Balancing the Cycle

Well the basic question was why is it easier to balance the cycle when it is moving fast than when it is stationary. So lets look at it.

First, why does the cycle fall?
For the cycle to be stable it should be in an orientation in which its centre of gravity is at a point directly above the line the joining the points of  contact of the tyres with the road. This is an unstable equilibrium, so even a slight disturbance would send it to the more stable equilibrium i.e. flat on the ground.

Next, why doesn’t it always fall?
Well when we are riding one, if there is a small disturbance towards one side then our body balances it by moving itself.

So why is it easy when the cycle is moving than when at rest?
This is the question I wanted to know the answer of. I simplified the case and made a physical model to study it.

The model has a wheel free to rotate about its centre along any axis. The centre of the wheel is fixed. And there is no gravity.
It is set to rotate about  the axis of the wheel. Let the moment of Inertia of the wheel be I and the angular velocity ω. The small disturbance can be modelled using a massless particle with momentum p doing an inelastic collision with the wheel.

If we look at the angular momentum about the centre of the wheel. Before the collision Iω along the axis of the wheel and pR perpendicular to it. During the collision all the external force is generated at the centre so there should be no change of angular about this point. Finally there is only the wheel left so it should have Iω along the axis and pR perpendicular to it. So it will have an angular momentum which is at an angle tan^-1(pr/Iω) to original direction. We can see that this disorientation increases with decreasing value of ω. So the same impulse creates a smaller disorientation when the wheel is rotating faster.

Randomness, entropy and freedom

The following article is a philosophical idea of the author and some of you may find it complete nonsense.

There are things that are supposed to be done i.e work, job or whichever term you hate (assuming you hate it). And there are things which need not be done. Now while the things of the first sort would lead to our progress of some sort, we end up doing things of the second sort. Now there are many theories which would explain this like we don’t like to be forced upon or  the things we are supposed to do are never things we like.

Although I agree with th above points, my argument is that there are lots of things that can be done and a very few of those is actually what we need to do. Now if we choose what to do in an unrestrained i.e. random manner we would do the un-needed jobs most of the time. This is like saying the unimportant jobs are done because they have more entropy. Hard work is just reducing the entropy. And random people do useless things. This gave me the idea:

Freedom = entropy of choices

Thank you for reading the nonsense.

Sriram’s Normals

Sometime in August 2009, I was studying normals. The knowledge of what is a normal had been taken for granted. However we didn’t have any rigorous definition. So we couldn’t define what a normal for some strange figure is. Then we got the definition that the normal at a point on a curve is perpendicular to the tangent at that point. The definition of the tangent at a point P on a curve was like this: Take two points Q and R on either side of P on the curve. Draw the rays PQ and PR. Move the points closer to P. If the union of the rays forms a line as Q → P and R → P, then that line is the tangent.

Then we came across a statement which said: The shortest distance between two curves is given along their common normal. Now if we have the following two line segments.

The lines appear to have no common normals but their shortest distance is along the line C. With this fact in mind I made a definition for a normal.

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Definition-Sriram’s Normal

Date: 25/8/2009

Statement

Let there be a curve S and a point P on it. A ray starting from point P would be called a normal if and only if there exists a point Q on the ray other than P and a neighbourhood of P in S such that the point nearest to the point Q among the points in the neighbourhood is P.

Consequences

• At a differentiable point, two normal rays can be found. They point in opposite directions and when combined represent a line. This line is perpendicular to the tangent and orthogonal to the curve.
• At an isolated point, all possible rays from the point can be termed as normal.
  
• At the point of intersection of a pair of straight lines, no normals can be drawn.
• In y=|x| at (0,0) infinitely many normals exist.
• At the end point of a line segment, infinite normals cover a half plane.

An observation (not yet proved)

To find out Sriram’s normals one can do the following.

1. Remove the point. This will divide the curve into many parts.
2. Draw the normals at the edge of the each part.( In most of the cases the normals at the end point will be similar to that at the end of a line segment.)
3. The common portion( belonging to normals of all individual parts) is the normal.

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