2 circles of radius r_{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?
Belts and Circles
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.
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 Circle_{2}, Point C is where the line is tangent to Circle_{1} and Point D is where the line is tangent to Circle_{2}.
We know that:
Angle ACD is 90
Angle BDC is 90
AB is length D1+D2
AC is length r_{1}
BD is length r_{2}.
Without knowing any of the angles is this enough to determine length of CD?
Assuming you don't know if any sides are parallel. Let say Point A is the origin, Point B is the center of Circle_{2}, Point C is where the line is tangent to Circle_{1} and Point D is where the line is tangent to Circle_{2}.
We know that:
Angle ACD is 90
Angle BDC is 90
AB is length D1+D2
AC is length r_{1}
BD is length r_{2}.
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.
Re: Belts and Circles
Oh yeah, American trapezoid Both radii are perpendicular to the tangent, thus parallel.
Re: Belts and Circles
oh yah. so [theta_{1}] = [theta_{2}] that makes it simple to calculate.