2] degrees. I also think that [theta1] = k*[theta2] where k is some function of D1 and D2. But I have no idea about where to start to evaluate the thetas.

Any ideas about where to go to get started?

TripleM
Posts: 382
Joined: Fri Sep 12, 2008 2:31 am

### Re: Belts and Circles

Draw in the radius for each circle which is perpendicular to the tangent, and join the centers of the circles with a line. You now have a trapezium (with two angles of 90 degrees), for which you can calculate the angles with trig.

drwhat
Posts: 41
Joined: Tue Sep 06, 2011 3:56 am

### Re: Belts and Circles

By trapezium do you mean quadrilateral with no sides parallel or with 2 sides parallel?
Assuming you don't know if any sides are parallel. Let say Point A is the origin, Point B is the center of Circle2, Point C is where the line is tangent to Circle1 and Point D is where the line is tangent to Circle2.

We know that:
Angle ACD is 90
Angle BDC is 90

AB is length D1+D2
AC is length r1
BD is length r2.

Without knowing any of the angles is this enough to determine length of CD?
Last edited by drwhat on Tue Nov 08, 2011 3:49 am, edited 1 time in total.

TripleM
Posts: 382
Joined: Fri Sep 12, 2008 2:31 am

### Re: Belts and Circles

Oh yeah, American trapezoid Both radii are perpendicular to the tangent, thus parallel.

drwhat
Posts: 41
Joined: Tue Sep 06, 2011 3:56 am

### Re: Belts and Circles

oh yah. so [theta1] = [theta2] that makes it simple to calculate.