Picture below shows two circles with centers M and N. The smaller circle touches the bigger one inside and has half the diameter. The smaller circle rolls inside the bigger one without friction. Center N moves over the green circle. The circles now touch in Q. Question: -
which way does P travel if it is fixed to the small circle?
This implies an arc on the bigger circle of vp/180 degrees, if the radius is 1. The smaller circle has half the radius, so it's rotation will be double the angle: 2v degrees. Rotation is clockwise. The new position P' of P after rotation is the sum of two rotations: -
1. center N to N', by v degrees counterclockwise
2. point P on the smaller circle, by 2v degrees clockwise Now look at the picture below The new position of P on the Y-scale becomes : Y = 0,5sin(v) + 0,5sin(-v) = 0,5sin(v) - 0,5sin(v) = 0 Conclusion: P moves over a straight line through center M. On the web you may find this nice movie named the "crazy-circle-illusion". You just read the explanation. |
||||||