Batch 2 - Class 94 - Of Bicycle Wheels and Tracks
Pre-Class Problem:
- Simpson Paradox: When one analyzes the voting records of democrats and republicans during the American Civil Rights Act, 1964, following facts emerge
- In the northern states, 94% of Democrats voted for the act, compared with 85% of Republicans
- In the southern states, 7% of the Democrats voted for the act, compared to 0% of Republicans
- Who showed more support for the act - Democrats or Republicans?
- Answer: Consider the following:
- What causes this non-intuitive answer? While there is abysmal voting by republicans in south, there are only 10 of them, so even if one vote had swung, it would look like 10%. On the other hand, there are lot of democrats in South, but very low support.
Attendance: Anisha, Smiti, Muskaan, Tishyaa, Abhiram, Nandini, Liza, Khushi, Arnav, Anishka, Tara, Navya, Anshi
Class Notes:
- Why are bicycle wheels round?
- Instructor Notes: Let kids think about and opine on this. Push them to go beyond the obvious.
- How do different points on the wheel move?
- How does a point on rim move? Trace it. Its called a cycloid.
- How does some other point on the wheel move? Trace some.
- How does the center move?
- Why is the axis at the center of the wheel?
- What would happen if you attached the axis at points other than the center of the wheel? Trace some.
- What would happen if a bicycle has square wheels?
- How would some of the points on the wheel move?
- How would the center of the wheel move?
- What would happen if you tried to ride such a bike?
- Can you think about how to straighten the path of such a bike?
- Can you actually construct a bike which rides like this? The shape of the road here is made up of Catenaries (Catenary is a hyperbolic cosine - the curve that results when you let a chain hang from two ends). Stan Wagon's bicycle illustrates.
- Interestingly, the points on the wheel rim here sometimes travel backwards instead of forward.
- What would happen if the bike had hexagonal tyres?
- Instructor Notes: Let kids think through this and come up with some shapes
- It happens to be the same as a catenary - its intuitive because at any point you only need to look at the flat surface in contact with the ground.
Which way did the bicycle go?
You are brought to a crime scene. You are told that a thief just escaped with a bag full of diamonds, on a bicycle. You come across the following pair of bicycle tracks in the snow, made by the fleeting thief's bicycle. But which way did the thief go?
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- Instructor Notes: Lets kids think about this for a bit before asking more specific questions
- Can you tell which could be the front wheel and which could be the rear wheel?
- Front wheels tend to be more wobbly than rear wheels
- Why?
- Which of the above lines could be front wheels?
- Also, the rear wheel always points to the front wheel. So which one is it?
- How do we determine which direction?
- We said that the rear wheel always points to the front wheel. What else is true?
- Rear wheel always points to the front wheel at a fixed distance!
- So what should happen if we draw tangents from rear wheel?
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- Can you also determine the length of the bike (distance between center of tyres)
- Are there cases where you can't determine the direction of motion?
- What must be true of these - the tangent from rear wheel track at all points must intersect the front wheel track at two points - in "front" and at "back". Which kind of tracks do that?
- Straight line
- Cocentric circles
- Any others?
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Homework
- Create your own bike that move on a catenary. See http://www.exploratorium.edu/square_wheels/square_wheels.pdf
References: