Two problems from a competition Please help

[wataru611]

1.Consider a circle A of radius 3 and a circle B of radius 1.The circle B is inside A and tangent to A. The circle B rotates so that,without slipping, it revolves once about A in the clockwise sense. Find the number and sense of rotations of B.

 

[lewando]

So, where are you having trouble? Show what work have you done, so we can begin to help.

 

[wataru611]

For number 1, I was thinking that when B revolves clockwise,A will also rotate by the same distance,so i was thinking that the answer will be 1.5 times and counterclockwise because then B will have traveled 3pi and A will also have traveled 3pi which adds up to the circumfrence of A..but the answer turn out to be two times which i do not get.

 

[Dick]

For number 1, I was thinking that when B revolves clockwise,A will also rotate by the same distance,so i was thinking that the answer will be 1.5 times and counterclockwise because then B will have traveled 3pi and A will also have traveled 3pi which adds up to the circumfrence of A..but the answer turn out to be two times which i do not get.

I don't think you are supposed to take A to rotate at all. Picture it fixed. The edge of B traces out a path that's 3 times its circumference and it rotates counterclockwise. But while it's doing that its center rotates once clockwise around the center of A. What's the total rotation of B about its center? Try sketching a picture.

 

[wataru611]

um..thanks Dick..but i do not quite get the significance of its center rotateting once clockwise around the center of A. Wont B still rotate 3 times if A is not moving at all?..Why will be B rotating twice? Thank you so much

 

[Dick]

um..thanks Dick..but i do not quite get the significance of its center rotateting once clockwise around the center of A. Wont B still rotate 3 times if A is not moving at all?..Why will be B rotating twice? Thank you so much

If a circle rotates 3 times counterclockwise and then you twist it back one turn clockwise, don't you get rotation two times? The rotation of the center of B around the center of A is also turning B. I'm not sure words explain it very well. Again, sketch a picture.

 

[wataru611]

Thanks Dick I just tried it with two coins..and you that's the case,but it is still a bit counterintuitive..haha.i think i need some more time to totally get it but you thank you!