Circle question

Jomo

Elite Member
You have a circle of radius R/3 that is rotating around a circle of radius R. How many revolutions does the smaller circle make to go around the bigger circle exactly one time?

MarkFL

Super Moderator
Staff member
Do you mean the smaller circle is rolling without slipping around the circumference of the larger circle? If so, 4 times.

Jomo

Elite Member
Do you mean the smaller circle is rolling without slipping around the circumference of the larger circle? If so, 4 times.
Your assumptions are correct. Before I say if your answer is correct or not can we please wait for at least one other response.

This problem was on a past SAT exam and 4 was NOT a choice.

MarkFL

Super Moderator
Staff member
Your assumptions are correct. Before I say if your answer is correct or not can we please wait for at least one other response.

This problem was on a past SAT exam and 4 was NOT a choice.
A slightly more general form of this question was given at a university math competition in which I competed as a student way back when. I thought of the Moon orbiting the Earth and was one of a very few who got it correct. It stuck out in my memory.

Harry_the_cat

Senior Member
Yes I agree with MarkFL. 4 times.

Subhotosh Khan

Super Moderator
Staff member
You have a circle of radius R/3 that is rotating around a circle of radius R. How many revolutions does the smaller circle make to go around the bigger circle exactly one time?
This is also a very common "dynamics" (kinematics) problem. The coin at the center said have a "pseudo-rotation".

Jomo

Elite Member
Yes, 4 times is the answer. I suspect that the author of the question and all of the reviewers thought the answer was 3 (as 4 was not a choice on the SAT exam). Do you guys ever make mistakes??

MarkFL

Super Moderator
Staff member
Oh yes, I make mistakes. Interestingly, I recently posted a variant of this question on a non-math related site I help admin, and no one got the correct answer, and didn't believe me until I demonstrated it with an animated image.

Harry_the_cat

Senior Member
My first thought was 3, but then thought that was far too easy for a SAT question. I drew a diagram, where P was the point on the smaller circle where the circles originally touched. The next time P touched the big circle (one third of the way around the big circle), the small circle had rotated 4/3 times. Lovely question!!

Jomo

Elite Member
My first thought was 3, but then thought that was far too easy for a SAT question. I drew a diagram, where P was the point on the smaller circle where the circles originally touched. The next time P touched the big circle (one third of the way around the big circle), the small circle had rotated 4/3 times. Lovely question!!
It would have been more lovely of a question if the SAT exam had 4 as one of their choices.

Harry_the_cat

Senior Member
Yes that error should have been picked up before it went to print. I wonder how much time was wasted on that question trying to get a "matching" answer. It invalidates the whole test really.

Jomo

Elite Member
Yes that error should have been picked up before it went to print. I wonder how much time was wasted on that question trying to get a "matching" answer. It invalidates the whole test really.
The SAT people just removed the question from the scoring once they were told that the question did not have a valid answer. I agree that the whole test was therefore invalid.