Why do couples at Marine Drive sit equally spaced from one another?

An interesting observation made by @TheToothsayer when we recently visited Marine drive was that the couples that sit at the sea front, sit equidistant to one another, quite precisely equidistant that too and over large distances. This nonchalant observation has more interesting insights than what is prevalent on the first look.

Photo courtesy: Kunal Bhatia - www.mindlessmumbai.com

Two odd things – firstly, there is no clear coordinator imposing that they should be sitting at a certain distance from one another and secondly, there seems to be no conscious effort to do this coordination. It just happens, it happens every single day.

How this formation happens is perhaps not too difficult to imagine if we piece this together in the chronology of the couples arriving at Marine Drive. Much like the XKCD Urinal Protocal Situation, we can postulate 3 basic rules that a logical couple would follow to find themselves a spot at the sea front -

1. The couples like privacy ^{[citation experience needed]} and they will optimize their distance from other couple such that they will maximize the minimum distance they have from any other couple.
2. There is a distance after which increasing the minimum distance from the next couple has no added value. Simpler way to state this is that a couple once a certain distance away from the next couple will just choose that spot because it’s not worth walking further.
3. Similarly, there is a distance that is the minimum acceptable distance from the next couple, that a couple can tolerate. This means that if the maximum minimum distance from any couple is less than this distance, then the couple will choose to wait for a spot over squeezing themselves awkwardly between the other couples.

Based on these the situation pans out to be something like this –

What is remarkable here is that all the 3 rules were for each individual couple and there was no coordinating agent that ensured that an equidistant pattern was formed. This is a purely self-organized pattern, which is a social trait of human beings. In fact, it would be safe to say that if there had been an established authority that made couples sit equidistant, then there would be resistance and it would not work with the smoothness it does naturally.

There is one situation when this logic does break down. When two neighbouring couples leave at the same time, where would the arriving couple choose their spot?

If the 3 logical rules are to be followed, then position A is the logical choice but Position B and C would keep the couples equidistant. I asked 30 random people on the train I am on right now, which position they would choose (I told them what I was doing after they chose). 27 out of the 30 chose position B. I would have chosen Position A because I would like to think I like logic over symmetry; however that is probably my frame of mind when writing this. Nature loves symmetry and it is clearly manifested in the human mind.

And THAT is why couples at Marine Drive are equidistant to one another.

Edit: This article has resulted in an overnight Phd into co-operative behaviour in the natural world – Seems like this is not only a human trait; interestingly here is a picture of ants self-organizing into equidistant pattern around their food.

Also surprisingly, Cognitive Biology is way simpler to make sense of over Photometry.

The Lunar Program - A Reflection

This is one of the most depressing plots I know -

Getting to the moon was significantly difficult to do in the 1960s than it is today, but still for some reason we don't go there. In the 1960s and the 1970s there was a cold war that caused enough motivation for J.F. Kennedy to pile up the country’s resources outside the NASA headquarters to get to the moon at any cost. Today, there is no cold war and economics has dominated the decisions to end the shuttle program. We need economic demand to go to the moon and we need to do this fast. I am going to do a few blog posts about a few ideas I have thought of. Consider these as Elon Musk style open patents, from someone who is significantly more lazier than him. We (humanity) must get some engineers out there to solve our problems.

What the world needs right now is a solution to the energy crisis. The Moon probably does not have oil, considering oil was formed from once fossilised life-forms. The Moon probably does not have coal or natural gas either for the same reason. What the moon has is a lot of barren land. What use is a lot of barren land that is visible from most of the Earth in the night time?

Meanwhile, the FIFA world cup has just concluded and there has been a lot of talk in Brazil, over whether it was worth hosting it. Sports are good. They are fun, bring the world together in friendly rivalry, boost local economies, provide great conversation starters and are a welcome distraction / entertainment source to the daily life. All of this comes at a cost and it is a good idea to bring these costs down. I am sure the FIFA world cup is expensive to host, one of the most visual expenses are the stadiums. The stadiums are large and extravagant. All of them now have flood lights which turn night to day at the stadium. This obviously is expensive to do, but night matches are economically worth it. It makes the game happen at prime time, and an artificial lights game is quite a spectacle.

It would be nice if we could play sport in the moon light, but unfortunately moon light is not enough.

This is an engineering blog so “not enough” is not an acceptable answer. Illumination can be measured. The SI unit of the Illuminance is ‘lux’. This is what light meters measure. When a cricket umpire holds out a meter, he is measuring the light per square meter. The SI unit of light is candela, which is the amount of light provided by one candle (a definition, a bit like horse power, I am amused by the idea that there is a standard candle/horse somewhere in the world which was used as reference). For cricket ‘lights’ is offered to the batsman if the illuminance is less than a 1,000 lux. However, the FIFA regulations are tougher, demanding at least an average of 3,500 lux for an International televised game. Moon light at an average is 0.4 lux. So moon light is “not enough” by 17,500 times.

Why is the moon light so low? Moon light is essentially Sunlight that is reflected by the Moon. The moon is at an average equally far away from the Sun as the Earth is, so the illumination received by the moon is the same as Earth on a sunny cloudless day, which is 1,20,000 lux. On a full moon night, the Earth receives a small fraction of light reflected by the moon, as this diagram will explain (Note the log scale).

The reflection co-efficient of the moon is 0.136 which means that 13.6% of the light incident on the moon is reflected. The fraction of the reflected light that the Earth would intersect is -

This value is ~.7 lux but the average illumination of the moon light measured on a clear night is 0.25, which means there is a 35.5% efficiency factor in there which accounts for light lost in the atmosphere (due to lack of clear skies) and averaging out of the fact that there is an angle of incidence involved both on Earth and on the moon.

To have moonlight soccer, we need to start working on making this more efficient. Let’s start with the reflection co-efficient - place mirrors on the moon. We clearly need to go to the moon to do this. We have managed to produce 99.99% reflecting surfaces but let’s cover the moon by 99% reflecting mirrors. However, even if we cover all of the moon, our new ‘Disco ball’ moon will only reflect ~7.2 times more light which is still nowhere enough.

Only a fraction of the reflected light actually hits the Earth, so the rest of the reflected light is wasted. We must have movable ‘sunflower mirrors’ that always point in the direction such that it reflects light to Earth. These movable mirrors will directly impact the light spread related efficiency factor. In fact, since all the light reflected by it is directed towards the Earth this efficiency factor can go up very close to 1. If the entire Moon surface is used to illuminate the entire Earth's night sky on a full moon day, the average illumination would be -

Adding the 35.5% efficiency factor accounting for atmosphere loss and angle of incidences, the average illumination of the Earth at night would be 3090 lux which is enough to host a cricket match anywhere on Earth any time of the day.

However, we still have all of the moon covered with reflectors that illuminate all of the Earth. This is still wasteful and very confusing to wild life.  We can consider reducing the effective reflector size. To illuminate the all of the Earth with 3090 lux we need the entire moon surface. To illuminate only a specified area on Earth with a desired illumination, the reflector area on Earth can be calculated by -

For a cricket field to be illuminated by 1000 lux, we need reflector area approximately 30 times smaller than a cricket field. For a FIFA match we can do with the total reflective area of 10 times smaller than a football field. They seem manageable, don’t they?

These reflectors can change their angle of incidence, which means the same reflectors can light up a football match in Australia, Europe and South America on a single day. Stadiums simply rent the reflectors on a need basis and pay by the hour. Beyond sport, this gives us the power of shining light on any part of Earth whenever necessary. Imagine how useful will it be to literally light up a rescue operation if a train were to have an accident at a remote location in the middle of the night. To light up 1 km of a 6 lane highway as per this, we need only ~35 meter square of reflectors.

Anyhow,  I'm looking forward to sporting events scheduled as per the lunar calendar, and a cricket match being interrupted by a lunar eclipse.

Hash-tag -> #GoToTheMoonAgain

P.S.: I also learnt that Photometry as a subject has one of the highest (Difficulty to understand / How simple it looks) ratio that I have come across. Also, solid angles are evil.

Suspension Design Kinematics - Degrees of Freedom

Suspension design starts off as a kinematic problem that the designer must solve. There are very easy methods to evaluate the basic sanity of the solution that one might have thought of. An independent suspension with its steering locked has one degree of freedom. This degree of freedom (DOF) is the travel of the suspension. When a design solution for the suspension is thought of, it is important to do the degree of freedom analysis.

For example, this double ‘a-arm’ suspension in the figure below has an upper and lower control arm with a toe rod which has ball joints on either end. 

The DOF of this suspension system can be simply analysed like this –

*The A-arms being joined to the chassis by 2 ball joins is actually an over constrain. The ideal solution would be to have a ball joint at one of the pick-ups and a ball joint in a slider on the other pick up. This is however not practical and usually the kinematic over-constrained is persisted with, to better distribute loads. Think is this in the same way as a door having multiple hinges when kinematically a single hinge would do. This is also the ‘a-arm’ must be manufactured with precision on a jig. Any misalignment will cause compliance in the structure.

** The 2 ball joints are usually rod-ends in a toe rod. The rod-ends are not ideal ball joints. The limited articulation does not allow the tie rod to spin about its axis and this constrains another degree of freedom associated with the spinning of the toe-rod within its place.

It has one degree of freedom which means that baring interference the suspension travel is easily achieved without any of the members flexing.

This degree of freedom analysis is important to determine if a suspension configuration would work or not. For, example these pictures below illustrate another solution to toe control in the rear where the tie rod is welded on to the control arm.

People who have read this blog post would be quick to point out that the toe base here is too small. The load path is not great because forces apply bending moments on the control arm. However, this suspension solution also does not fare well with the DOF analysis.

Such suspension solution might function like a 1 DOF system if the over-constrains are redundant. However, slight misalignment or manufacturing tolerance error would cause components to go through high stress cycles and eventually break. For the sake of completion the correct way to get the DOF correct in the above solution is to have a control rod, with ball joints at either end as the lower control arm instead of an ‘a-arm’.

DOF is only one of the criterions that must be kept in mind. The DOF solution in the suspension system in the picture below is good. However, there can still be an issue with the toe compliance here. Even though the toe base is reasonably large the toe rod pick-up is a cantilever on the upright. It will see a lot of bending moments and cause compliance due that bit flexing.

This blog post is originally written for the Formula Student India website and has been cross posted from here.

Toe Compliance Control - Toe Base

We have a seen a lot of teams show up with a rear suspension like the one in this picture –

What I am specifically pointing out is the method of toe control in the rear. There are 2 control arms and the upper one what looks like an implementation of a revolute joint through a bushing. This works in theory but practically there is bound to be compliance in the joint. This is not only form the compliance from the bushing but also from the compliance that arises out of having a tiny toe base.

The definition of the toe base of the vehicle is varied but in general it is the length of the moment arm of that opposes the re-aligning torque that the tyre produces. In this above picture it is the length of the cylindrical section at the outer end of the control arm. This is clearly small as is a definite ‘No No’ for a Formula Student car. The suspension must have a toe link.

In a more conventional suspension system there is a toe bar that prevents the rear suspension from steering. Here the toe base can be more clearly defined as the perpendicular distance of the point where the toe arm meets the upright from the king pin axis. The diagram clearly demonstrates this –

It is important for the toe base to be large. The compliance (or the ‘play’) in the suspension system dramatically reduces as the toe base becomes bigger. This can be simply illustrated by this simple thought experiment. Support you have a spherical bearing with 1 mm of compliance in it. Note: 1 mm compliance is chosen for ease of calculation, if you really have a bearing with 1mm compliance you must throw it away. This compliance will translate to the wheel being compliant when turned in the top view of the car. Here is a plot of the compliance in the wheel in degrees vs the toe base length.

The graph clearly illustrates the importance of having a large toe base. Simply having the toe link will NOT do. The suspension in the next picture will be compliant of this reason.

Rear toe compliance on a Formula Student vehicle will make it handle like a super-market trolley and will lose points in design. Which is why it finds second place in Pat’s “Seven Deadly Sins of FS Design” 

Moral of the story : Toe compliance is evil. Evil can be fought with larger toe-base. Hero wins only with a larger toe base, a little like this one -

This blog post is originally written for the Formula Student India website and has been cross posted from here.

Bangalore Rain Vs Bombay Rain

Dear Bangalore,
If you want to be the next Bombay, you HAVE TO figure how to handle the rain.
Sincerely,
Almost missed the train.

I would have uploaded this earlier if the power did not go away every time it rained.

Mumbai - Rajasthan NRR situation over-analysed

Since posts related to the IPL seem to get me a lot of hits on this blog here is another one. Yesterday, while I was watching a rather boring Indy 500 race, I heard shouts and screams from our neighbors who are cricket fans. This quickly distracted me and thankful I was because I witnessed one of the most bizarre cricket match endings I have ever seen.

The confusion was huge, mathematical and needed some time to get round to. I believe that the teams, even after the scores were level at 14.3 overs had not a true idea of the situation. As this facebook post by Mayur suggests that even if 14.4 where a dot ball, the Mumbai Indians still had a shot at qualifying if they hit 14.5 for a boundary.

Post by Mayur Srinivasan.

In fact, cricinfo has further gone ahead to explain that even if 14.4, 14.5 and 14.6 were a dot ball, Mumbai could still make it to the play off by hitting a six off the 15.1 delivery. Assuming of course sportsmanship from Rajasthan who would not ball a no-ball post 14.4.

Further analysis from an over the coffee table debate has spurred an interesting idea. I am going to argue that even if 15.1 were a dot ball - on paper, Mumbai still had a chance to make it to the play offs. All they had to do was a little rain dance.

No complication in cricket is complete without a discussion on the Duckworth Lewis system^{[citation needed]}. In fact with all the confusion that was yesterday, I am surprised it was not mentioned.

Cricinfo's "net run rate explained" page tells us that -

"[if] a result is achieved under Duckworth/Lewis, for net run rate purposes Team 1 will be accredited with Team 2's Par Score on abandonment off the same number of overs faced by Team 2."

Which means that if it rained when the scores are level at 14.5 and the rest of the game abandoned, then Rajasthan's score would have been reduced. By how much? The D/L system calculation is clearly out of the scope of this blog post, But we do have this handy tool to help us.


These calculations suggest that Mumbai would have won by 58 runs if rain poured in at 14.5. Which would make the net run rate look like this -

	For	Against	Net Run Rate
Mumbai	2175/270.8	2109/268	0.146568
Rajasthan	2094/264.2	2158/272.5	0.012225

This would qualify Mumbai for the play-offs and cause Twitter to accuse the Rain Gods to be Ambani's agents.

If no wicket falls this can go on till ball 16.4 where Rajasthan would be safe even from the rain Gods. Here is a plot for who would have won given ball when it rained, and Rajasthan did not concede an extra.

As the plot shows, Rajasthan could take 2 more wickets to seal the match earlier.

$2 Undecillion Lawsuit - An alternative solution

XKCD What-ifs are a pleasure to read. This week Lord Randall wrote about a $2 Undecillion Lawsuit. This what-if got me thinking in an alternative direction, and here is an alternative What-If to the same question.

If you have not read the XKCD article - in the $2 Undecillion Lawsuit a man called Anton Purisima has filed a lawsuit against Au Bon Pain Store, which if he wins will fetch him 2 Undecillion Dollars which is ($ 2 x 10³⁶). Everything in the world can be purchased for $ 7.7 x 10¹² so that is significantly more than all the money in the world.

As the xkcd article suggests, 2 undecillion dollars is comfortably higher by magnitudes to the sum total of the money that exists in the world. In modern times though, the sum total of money in the world is not such a clear concept. This is because day by day the world is becoming more of a credit based system than a cash based one. When you have a cash based system the sum total of money in the world is a finite. While things are more flexible when it comes to credit -

In case the Au Bon Pain Store did lose this lawsuit to Anton Purisima - in a cash based system Au Bon Pain simply could not pay. The best they can do is surrender all the cash that they have and everything that they own to Anton. Anton would be rich but not be the undecillionaire which he hopes to be with this lawsuit. However, in the credit based system there is still hope for Anton. All he has to do is convince a bank to loan 2 undecillion dollars to Au Bon Pain so that Au Bon Pain can pay him.

Why would a bank sanction such a sub-prime loan? For a bank to make such a loan, it must see a profit. The bank takes on risk when it gives out a loan and they like to bundle in the cost associated with this risk into the loan agreement via the interest rate or insurance. To offset the risk associated with a 2 undecillion dollar loan, there needs to be a massive payment as an incentive to make the loan. There is not enough money to make this payment but the bank knows that soon there will be one person who will be rich enough to pay. We are talking about Anton here. Anton is not going to be rich if the bank does not make a loan and the bank is not going to make a loan if Anton is not going to make the bank rich.

They strike a deal. Bank will make the loan to Au Bon Pain (with everything Au Bon Pain owns as mortgage), the court will send the money to Anton. In return, Anton must become a 'hyper-premiere' customer of the bank for which the bank will charge a modest one-time fee of 1.9 undecillion dollars. Anton will agree because it's better than nothing and still he will have more money than he can possibly spend.

This leaves 2 ridiculously rich entities in the world - the bank and Anton. They can buy anything and still have almost the same amount of money left over. Anton can do what he pleases with the money. The bank has an agenda though; it still has to make the other 0.1 undecillion dollars to balance its sheets. It made a loan against everything that Au Bon Pain has which is worthless compared to the loaned amount. It must raise the value of everything Au Bon Pain owns in that case, it has more than all the money in the world to do this.

If Au Bon Pain owned all its stores, then the bank can buy property around the stores at ridiculously high pricing driving the price of the property up. Not only has the price of the property gone up but the neighbours of the store are now suddenly ridiculously rich and Au Bon Pain can start charging astronomical prices because people don't know where to spend. This will not only happen to Au Bon Pain but to everything.

People will spend because everyone suddenly has a lot of money and the value of this money is falling fast. As more people have access to preposterous sums of money other people want some of it as well and prices will go up. Moreover, the government will be collecting huge taxes on all these transactions which means that the government will get spending and the inflation will spread like wild fire. The bank and Anton will have a gala time for a short while when they can buy anything they want. If they are fast enough, they can be the 'king / corporate entity of the world'. This will not really happen because people can refuse to sell but they can capture a significant portion of the world's stuff.

Something similar has actually happened before. This is the inflation chart of the Zimbabwean Dollar (If that looks steep then look again and find the logarithmic scale) -

When the Zimbabwean dollar was introduced, it was arbitrarily valued greater than the American Dollar, which was highly optimistic. The Zimbabwean economy was not in good shape and the exchange rate immediately went into freefall. People lost complete faith in the currency and dumped all the Zimbabwean dollars they had, making the fall worse. What eventually happened was that the country moved to using foreign currencies and cell phone talk-time as trading units.

So Anton and the bank will plunge the value of the American Dollar and the country will be forced to move to a foreign currency like the British Pound? Hold on. The banks would have thought of this, they will quickly realise that the USD will fall and will invest in a foreign currency. They will buy out all of it and take the foreign currency down with the dollar. This will go on until a new steady state is reached and the new crazy sum of money is significantly distributed. The world will temporarily move to barter before a new cash based (or limited credit) economy is set up.

The people who were opportunistic during this turmoil will become the new rich and the people who went on a tech detox vacation in the middle of a desert will be the new poor. There will be a complete redistribution reshuffling of the world's wealth. There is a good chance Anton and the bank will end up on the rich side of the new world. If you were a bank would you sanction this loan? I bet some bank would.

Side-note: The bank as per law will have to maintain minimum reserve, but it has to do that only at the end of the day. It can do this in two ways - act so fast that all the turmoil will happen in one day and they own enough barter at the end of it or they can also meet the minimum reserve with a deposit at the central bank, which can make the central bank panic and look for reserve.

G-Train through the Center of the Earth

Humans have always looked up in the sky, fascinated more by birds than moles, we built hot air balloons, the Wright brothers flew and finally NASA took us to the moon (Which is almost 5 decades ago, but that is a rant for another blog post). Humans have travelled 384,400 kms to the moon and the voyager is away from the Earth. What we have not done is scratched the Earth’s surface.

An old QI episode (G for Gravity) talked about drilling a hole through the center of the Earth and the travel time if the tunnel was frictionless through it would be 42 minutes and 12 second. This is called a “Gravity Train” and is quite a popular concept. The more interesting thing about the “Gravity train” is that it does not need to pass through the center of the Earth. Even if we cut a cord across any point of the Earth to another, the train will get from one end to the other in ~42 minutes in frictionless conditions.

A gravity train route that cuts through a chord of Earth looks like this –

If we move things around such that Gravity always points down then the gravity train looks like a massive roller coaster –

This seems to be a great solution to the world transportation problem. Get from anywhere to anywhere in less than 43 minutes and the only energy that is required be used to overcome friction and drag. You start at the top of the roller coaster, build momentum, go as far as you can up the other way and top up with some power when steam runs out due to friction.

However, it is difficult to drill a hole through the center of the Earth. It is surprising how quickly we will get to magma if we start digging downwards, after about 20-30kms of digging and you have almost created a man-made volcano! (More on this evil plan in a future blog post) Meanwhile, we will dig sideways. The furthest we have gotten in terms of drilling deep is the Al Shaheen Oil Field  ~12.5kms which is a surprising 0.19% of the distance to the center of the Earth, and this was a bore hole and not quite a tunnel. Suppose we could tunnel this deep – How far can our gravity train travel?

So from the plot above if we can dig a tunnel that has a max depth of 12.5 kms which is close to the deepest drilled hole we have today we can travel ~810 kms on the surface of the Earth. That is Bangalore – Mumbai, New York – Detroit and London – Zurich in less than an hour. That is an average speed of ~1000km/hr. What’s more, there is heavy return for development of technology, for every 1 km we can go deeper, we can travel ~ 30 kms more on the ground in the same time.

All we need is some focused energy to get this done. Half a century ago people thought going to the moon was a ludicrous idea. It’s difficult to convince investors why it is worthwhile to go to the moon today, but we can convince them that this will change the world and be profitable. It’s been almost 50 years since we were on the moon, come on engineers – Challenge Accepted?

Shank Lengths of Bolts

Bolts and nuts are the most crucial part of any assembly. They hold the different components of the vehicle together by providing the structural strength required. Selection and the usage of bolts, nuts and other fasteners are a very critical aspect of engineering design and should be considered with high importance at the design phase of the vehicle.

A very common mistake on Formula Student and Baja cars that is seen is the incorrect selection of bolts in terms of their sizes and shank lengths. Before proceeding here is the clarification of the shank length and threaded length -

Bolts are designed to carry axial loads. It may carry bending loads as well however it is important to understand which areas of the bolt should see the bending load. To get a better idea of why this is - consider a rod that needs to be broken. A force is applied in bending because it is the weakest then (Why? -  Load PathsDesign Error Blog).

If the rod still does not snap the next best thing to do is create a notch like in the picture below. It should be easier to break this after that. The notch can be really tiny, but it has an associated stress concentration factor which will cause it to see higher stress in that region and break faster. The notch is also a perfect crack initiation point for the rod to break.

The threads on the bolts are nothing but a series of notches when the bolt comes under bending and the bolt is much likely to break when loaded in bending at the threaded region. 

The threaded length of the bolt is the section which should be used to fasten the nut onto the bolt and for that reason only, all the other bending load must be taken up by the shank region. Which means that a bolt that connects a rod-end to the chassis mounts must look like this -

The clamps and the spacers and the rod end see the shanked length of the bolt and the threaded length of the bolt only starts under the final washer after which the nut is tightened and torqued up. At least 2 threads should be visible on the other end of the nut to ensure that the bolt is properly fastened.

This not only applies to the suspension bolts but any critical bolt that might be used in the vehicle. This mistake is also often seen in brake balance bars there bolts or even studs are used and the brake forces travel right through the threaded region of the bolt / stud. That will break. Custom engine mounts often ignore this as well and we see these bolts bend and break with time.

This blog post is originally written for the Formula Student India website and has been cross posted from here.

On the Boundary Lynn Catch

I am not big on following the Indian Premier League (IPL), but have noticed this fantastic match winning catch that Chris Lynn took to get AB DeVilliers out. If you have not seen it yet here it is on the StarSports website. [If you are in America - here is the YouTube link]

This catch is one of the most athletic ones we have seen in cricket and also he seems to pull off a physics defying jump where he catches the ball and still manages to land within the boundary line. There are certain stills of the catch (like the one above) that would convince you that he had to fall over the line and divine intervention or a extra strong draft of wind is the only thing that made him land well within the boundary limits. This obviously prompted this question on Quora - "Why didn't Chris Lynn fall on the boundary rope while making the best catch in cricketing history?" Here's my answer -

Let's start with a slight detour - There is a very famous (& I think very cool) physics fact about pole vault. When a pole vaulter jumps a high bar, her Center of Gravity (CG) need not go as high as the bar she needs to clear. In other words, at the top of the jump the pole vaulter needs to go over the bar and not her CG. To achieve this the pole vaulter arches her back forwards or backwards to be in a situation where the body is clear of the bar but the CG passes under for the most optimal jump. The idea being that Newton's laws only dictate the trajectories of the CG and not the entire body.

Lynn's miraculous falling back within the boundary rope is something similar. After the little stumble he has just before he makes his jump, he has his eyes on the ball. Accesses the trajectory of the ball and it looks something like this -

Now Lynn considers jumping straight up and try get the ball. Is there a chance he can make it? His jumping ability is limited and there is only a limit to which he can raise his CG. He would not have caught the ball.

However, if he jumped and turned back at the same time things could look very different even if the his jumping ability is considered to be the same. As this diagram illustrates -

At this point if a snap was taken it seems as if he would fall on the boundary rope, but he has some angular (spin) momentum on his side which a still would not capture.

He does exactly that makes the catch and after which since his CG is still very much inside the boundary region he can easily use his flexible and agile body to bring his hands inside the boundary rope. Eventually using his hands to safely land well within the boundary.

I am sure Lynn did not go through all this physics in his mind before making the catch, but intuition is a powerful physics engine due to the world experience it has.

Bove-torch

Cow fart is essentially a lot of flammable gas (Methane and Hydrogen Sulphide to be precise) coming out of a small hole (The anus to be precise). Any engineer in his right mind will think of only one thing about such a situation. Yes, you are right – Blow torch. So the question is can one hook up some piping to the correct places to build oneself a Bovine Blow torch or a Bove-torch?

First thing is first. We cannot be rude to the cow and set it’s behind on fire. Apart from facing a lot of scorn from animal activist, there is a very high probability of being kicked in the face by a cow on fire.  The revenue generated by this youtube video might make this a good business model, but we will be kind and attach a pipe with a nozzle at the end of it.

 I’ll start with a simple assumption which is probably true (Being from India, unfortunately I have rich experience with being in close proximity to cows) - Cows fart all the time they are awake, continuously. According to this research 13 million cows in USA in 2001 produced 2 million metric tons of Methane which amounts to 160 kg per cow at an average which is 0.0105g/sec. The Enthapy of combustion of Methane is 0.891MJ/mole therefore 0.0105g of Methane produce 584.7J/sec (Watt).

This is only half the story we have some (8.6kg/year) Hydrogen sulphide too being produced which is combustible at 0.519MJ/mole. This is 0.00055g/sec of Hydrogen Sulphide which is 8.8 Watt of energy. OK so this half of the story is not that significant.

Let us assume we are using the blow torch to do some brazing with copper as filler. Assuming 50% efficiency of the torch, the energy required to melt 1g of copper is 509J, in other words - the bove-torch can melt 0.6g of copper every second. You need about 20g of copper to braze a 1 inch diameter pipe joint so you might take about 35-40 seconds to do one which is impressive but not quite blow torch.

Some simple research (read: google) shows that a blow torch must be at least 50,000Watts to cause an American to boast about it. If we hook up say (50000/590 ~= 85) cows to a single tube we might have a proper blow torch that will operate 12 hours a day! So if you have 80 or more cows then you can run a small scale metal blazing / fitting industry at zero operating costs.

More interestingly, these farts also have a LOT of ammonia in there, which is unfortunately not combustible. But if you remember your 10^th grade chemistry lesson well and your mind function is evil by nature you will realise that ammonia will combust if you add platinum to the mix as this video will illustrate.

So if we have a platinum nozzle bove-torch (turbove-torch?) things can get really interesting. We have 54.3kg of ammonia per cow per year which boils down to 0.00344g/sec burning with 382.81kJ/mole which is an extra 77.5Watt per cow, that’s a full 15% extra!

May I remind you that you are also contributing to saving the world by doing this as Methane is 50 times more of a green house gas compared to Carbon dioxide.

Welded Rotating Components

We see a lot of welding in Formula Student and Baja Cars, which is not surprising. However, design judges always come across a few horror welds that are easy to label as 'GTB' or 'Going to Break'. Here is one example -

Both the half shaft and the universal joint are off the shelf components which are not designed to be mated with one another. The universal joint on the welded end has an internal spline and a mechanism where one can use a threaded bolt to tighten the grip of the universal joint on the splined shaft. Ideally there was no need for welding, if the shaft could have had splines of the correct size.

This team for some reason has decided to attempt a "butt-weld" to couple the two. Butt-welded shafts have been a source of debate many a times on technical forums, but they are risky but if done correctly there are championship winning cars out there with butt-welded half-shafts. However, the weld pictured above is probably not going to make it for very long.

When rotating parts are welded together such that they need to have their individual axis of rotations collinear, extreme care is required to ensure that the components being welded are held together tightly in place via rigid fixtures and jigs. Slight nonlinearity can cause many issues. If for example, the co-linearity is off by half a degree on a meter long half shaft then the other end of the shaft would move around ~9mm either way during rotation causing, compliance in the joints, vibration and high cyclic loading.

The shaft in the first picture above is clearly misaligned by more than a couple of degrees. In fact even when care is taken when doing such a weld there is invariably going to be some misalignment because of the welding process not being symmetrical (You have to start somewhere). In this video a shaft is welded and it is interesting to observe the amount of effort and precision put into making sure the shafts are aligned after the welding is complete.

This is not only true about half shafts. Any part welded about a rotating axis must take this tolerance into consideration. This includes wheel hubs, steering columns, their connection to joints, brake disk mounts and other rotating components. These welds are many times unavoidable as may have been in this case. here are some ideas to improve the quality of the design -

• Rigid and tight fixtures set up during the welding process. The weld should be completed fully before the components are removed from the fixtures to avoid bending and deformation during the welding process.
• In the case of chrome alloy and there metal being used that require heat treatment after the welding is complete. the component should be heat treated with the fixtures such that the reheating or cooling does not cause any deformation.
• If possible especially for welded hubs, have some excess material on the welded on bits such that after the welding is complete the entire component can be placed on a turning machine and made radially symmetric. A welded wheel hub like in the picture below is bound to have some misalignment.

• Very small misalignments (not visible without measurement) can still cause vibration and large cyclic loading of the component.
• It is a good idea to have grooves and steps in the weld areas such that the components fit into one another when on the welding fixture. This drastically reduces the chances of misalignment

Apart from misalignment weld strength is also questionable in some of these designs. Specially for rotating parts the weld area see high shear forces, which is not the best place to have them. Special care must be taken to ensure the filler material used during welding is the correct one. Small differences in the material composition in the weld area can cause minute differences in the coefficient of Thermal expansion that can cause tiny cracks to be formed in the area. These cracks will propagate during cyclic loading.

A couple of ideas to improve the weld strength -

• Find a method to increase the weld area. This can be done by having grooves and flanges or have the 2 components to be joined to be cut at an angle. Cutting at an angle not only takes the weld seam out of plane with the shear forces but also increase the weld area significantly.
• Support the butt-joint with an internal or an external sleeve. This distributes the shear force over multiple welds thereby effectively increasing the welded region and the strength of the component.

This blog post is originally written for the Formula Student India website and has been cross posted from here.