Brain Teasers
Wheel 2
Consider a vertical wheel of radius 10 cm. Now suppose a smaller wheel of radius 2 cm, is made to roll around the larger wheel in the same vertical plane while the larger wheel remains fixed. What is the total number of rotations the smaller wheel makes when its center makes one complete rotation about the larger wheel?
Answer
The correct answer is 6 rotations. Suppose you rotate the smaller wheel and the larger wheel together without rolling through one rotation clockwise, the small wheel has rotated through 1 turn clockwise. Now rotate the larger wheel back to its original position counter-clockwise through one rotation at the same time holding the center of the small wheel in the original position but allowing the wheel to rotate without sliding. It will rotate through 5 more turns due to the ratio of the circumferences of the wheels. The total number of rotations of the small wheel is 5 + 1 = 6.Hide Answer Show Answer
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Comments
Well I liked it even if nobody else did.
I'm sorry, but I'm not seeing where you're coming up with the +1 to get 6 rotations. This teaser's simply a matter of comparative lengths. The 2cm wheel's circumference is 12.57cm and the 10cm wheel’s is 62.83cm. 62.83/12.57 = 5. In other words, you have to repeat the 12.57cm length 5 times in order to equal 62.83. The 'repeat' of the length is, in other terms, a rotation. If you rolled the small wheel 6 rotations, you would have traveled a total distance of 75.4cm, which is beyond the starting point defined. Draw out a diagram - I think you'll see what I'm talking about. Your explanation confuses me, as it mentions rolling both the small and large wheels together, which is contrary to the stipulations provided in the teaser of the larger wheel remaining fixed. The second half of your explanation I completely agree with, but I don’t know why you started clockwise, then reversed counterclockwise to get 6 – it seems to me that would make the small wheel retrace some of the same path. I liked the quick challenge, but I want to know where you got the +1. Thanks!
The answer is correct, even if the explanation may be difficult to follow.
The easiest way to convince yourself is to take two coins of the same size. Hold one fixed and roll the other one around it. It will go through 2, not 1, complete revolutions when going around the fixed coin once.
Rolling around a circle add one revolution to rolling along a straight line as long as the circumference of the fixed circle.
The easiest way to convince yourself is to take two coins of the same size. Hold one fixed and roll the other one around it. It will go through 2, not 1, complete revolutions when going around the fixed coin once.
Rolling around a circle add one revolution to rolling along a straight line as long as the circumference of the fixed circle.
I have to agree with the answer, though the explanation is beyond tedious. The coin analogy above is very useful. Using the same sized coins, while rotating only 1/2 way around the stationary coin, the rotating one makes one full rotation. It's only traveled 1/2 of it's circumference, but it's point of contact has moved 1/2 way around the stationary coin, making it oriented again exactly as it was at the start - thus one rotation.
Cool! I like it when the answer surprises me!
Paul's explanation of going half way around gave me the clearest visual to see what I missed when solving it. Very nice!
Paul's explanation of going half way around gave me the clearest visual to see what I missed when solving it. Very nice!
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