a little diversion

I thought this may be of interest.

https://www.practicalmachinist.com/vb/general/ot-math-question-about-ratio-old-sat-exam-395259/

Edited By david bennett 8 on 09/10/2021 18:08:15

Edited By david bennett 8 on 09/10/2021 18:11:49

Very nicely demonstrated.

MichaelG.

Its like asking how many times does the earth revolve on its axis in a year if viewed from outside the solar system. The answer is 366 because on earth one rotation around the sun looks like no rotation as it would look if tidally locked, we only see 365 days. This is also why the sidereal day is slightly less than 24 hours.

Martin C

None of the explanations seemed instantly self evident. Here is my attempt.

Start off by fixing gear axes. Then if large gear is rotated 1 rev clockwise, small gear rotates 3 revs anticlockwise. Then to get the large gear back to the beginning the whole lot must be rotated 1 rev anticlockwise in which case the small gear will have made 4 revs anticlockwise. Is there a fallacy given that 4 is not one of the listed answers ?

I like Macolm's explanation (3)

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I think “4 is not one of the listed answers” was the point of the original post, wasn’t it ?

MichaelG.

Edited By Michael Gilligan on 10/10/2021 16:22:00

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Tidier than the video, but essentially the same information

MichaelG.

Michael, how did you like post #32 in the link?

Dave8

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Sorry, I have to admit I didn’t read that far … but I have now done so

MichaelG.

Hi, as can be seen in Macolm's explanation though, the circumference of the small circle only makes contact three times on the circumference of the big circle, which is why anyone would give an answer of three.

Regards Nick.

I have never seen that explanation before. Interesting.

Dave8

I don't think the video is correct. If you look at Macolms drawing, there are only 3 points on the large circle where the A on the small circle touches which means it revolves 3 times around the large circle, not 4 times.

I tested it using back gears from my lathe which ensures there is no slippage between the circumferences. I took 2 40 tooth gears, marked the teeth where they meshed and rotated one gear around the other until the marks lined up again. It took one revolution, not two as suggested in the video. I then took a 28 tooth and a 56 tooth gear and rotated the 28 tooth gear around the 56 tooth gear. It took 2 revolutions for the small gear to return to its original position, not the 3 suggested in the video.

Regards,

Alan C.

Hi Alan, if you look at Macolm's explanation again, you'll see that each position that the small circle is in, the letters A, B, C and D are all in the same position, thus for every quarter of roll around the big circle, the same distance has occurred to the circumference to the small circle, which is 3 / 4 of it, but the small circle has rotated once and this happens four times. As I've pointed out, the full circumference of the same circle only makes contact three times. Each of the letters on the small circle will line up with the same letters on the big circle, so when the two A's on the first quarter line up, the small circle has rotated more than once, so you get four times 3 / 4 plus four times 1 / 4.

Regards Nick.

Edited By Nicholas Farr on 11/10/2021 07:44:47

Hi Nick,

Try looking at it in reverse. If the small gear turns 4 times when it travels around the large gear, then it should also turn 4 times if the large gear is turned once. That wouldn't happen as the small gear would turn once each time an A came past - i.e. 3 times.

Regards,

Alan C.

Hi Alan, yes the ratio is three to one, but in the question, the big circle is stationary and the small circle Rolls around the big circle, hence four revolutions of the small circle and if the small circle was stationary, it would take four quarters of the big circle to revolve around the small circle once.

Regards Nick.

Edited By Nicholas Farr on 11/10/2021 09:11:53

Hi Alan, a couple of photos and Macolm's sketch with a 60T and 20T back gear (3 to 1 ratio) first photo shows start position and second photo shows first quarter on 60T gear, which shows the first rotation of the 20T gear, note that the small hole in the 20T gear in both photos is at the bottom, which will always be in position of A on the small circle.

Regards Nick.

Damm I didn't want to get involved in this, obvious answer is 3 but I now have to go and prove it to myself.

Tony

Yes it's 3, I can sleep easy tonight.

Tony

Sorry to disturb your sleep but it's 4, Chris LH explained it