Weekly Brain Teaser | Consider a vertical wheel of radius 10 cm
Every week we feature a Brain Teaser or a thought-provoking question.
If you have one that you would like to share with fellow engineers, please email it to info@engineeringdaily.net.
So, here is this week’s,
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?
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Sorry admin, the correct answer is 5 rotations. The circumference of the 10 cm circle is 20 pi. The distance the 2 cm circle has to travel to complete one complete rotation is 4 pi. 20 pi / 4 pi = 5 complete rotations.
5 is the correct answer.
If you can not visualize it, make yourself some cardboard cut-outs and give it a spin. Empirical evidence will always trump endless babbling.
I looked at this problem again and it seems that I misinterpreted the question.
I assumed that, from the initial point of contact between the two wheels, one rotation would equal the time it takes for that initial point on the smaller wheel to make its way back to the larger wheel. This resulted in one rotation becoming a 432 degree turn, while the question defines a rotation as a 360 degree turn.
When recalculated, I found that the former solution was correct, with exactly 6 rotations to encircle the larger wheel.
Are you sure it isn’t 5?
I was trying to figure out the brain teaser and I think that there is a mistake in the solution.
“The center of the smaller wheel is 12 cm from the center of the larger wheel, so the center of the smaller wheel has to travel a distance of approximately 75.4 cm (circumference of a circle = pi x radius x 2) before returning to its original position. The center of the smaller wheel, with a radius of 2 cm, travels a distance of approximately 12.6 cm with each rotation, so that the smaller wheel has to go through six revolutions before its center returns to its original position.”
The center of the smaller wheel would move 12.6 cm if it were moving on flat ground.
Since the question implies that it is moving around the larger wheel, it is actually moving along a curve. As the edge of the smaller wheel moves around the circumference of the larger wheel, the center of the smaller wheel moves at a pace that is a 6:5 ratio to its own edge. So if the edge of the wheel makes a single rotation and moves a distance of 4pi cm (or 12.6 cm) the center moves a distance of [(4pi)(6/5)] cm, which is about 15.08 cm. With 5 rotations, it covers the distance of 75.395 cm, or rounded to 75.4 cm which from your calculations that is the distance must travel.
Its like taking an unwinded yoyo and swinging it in a circle. The farther away from the center a point is, the faster it will travel. But no matter where on the string a point is, it takes the same time to make 1 rotation as any other part of the string.
With one rotation of the smaller wheel, the edge travels a distance of 4pi, and it must cover a total distance of 20pi, therefore with 5 rotations the smaller wheel makes its way completely around the larger wheel, bringing its center point with it. Even though the center point travels a longer distance, it also travels at a speed proportional to the distance it must travel. That proportion as I said earlier is 6:5, meaning the center of the smaller wheel is 12cm from the center of the larger wheel, while the edge of the smaller wheel is only 10cm away. 12:10 is reduced to 6:5.
Sorry Antonis, the correct answer is 6 rotations. The center of the smaller wheel is 12 cm from the center of the larger wheel, so the center of the smaller wheel has to travel a distance of approximately 75.4 cm (circumference of a circle = pi x radius x 2) before returning to its original position. The center of the smaller wheel, with a radius of 2 cm, travels a distance of approximately 12.6 cm with each rotation, so that the smaller wheel has to go through six revolutions before its center returns to its original position.
Congrats Shane for being the first one back with the correct answer.
-> 5 ?
Looks like the ratio of perimeters
is the same to the ratio of radiu’s
(Perimeter=2πR),
Each time the 2cm-radius circle rotates..
it’s like being unfolded on the 10cm-rad one..