Make a wobbler

[Cubist]

This is an unusual shape that keeps it's centre of gravity at the same height while it wobbles. Therefore you can blow it along a floor/table, or it will wobble down a gentle slope...



Just cut two equal discs out of card and make a slit in both of them that goes from the edge toward the centre, length of slit ≈ 29.3% of the radius. Slot them together and away you go! Be careful with your scissors ?

I came across this shape in the book, "Things to make and do in the fourth dimension", by Matt Parker. (I found this to be a nice, easy, interesting read.) The author found an early reference to it in 1966 as a Two-circle Roller in the "American Journal of Physics" by A T Stewart.

The helpers might want to try proving that the COG stays at a constant height! Apparently it's difficult.

 

[AAA]

what's your question here? the shape looks interesting

 

[Deleted member 4993]

what's your question here? the shape looks interesting

This is a post in "Math Odds and Ends" - for most of the posts it is a point of discussion - not a query for "particular answer" for "particular question".

By the way, if you had noticed the number of "posts" in Cubist's profile - you should have guessed that s/he is a tutor here (and super-competent at that).

 

[Cubist]

- you should have guessed that s/he is a tutor here (and super-competent at that).

? thanks! ( I just tend to spend a long time thinking before I post, which makes me seem more clever than I am )

what's your question here? the shape looks interesting

@Subhotosh Khan is correct - it seemed like a quiet day on the forum so I posted something that people might find interesting (and perhaps even have a go at making one). Your comment has however motivated me to have a go at proving that the COG stays at a constant height. It probably isn't that difficult. I've started a diagram. Will post it soon.

 

[Cubist]

Proof, part 1

The diagram below shows the wobbler standing vertically. There's a view from the front and side...



Obviously r is the radius and x is the length of cut in each disc. The red dot indicates the centre of gravity.

The next diagram shows the wobbler after it has undergone a clockwise rotation about M (while being viewed from the front). The rotation leaves the top disc at angle θ to the horizontal...



Now, if the wobbler was to topple straight backwards (from the "front 2" view), it would make contact with the ground at the point labelled L. The line OL is tangent to the ellipse shape CGFH. Therefore the centre of gravity would be at the height labelled y (distance DK).

Now I have to prove that, for some choice of x, that y will remain constant as θ changes (within its range of motion).

I'll aim to post more work tomorrow, but any comments or suggestions for alternative (easier!) strategies are welcome.

 

[AAA]

Im not that smart but thats really amazing! You broke down that shape into a mathematical problem