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.

Reflector

If you’ve thought what I was doing two days before IIT-JEE (If you don’t know IIT-JEE is the entrance exam to the most prestigious engineering institutions in India, IIT). Here’s it:

Date: 8/4/2010

Work: It reflects the light, incident on it, to the direction from which it came. I think it is used in the glow in the dark signboards on the road.

Working: It uses three mutually perpendicular reflecting surfaces.

I assume my axes to be along the lines of intersection of the planes of the reflecting surfaces. If (α, β, γ) are the direction ratios of the initial ray. After reflecting from the mirror parallel to the x-y plane the direction ratios will become (α, β, -γ). After the other two reflections it will be (-α, -β, -γ) i.e. opposite the original ray.

Making: First, I took a cardboard sheet. And cut squares (2 cm × 2 cm) from it. Then I made a half-thick cut along a diagonal, and then folded along it.

Now I made the outer side reflecting. For this I used Aluminium foil. Now I made two more of these and joined them like this.

I joined surfaces 1 & 5, 1 & 3 and finally 3 & 5 such that 1, 3 & 5 formed three surfaces of a tetrahedron (something like the edge of a cube). The reflecting surface is the inner surface of the tetrahedron. Then I used nine more squares to make something which looked like:

It worked well but I think there are better reflecting materials than Aluminium foil.

Things used

1. Scissors
2. Blade
3. Tape
4. Glue
5. Cardboard
6. Aluminium Foil

Here are some photos of the one I made:

And it works, Can you spot it?

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.

A way of finding the coefficient of friction

It feels really nice when you propose a theory and later find out that it is right. This is what happened with me. Here’s a page from Journal de Brouillons.

“

Another method of finding co-efficient of friction μ

Date: 1/3/2009

I’ve found a new method to find the co-efficient of friction of bodies cylindrical in shape. It requires a thread and something which can measure force like a spring balance and some weights.

The things are arranged as in the diagram.

The weight of the weight when it is free is found. Let it be T₁.

Now the maximum and minimum tensions(for which the system is in equilibrium) of the string connected to the spring balance is found. Let them be T_2max and T_2min.

ΔT = T_2max – T₁ = T₁ – T_2min
Consider a small section of the string wrapped around the object. Let it subtend an angle dθ.

Then the normal force due to this segment

= T sin(dθ/2) + (T+dT) sin(dθ/2)

≈ Tdθ

Under limiting conditions;

f= f_max =μ_sN = μ_sTdθ

For the string to be equilibrium

T + dT = T + f

dt= f = μ_sTdθ

_T2∫^T₁ dT/T = ₀∫^θμ_sdθ
ln(1+ΔT/T) = μ_sθ
μ_s=ln(1+ΔT/T)/θ

if ΔT/T is small enough (if ΔT/T =1/50 the following approximation gives 1% error.)

μ_s=ΔT/Tθ

However the spring balance may have some friction and it will change the values. So it should be taken into consideration.

I would also like to ….

…

”

The remaining text has some other methods of finding the coefficient friction. And now these are the results I got

T1	T2
50	25
100	58
150	85
200	117
250	141
300	175
350	209
400	246
450	284
500	293

And this is the table of values which tell about the relationship between T2 and θ. It is exponentially decreasing.

Angle of Wrapping	T2
π	175
3π	80
5π	30
7π	13
9π	5
11π	3

And ΔT/T was never small enough.

Cracking The Bitmap

I thought of looking at how information is stored in the Bitmap format. So I made the following C++ program.

#include<fstream.h>

int  main(void)
{
	ifstream onfi;
	onfi.open("untitled.bmp",ios::nocreate);
	while(!onfi.eof())
	{
		char a;
		onfi.read(&a, sizeof(char));
		cout<<'\n'<<int(unsigned char(a));
	}
	return 0;
}

The program simply outputs all the bytes in the file. One byte in one line. After much analysis I found out that the first 2 bytes store “BM”. The next 52 are used to store the various properties of the image (like dimensions). Then the next bytes store the colours if the file is not 24 bit. Then all the pixels are stored in a linear fashion. I won’t give the details of the file here (like which byte stores the width of the file). If you want them contact me. Now I have also made a program which can make .bmp files.

A Strange Ocular Observation

I don’t know if it’s just my eyes. I observed something strange while sitting at the bus window last week. But I confirmed it a few days back. I looked at an object illuminated by the setting Sun. To my left eye it seemed to be illuminated by a bluer source while to my right eye the source seemed redder. I think the following images will explain better.

This is an edited photo. As of now I don’t have any explanation. I don’t even know if it has to do with optics or physiology. I would like you to try it out and do comment if you also observe it. But I won’t be able to tell you the exact conditions. It is not always visible to me and the difference is very small.

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.

“

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.

”

LPS and the 3D image

I did this on 15, January, 2010. I was flashing a torchlight at a light bulb. If I kept the torch and the bulb at some place and moved myself about in the room, the position of the image of the torchlight on the bulb changed.Now if I was standing at A, I would see the reflection of the torch at P and if I were at B it would be at Q.

(of course the bulb wasn’t switched on)

Voila!

Well here’s the LPS. The Local Positioning System. All I have to do is look at the position of the image of the torchlight in the bulb and I can know my position in the room. ( Can I? )

That’s the theory. Then I tried to make it work. I took a bulb and painted red, blue and green stripes on it hoping that the colour image would tell me the position of the image. But the image was still white. So my brother darkened the colours. Now the reflection was not there.

But I noticed one more thing. If you get close to the bulb and flash a light on it, you can see the image formed by the interior curved surface of the bulb, the image of the filament. A great 3D image. Yes 3D! Try it if you don’t believe. Hold the bulb in your left hand and the torch on the right as shown and look from an angle similar to the one shown in the figure.

Here is a photo of the image which I took.

The Digital Multimeter

Sometime in the third week of April, 2010 I bought a Digital Multimeter(DMM). It cost me Rs. 150. But the experiments I did with it were very interesting. Apart from the usual find the voltage, find the resistance experiments I did a lot of interesting things.

Measuring the resistance of the Human body.

Well I just turned the knob to the 2000kΩ and held the two terminals in my two hands and pressed hard. I got a reading of about 500kΩ. Then I held the two terminals at two points on the same finger and I got a reading of about 400kΩ. Such a small change in resistance. The resistance is clearly not proportional to the length here. Can you guess the reason? Well here’s another point. I made my hands wet and got a reading of about 80kΩ.

Brain waves or what

Now I did the same thing with the millivoltmeter. And to my surprise I got a reading of 10-20mV. So there was a potential difference between my hands. And between two points on my head it was about hundred mV.

Handshake detector

I went back to the megaohmmeter. My brother and I held a terminal each in the left hand and then shook hands. As we shook hands the multimeter showed a reading i.e. it had detected our handshake.

Now the reason I think why I got a strange reading in the first experiment. Our body is a somewhat good conductor on the inside(Don’t ask me why, but I think it is due to all ions in the blood). But the skin is a bad one. In both the cases, viz. the one in which I held the terminals across the length of my hands and the one in which I held them between two points on the same finger, the current had to pass through two layers skin and I think that explains everything.

Electrolysis and Galvanic cell

Well then I and my brother made a galvanic cell. We took a table salt solution in a small cup made of I don’t know what but something insulating. And then measured the voltage across the terminals, it was something small and soon fell to zero. Then we used a different electrode: an aluminium foil and generated about 500mV. Then we started testing everything that was there in house. Later we even added some vinegar and aluminium sulphate(which had no visible effect). Finally after experimenting so much the maximum voltage we got was about 900mV with aluminium foil as the anode and the pencil as cathode.

Now we started doing electrolysis. We started with a pencil cell. And a gas started evolving at the cathode lost metal at anode. And now to quicken things up we used the voltage across the mobile phone charger with pencil anode and Al cathode. I don’t remember the value of the current measured but I think it was more than 1A. And the surface was swarming with bubbles. Then I did the test about which I had only read so far; I took a lighted matchstick and brought near the bubbles. And it did explode with a pop sound. This time there was some gas at the anode also. And it smelled bad. The only gas which I could think of was Chlorine. It must have come from NaCl. But I don’t know any method of testing it.

The first post

In this blog you will mainly find the various experiments done by me and the strange results that I have seen. There might be something else also. Do try them out if you can and comment what you found.