The Awesome Number 2 : Puzzles, Games, The Power of Doubling

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There are many games and puzzles based on the number 2; I shall discuss some in this blog.  

Number 2 is the only even number that is also a prime number. It is also one of the factors of all even numbers. 
(By the way - did you know that every even number may be expressed as sum of two prime numbers - amazing!)

What I find interesting is that many things around us manifest duality - in cultures, languages, philosophy, evolution.  One talks about 
mind and matter; good and evil; high and low; right and left; up and down; front and back; right and wrong; loss and gain;... the list goes on - see if you can think of some yourself. 
It may be that, to make sense of the complex world around us, we need to look at it in the simplest way possible.  Defining two extremes is a convenient way to set reference points for managing complexity. 
Politicians make important points in groups of three and look at the mess they have created - not able to cope really. 

The Power of Doubling:  The following example demonstrates the power of the number two:
Take a sheet of A4 printing paper.  Let us assume that its thickness is 0.1 mm. 
Fold it over once to make its area half - thickness is now 0.2 mm.  
Fold again (second fold) - thickness is 0.4 mm.  
If we continue to fold the paper 40 times (can certainly be done in a thought experiment), then the thickness will increase - say, to a value H.  
What do you think - make a guess how big H is?
Would you believe that the thickness will be ~ 100,000 kilometres!!
That is the power of doubling! It is the basis for understanding the ideas behind exponential growth (some people think it should be called runaway growth).  
For some of my favourite examples of the power of doubling, click 1, 2.

(The next slide sets out some simple mathematical background.  You can miss it out without loss of continuity but we shall use some of the results; click on the slide to see full page image)

We have looked at an example of doubling in folding of an A4 paper.  
Let us consider one more example of the interesting fable to further demonstrate the power of doubling. 

Pleased with the musician, the king asked him to choose any prize he wished for.  The musician asked for some grains of wheat.  He asked that on a chess board a grain be placed on the first square and the number of grains is doubled on each subsequent square.  the king laughed at the naivety of the musician and granted his wish.  This is what happened:


 
All the wheat in the kingdom was not enough to fill the board!

Let us now consider an example of the second case (eq. 3) where each successive term halves.

Example:  A bouncing ball is dropped from a height of 1 m on a concrete floor.  The ball bounces back to half its original height viz. 0.5 m.  In the next bounce its rise is halved again to 0.25 m; and so on.  The ball bounces about 20 times before coming to rest.  What is the total distance the ball bounces before coming to rest?
Solution:  The ball will travel a total distance S as follows: The ball travels the initial 1 m and then it  rises and falls in each of the 20 bounces)

S = 1 + 2 x (1/2 + 1/4 + ...   20 terms)  i.e. n = 20

From eq.3 in the slide S = 1 + 2 x (1 - 1/2^19) ~ 3 m
(1 in brackets is shown red to point out that the first term in the sequence is absent and is accounted for by subtracting 1 from the sum)
The value of the term  1/2^19 is very small and may be neglected for ease of writing the result.

You can also solve the puzzle by simply summing the heights traveled by the ball in successive bounces:
S = 1 + 2 x (0.5 +0.25 +0.125 + 0.0625 +...) 
and arrive at the same answer.  I think the first method is more elegant.

Doubling is the basis for understanding and planning in situations like population growth, increase in bacterial populations, nuclear power production, inflation, banking and much much more.  

I now look at a couple of mathematical puzzles where the number 2 plays a crucial role:

Puzzle 1:   What is the lowest number of weights you need to weigh objects from 1 kg up to 50 kg. Weight of each object is an integral number of kg.

Solution:  You might remember that in the series 
1, 2, 4, 8, 16,.. for any term, the sum of preceding terms is one less than the value of the term.  
For example: 1 and 2 add to 3 that is one less than 4
1, 2 and 4 add to 7 that is one less than 8
1, 2, 4 and 8 add to 15 that is one less than 16
This holds for all terms in this sequence (an example of a geometric series).
To measure integral kg weights up to 50 kg, we need weights 1, 2, 4, 8, 16 and 32 - altogether 6 weights.  You can check that these will actually work fine.  In fact, the weights will measure objects up to 63 kg.

Puzzle 2:  What is the lowest number of weights you need to weigh objects from 1 kg up to 50 kg. Weight of each object is an integral number of grams.

Solution:  This is an extension of puzzle 1.  We still need to weigh integral kg, so we need the 6 weights as before.  But we now also need to weigh from 1 gram to 999 grams.  This means that we should have weights of 1, 2, 4, 8, 16, 32, 64, 128, 256 and 512 grams - 10 additional weights.  
This makes 16 weights altogether to be able to weigh object up to 50 kg with 1 gram resolution.  Not bad.

Number Puzzles for Mental Arithmetic Enthusiasts

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The following puzzles are based on some interesting situations that use least common multiple (LCM) of a set of numbers. 

LCM of a set of numbers is the smallest number that is completely divisible by each of them without a remainder.  

For example, LCM of 2, 3, 4 and 6 is 12.

You can also see more mathematical games here and here

(you can use a calculator if you wish; Answers at the end)

Puzzle 1:  In a party, the host wants to arrange exactly the same number of guests at each table. He tries to seat 4, 5 and 6 guests per table but finds that, in each case, the last table has only 3 guests.  However, if he tries 7 guests per table then they fit exactly.  How many guests did the host invite?

Some variations of the above puzzle, slightly more difficult, are:

Puzzle 2:  In a party, the host wants to arrange exactly the same number of guests per table. He tries to seat 4, 5 and 6 guests on each table but finds that, in each case, the last table has only 2 guests.  However, if he tries 7 guests per table then they fit exactly.  How many guests did the host invite?

Puzzle 3:  In a party, the host wants to arrange exactly the same number of guests per table. He tries to seat 4, 5 and 6 guests on each table but finds that, in each case, the last table has only 1 guest.  However, if he tries 7 guests per table then they fit exactly.  How many guests did the host invite?

Moving on to a different situation, consider the following puzzle

Puzzle 4:  Jack and Debbie want to combine their substantial stamp collections and prepare a new folder.  Sticking only odd number of stamps in a row, they find that when they use 

5 stamps in a row then 4 stamps are left in the last row

7 stamps in a row then 6 stamps are left in the last row

9 stamps in a row then 8 stamps are left in the last row

11 stamps complete all the rows perfectly with none left over.

How many stamps in total do they have?

Puzzle 5:  The priest wishes to arrange his congregation is rows such that all rows have exactly the same number of people.  He does not like 13 people, not an auspicious number, in a row.  

He finds that if he sits them in rows of 7, 8, 9, 10, 12, 14, 15 or 16 then he is left with 6, 7, 8, 9, 11, 13, 14 and 15 in the last row.  Not what he wants, but rows of 11 fill properly.

How many people are there in the congregation?

Answers:
1.  63;  2.  182;  3.  301;  4 and 5.  2519



  

The Science of Hypothermia, Windchill, Frostbite - When Body's Thermal Regulation Mechanism is Unable to Cope...

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We are endotherms (warm-blooded animals) and must maintain a nearly constant temperature (~37C) of blood (the core body temperature) circulating to the vital organs like brain, heart, liver and kidneys. Most of the heat loss happens from the skin which is held at a slightly lower temperature of 33C to 34C.  Mild hypothermia is defined when core body temperature falls to 35C but serious complications occur if the blood temperature drops further.

In hypothermia, skin temperature particularly at the extremities is much lower. It can be almost as low as the ambient temperature with extremities receiving very little blood flow (vasoconstriction).

Vasoconstriction is mediated by restriction of the capillaries in the dermis/subcutaneous tissue as shown in the slide

We consume 2000 Calories per day to supply metabolic energy needs of the body.  This generates energy at the rate of ~100 W which must be removed to maintain a constant body temperature.  Heat is generated mainly in the following activities:

a.  Metabolism of all the cells of the body
b.  Muscle activity - exercise etc.
c.  Effects of hormones such as thyroxine, testosterone
d.  Extra metabolism need for digestion, absorption, storage of food

The body's thermal regulation mechanism easily gets rid of this heat energy if the ambient temperature is around 23C - the main mechanisms of heat loss are:

a. Radiation of infrared waves (most important)  
b. Convection by air 
c. Evaporation of water (the sweat) on the skin by moving air.  
d. Respiration - moist warm gases given out during breathing
c. Conduction - through contact with solid material

The question we address in this blog is what happens when the ambient temperature is much lower than 23C. Heat loss by radiation increases rapidly at lower ambient temperatures and easily exceeds heat production to cause hypothermia.  
(Click on the slide to see bigger image; Press ESCAPE to return to text)

At the onset of hypothermia, as a first response, the body starts shivering - an involuntary oscillatory muscular activity that produces additional heat energy to warm the body.

Different stages of Hypothermia:  Hypothermia may be divided into several stages depending on its effect on the body:
Mild Hypothermia:  This results in symptoms such as shivering, numbness in hands and other extremities and reduced manual dexterity.  Complex skills become more difficult, the victim may also feel tired, may argue and become  uncooperative.  Difficulty in performing tasks such as fastening up clothing, putting on gloves, a hat etc. or taking them out of a rucksack may result in the victim getting irritated and ending up not bothering. This will of course make them get even colder.
How the physiological system is affected by progressively falling temperatures is discussed in detail here (see table 11-1, page 361 of the reference)

The final stage is profound hypothermia:  In this stage the body has effectively stopped trying to keep itself warm and some final steps are taken to avoid death. The heart rate and breathing slow so that they are hard to detect at all. Only one or two breaths per minute may be taken. The skin is very pale and icy cold to the touch. Metabolism has slowed so far that they are almost in a state of hibernation.

At a core temperature of around 28°C heartbeat irregularities may occur - called cardiac arrhythmias - this can lead to an uncoordinated twitching of the heart muscle preventing it from pumping blood properly and can result in death. Even if this does not happen, the heart will stop beating completely at around 20°C causing death.

When Thermoregulatory system is unable to cope:  If someone is exposed to cold and inadequately protected, their body will first try to generate more heat through shivering to maintain a normal temperature. If this does not solve the problem, the body will start decreasing blood flow to the extremities to curtail heat loss. Extremities will turn cold and appear blue.  If the loss of heat carries on despite these measures, in the final stages the body will slow its metabolism to minimize its need for fresh blood flow and oxygen supply.

Actually, the sooner the body reaches this final step,  better is the chance of survival as the organs won't be starved of oxygen. 

The Physics of Heat Loss from the Skin:  I now look at the physics of heat loss by the skin.  In hypothermia, the main mechanism of heat loss is by radiation.  The body has effectively switched off convection (vasoconstriction makes exposed extremities much colder, thereby reducing heat loss, also one wears woolen clothes to reduce air flow across the skin) and evaporation (switch off sweating).   

Radiation:  Every surface radiates and absorbs energy - how much? - depends on the area of the surface, its nature (mainly colour and texture), and temperature.  The law that governs the radiation of energy is called the Stefan-Boltzmann Law - heat loss varies as the fourth power of the absolute temperature (T in Centigrade + 273) of the radiating surface.  If we consider that human skin temperature is 33C and has an effective area of 1.0 square metre (actual adult body area is nearer 1.7 square metre but not all skin is exposed to air; clothes also affect radiation loss), then the energy lost per second is estimated as (remember that the body also absorbs heat radiated by the surroundings and it is the difference between heat radiated and heat absorbed that defines the heat loss by radiation)
   Ambient temperature    Energy Loss per second
             (degree C)                   (Watts)
                  30                              20
                  23                              65
                  15                              111
                  10                              137
                    0                              188
                 -10                              235
                 -20                              275

The net heat energy loss increases with decreasing ambient temperature; even at outside temperature of 15C, energy lost is greater than the heat generated by the calories consumed.  The above calculation actually overestimates the heat loss - the skin temperature falls below 33C as hypothermia sets in and the heat loss by radiation is reduced.  Considering an average skin temperature of 23C (extremities may be much colder but rest of the body will still be near 30C) and an ambient temperature of 0C (not uncommon during winter months), radiation heat loss will still be about 230W - too large for the body to compensate through thermo-regulatory mechanisms.

Convection:  is transfer of heat between our skin and the surrounding air.  How much heat energy is lost from the skin depends on the temperature difference between skin and surrounding air; it also strongly depends on whether the air is still or moving.  
Heat Loss per second = K A [T(skin) - T(air)]
K is convection coefficient and its value depends on the speed of air - the largest difference being between still air and moving air. 
Some values of K in units of W/m^2/C are listed below:
K =   3  for air speed =  0 m/s  (still air)
K = 26  for air speed =  2  m/s
K = 37  for air speed = 10 m/s
K = 41  for air speed = 20 m/s
The way it works is that our body warms a thin layer of air next to the skin.  This boundary layer acts as an efficient insulation reducing heat loss.  In convection, wind blows this boundary layer away and heat from the skin creates another boundary layer.  As wind continues to blow away such boundary layers, body looses heat energy and cools. 

For a 5 degree difference in temperature, power loss is about 20 W in still air, but increases to 200 W in moving air!  Convection is a very efficient method of heat transfer when there is a breeze present.

On a cold day, if you are indoors and properly clothed then heat loss by convection is not significant.  Outdoors with wind blowing is a different matter, and one can lose additional heat due to convection - the ambient temperature will feel much colder - this is called the wind chill factor that I discuss next. 


Wind Chill and Frostbite:  When outdoors, on a cold day, one feels that the temperature is lower than its actual value.  This is due to the combined effect of heat loss by radiation (indoors and/or outdoors) and convection (important when outdoors in the wind).  The wind chill factor (or wind chill temperature) is a measure of how it feels when the effect of convection due to wind is included. It feels that the temperature is lower than the thermometer value.  Note that your body temperature can never fall below the actual temperature even in windy conditions - only difference is that the body cools quicker when the wind speed is higher.
Frostbite happens when the body organ actually freezes (temperature falls below about -5C for an extended period).
The table below (adapted from) provides windchill temperatures (how it feels) and estimated frostbite times of peripheral organs like tip of nose, ear lobes, fingers and toes. (click on slide to see full page image)

Notice that even mild wind speeds cause serious wind chill.  For example - At a temperature of -20C, wind chill is -26.2C for wind at 5 mph; -33C at 20 mph and -38.2C at 50 mph.  This is because the heated boundary layer at the skins is swept away by even modest wind speeds and further increases in wind speed do not make a proportionate effect.

Frostbite mainly affects organs at the periphery that experience vasoconstriction (narrowing of the blood vessels) and do not receive much heat energy from the core of the body.  Below about -5C, freezing causes ice crystals to form in the tissue.  Ice crystals can damage cell membrane and small blood vessels in the frozen tissue.  I remember when I was living in Saskatoon, Canada, the advice was never to walk outside for more than 10 minutes as the ambient temperatures would go down to -44C in winter months! 
Most at risk of frostbite are people who are away from sheltered places - out in the open; e.g. military personnel, homeless people, sports enthusiasts.  Alcohol beverages cause your body to lose heat faster.  
Management of frostbite is a complex subject - please click here for more details.

End Note:  Human body maintains a state of homeostasis - parameters like core body temperature, blood sugar level, blood pressure and many more, are regulated in a narrow range by the hypothalamus in a negative feedback mode.  Hypothalamus does a wonderful job but it has its limits beyond which the regulatory mechanisms cannot cope.  
Human body produces 100W by metabolizing the food we eat (click here).  The regulation of temperature fails when the ambient temperature is too high (click here) or too low (covered in this blog) - this can be life threatening.  The science is well understood but the responsibility of keeping oneself safe rests with the individual.   

Finally, there are a couple of YouTube Videos (1, 2) which might be worth watching.    

The Science of Heatstrokes (Hyperthermia) - When Body's Thermal Regulation Mechanism in Unable to Cope...

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We are endotherms (warm-blooded animals) and must maintain a nearly constant temperature (~37C) of blood circulating to the vital organs like brain, heart, liver and kidneys. The skin, from where most of the heat loss happens, is held at 33C to 34C.  Heatstroke is defined as blood temperature above 40C; it is dangerous and if left untreated then serious damage to major organs can happen or even death is likely.

Consumption of 2000 Calories per day produces energy at the rate of ~100 W.  The body's thermal regulation mechanism easily gets rid of this heat energy if the ambient temperature is around 23C - the main mechanism is heat loss  by radiation (~65%).  Most of remaining energy is lost by convection (~12%), evaporation (~12%) and respiration (13%).  The question we address in this blog is what happens when the ambient temperature is higher.

Thermal Regulation Mechanism:  The hypothalamus in the brain regulates the core body temperature around 37C mark.  It senses the circulating blood temperature in the brain and using similar data from skin sensors monitoring external environment, the hypothalamus operates like a thermostat in a negative feedback loop to maintain a nearly constant value around the set point of 37C (homeostatis - status quo). 
In negative feedback, a deviation from the set point automatically starts a corrective mechanism that reverses the original change and pushes the system back towards the set point.
The action of the hypothalamus is involuntary and happens automatically.  In addition, humans can act to change the environmental effects by voluntary action such as cooling the body by moving into shade, wearing lighter cotton clothes, increase intake of water or even stretching out to increase surface area for heat loss.
The hypothalamus automatically activates what are called effectors.  For example:
1.  Sweat Glands:  secrete sweat on the surface of the skin where the water evaporates using heat energy from the body.  Since water has a large value of heat of evaporation (2260 J/gm), it is very efficient in cooling the skin.
2.  Smooth Muscles:  relax to allow greater blood flow.  This transports more heat to the skin allowing it to be lost by convection and by radiation (if the outside temperature is less than 33C).  Skin appears  more red due to increased blood flow.
3.  Erector Pili Muscles:  muscles relax to flatten the skin hair, allowing better air circulation over the skin which increases heat loss by convection and evaporation.
4. Adrenal and Thyroid Glands:  Glands stop secreting adrenaline and thyroxine.  Metabolic activity is slowed down to generate less heat energy in the body.

As ambient temperatures increase towards the skin temperature, loss of heat by radiation is reduced and the body relies more on sweating and evaporation for losing heat. Prolonged exposure to high temperatures or physical exertion can cause heat exhaustion.  Left untreated, heat exhaustion can quickly lead to heatstroke - a potentially deadly condition.

Heat Exhaustion:  The symptoms of heat exhaustion include muscle cramps, heavy sweating, weakness, fast heart rate, nausea and pale/cool skin. To treat heat exhaustion, move to a cooler location, sip water, cool body by applying wet cloth.

Heatstrokes:  is defined as a core body temperature of 40+C.  The fever is accompanied by confusion, headache, rapid breathing, lack of sweating, nausea/vomiting, muscle cramps, weakness.  There is dysfunction of the central nervous system resulting in symptoms such as fainting/unconsciousness or seizures.  

Lack of sweating also means that the body can not use the two most effective means of losing heat energy viz. radiation and evaporation. Heatstroke is a serious condition and must be treated as an emergency.

With global-warming (leading to many more extremely hot days - more countries are reporting record high temperatures of 47 C or even 50+C), greater urbanization (heat-island effect), rise in obesity and an aging population (less robust), heat related incidences are on the rise.  
In India, 2500 heatstroke deaths per year are reported and represent a 61% increase in 10 years. 
Following the 1995 Chicago heat wave, half of the patients admitted to intensive care unit died within a year.  Many experienced severe functional impairment after discharge - with no recovery reported. 

The Physics of Heat Loss from the Skin:  I would now look at the physics of heat loss and heat gain by the skin.  The three main mechanisms are radiation, convection and evaporation or vaporization (sweating).  

Radiation:  Every surface radiates and absorbs energy - how much? - depends on the area of the surface, its nature (mainly colour and texture), and temperature.  The law that governs the radiation of energy is called the Stefan-Boltzmann Law - heat loss varies as the fourth power of the absolute temperature (T in C + 273) of the radiating surface.  If we consider that human skin temperature is 33C and has an effective area of 1.0 square metre (actual adult body area is nearer 1.7 square metre but not all skin is exposed to air; clothes also affect radiation loss), then the energy lost per second is as follows
   Ambient temperature    Energy Loss per second
             (degree C)                   (Watts)
                  23                              65
                  30                              20
                  33                                0
                  40                              -50
                  45                              -90
                  50                             -120
                  55                             -165
Negative energy loss means that skin is absorbing heat energy from the environment.

If you are outdoors in the Sun, then you actually receive extra energy from direct sunlight.  On a bright day this can be as much as 250 W.  So, on a hot day at 40C, one can be absorbing up to 300 W energy. This can result in a sun stroke (same as a heatstroke but can come on quickly due to exposure to direct Sun). Skin can get very hot locally and cause sunburns.   Physical activity like sports will produce even more energy. On such hot days, it would be better to stay in shade or indoors and avoid strenuous activity.

Radiation loss is an effective heat loss mechanism only if ambient temperatures are less than 30C. 

Convection:  is transfer of heat between our skin and the surrounding air.  How much heat energy is lost from the skin depends on the temperature difference between skin and air and strongly depends on whether the air is still or moving.  
Heat Loss per second = K A [T(skin) - T(air)]
K is convection coefficient and depends strongly on the speed of air - the largest difference being between still air and moving air. 
Some values of K in units of W/m^2/C are listed below:
K =   3  for air speed =  0 m/s  (still air)
K = 26  for air speed =  2  m/s
K = 37  for air speed = 10 m/s
K = 41  for air speed = 20 m/s

The way it works is that our body warms a thin layer of air on the skin.  This boundary layer acts as an efficient insulation reducing heat loss.  In convection, wind blows this boundary layer away and heat from the skin creates another boundary layer.  As wind continues to blow away such boundary layers, body looses heat energy and cools. 

For a 5 degree difference in temperature, power loss is about 20 W in still air, but increases to 200 W in moving air!  Convection is a very efficient method of heat transfer when you are standing in front of a fan.

If the surrounding air temperature is 35C or greater, then heat is actually transferred to the skin and the evaporation of sweat will need to be increased to compensate this.  
It is best to stay out of the wind on hot days as convection is ineffective in cooling the body.

Evaporation:  Sweat brings saline water to the skin, and forms a thin damp layer.  The water in the sweat evaporates to cool the body.  Water has a high value of latent heat of vaporisation - it takes 2260 Joules of energy to evaporate 1 gram of water. This energy is drawn from the skin which cools as water evaporates.
Humans sweat a maximum of about 0.7 to 1.5 liters per hour - approximately 0.2 to 0.45 g per second.  If this water evaporates, then body will lose heat at the rate of 500 to 1000 W.  This is a large amount of energy loss and is very effective in keeping the body cool as the ambient temperature rises beyond 33C when radiative and convective cooling becomes ineffective.  
Evaporative loss is independent of ambient temperature and depends only on the amount of water that vaporizes.

Water is evaporated from the skin more efficiently when the air is dry and holds only a small quantity of water vapour.  On a hot and humid day, when the air might be nearly water-saturated, evaporation from the skin becomes less efficient and the heat loss is reduced.  The result is that the body retains more heat being unable to vaporize all of the sweat. More sweat on the skin gives a clammy and uncomfortable feel. The skin temperature also rises towards 35C.

The following slides represents the situation - I have redrawn the slide from the paper by Hardy and DuBois.


On hot days with ambient temperature exceeding 40C, the water loss from evaporation must be compensated by drinking more water.  If ambient  temperature rises much higher - to say 47 or 50C - then evaporative loss may not be sufficient to keep the body core temperature below 40C and that is when the danger of heatstroke becomes serious.  Hypothalamus starts to mysfunction and sweating can actually reduce or even stop - this a dangerous signal and urgent action is required.

Respiration: or breathing exhales water vapour that carries energy.  This rate of energy loss is estimated int he following slide

End note: The body can and does change set point from its normal value of 37.5C in the special situation when a more intense response is required to fight infections and/or increase the efficacy of the immune system.  In such situations, chemicals called pyrogens are released by white blood cells.  Pyrogens raise the set point of the thermoregulatory centre by a couple of degrees.  Hypothalamus then tries to maintain the core body temperature at a higher level that helps to kill bacteria, inhibit viruses - effectively fight pathogens.  The body at a higher temperature loses more heat and shivers to generate extra energy. 
Maintaining higher body set point over long periods can be harmful to organs and there is an interesting ongoing debate whether one should reduce the fever by using medication (paracetamol or ibuprofen) or let the fever ride. The topic is somewhat outside the scope this blog but you can follow the reference cited here for details.  

Heat given off by a human body is about 100 Watt

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Humans are warm-blooded mammals and their body temperature is maintained constant at about 37C. Comfortable ambient temperature is much cooler at about 20C.  We are constantly losing heat energy from the body at higher temperature to the  surroundings at a lower temperature.  The heat energy is lost by radiation, sweating and respiration.  That is why when a lot of people are present in a room, it feels warmer.

The body temperature is maintained at 37C by compensating the energy lost to the colder surroundings by converting the food we eat into energy. Heat is generated in biochemical reactions in cells (we say that food is burnt to produce energy). Each gram of carbohydrate generates 4 Calories, while fat and protein generate 9 and 4.5 Calories respectively.  In the end, all energy generated by  digesting food is converted into heat which needs to be lost to maintain a constant body temperature.

Therefore, energy given off by a human body in a day is equal to the energy produced by the food we eat per day.  We shall calculate this in the following:

Suppose, that every day one consumes 2000 Calories (Some eat much more and get obese!). We can calculate, using physics, the amount of energy produced in a day.
The Calorie used by nutritionists is actually equal to 1000 times the calorie used by physicist to measure energy. Note C and c in the two definitions of energy.


1 Calorie (used in nutrition) = 1000 calories or 1 kcal (used by physicists)

In physics the usual unit to express energy is a Joule (J), and 1 calorie = 4.18 J 
(One Joule is not a lot of energy - if we lift a 100 gram tomato from ground to 1 meter height then we have spent 1 Joule of energy.  
If we took one second to do so then the rate of energy used is 1 J/s or 1 Watt; take 2 seconds, the energy spent is still one Joule but rate of energy used is 0.5 Watt)

So we consume 2000 Calories or 2000 x 1000 calories/day
= 2,000,000 calories/day
= 2,000,000 x 4.18 J/day
= 8,360,000 J/day

To make more sense of Joule per day, it is better to talk about power that is J per second or Watt (W). We divide J/day by the number of seconds in a day to obtain J/s or Watt.

A day has 24 x 3600 second = 86,400 seconds


We produce 8360000/86400 J per second (W) = 96.76 W (nearly 100 W) and this is eventually given off to the surroundings as heat.

10 humans will give off 1000W - that is like one bar of electric fire switched on!

A human gives off heat equal to that of a 100 W incandescent light bulb (old style light bulbs) which actually converts most of the electricity to heat.
The new style LED lights are much more efficient in converting electric energy to light and that is why they do not get hot when in use.  This also makes LED much cheaper to use for the same light output.

Your calorie consumption goes up if you are physically active, like running or climbing stairs than when you are sitting calmly. 
In one hour, a 68 kg man would burn, approximately, 70 Calories sitting; 85 Calories standing; 340 Calories walking at 6 km/hr and 700 Calories running at 10 km/hr.

Now you know why the room feels very warm when a lot of people are present and talking animatedly.
Or why you sweat and breath faster when jogging - you are burning extra Calories and producing heat faster.  The body has to get rid of the extra heat by sweating and respiration.

Final Word:  In this blog, we have calculated the heat produced by the food (2000 Calories per day) we consume  - essentially our metabolic rate (BMR) - and to a good approximation it is 100 W.  Under ideal conditions of comfortable environmental temperature, this energy is lost to the environment. The question I would like to address in my next blog is what happens when the outside temperatures are either too cold or too high for our thermal regulation mechanisms to work efficiently or not very well at all.  With climate change, heat strokes have become more wide spread and a good understanding of the science of heat strokes will be very useful - this is the subject of my next blog. 

Galileo Galilei - Family Tree, Friends and Foes

Index of Blogs

Galileo Galilei lived in Italy four hundred years ago.  He made some astonishing scientific discoveries which would be sufficient to rank him among the very best.  His impact on the scientific method - how a theory or hypothesis becomes established - was far-reaching.  Galileo emphasized the role of experimental verification of what a theory predicts as paramount for its acceptance.  Stephen Hawking calls Galileo the father of modern science.

Italy of the 16th and 17th century was a turbulent place with the Catholic Church fighting for its pre-eminent position against the onslaught of the forces of Reformation.  Having launched the Counter-Reformation, the Church was keen to suppress any dissenting voices and Galileo, with his research in astronomy, was sucked into this intrigue.  He paid a heavy price for demonstrating faith in his work and standing up for what he believed to be the real truth.  

I start by presenting the family tree of Galileo Galilei.   I also present a very brief political background of sixteenth century Italy and point out the relevant regions and cities.
(Click on the slide to view its full size image; press Escape key to return to the main text)

The history of Italy in the 16th century is characterized by foreign domination. Following the Italian Renaissance Wars (1494–1559), the south of Italy and the Duchy of Milan were controlled by Catholic Habsburg Spain.  The Republic of Venice, the Duchy of Florence, the Papal States and the Republic of Genoa remained independent.  The House of Medici was recognized as the ruling family of the Grand Duchy of Tuscany by Pope Pius V.

The Papal States launched the Counter-Reformation, which lasted from the Council of Trent (1545-1563) to the Peace of Westphalia in 1648 to end the Thirty Years’ War fought in Central Europe resulting in eight million fatalities. Initially a war between various Protestant and Catholic states in the fragmented Holy Roman Empire, it gradually developed into a more general conflict involving most of the great powers. The war became less about religion and more of a continuation of the France–Habsburg rivalry for European political pre-eminence.

The Inquisition was a group of institutions within the government system of the Catholic Church whose aim was to combat heresy. With the Protestant Reformation, Catholic authorities in the Vatican became much more ready to suspect heresy in any new ideas. In 1542 Pope Paul III established the Congregation of the Holy Office of the Inquisition. It had the tasks of maintaining and defending the integrity of the faith and of examining and condemning/outlawing errors and false doctrines; it thus became the supervisory body of local Inquisitions. Arguably,  the most famous case tried by the Roman Inquisition was that of Galileo Galilei in 1633.  

There are some detailed accounts online (1, 2) and good books (3) describing Galileo's scientific work.  I give a brief introduction in the following two slides.

Most scientists work very hard on the subject they love but by and large they operate in a benign environment.  May be not well paid but they are able to have a lifestyle that suits them best to pursue their research without interference from politics and other social/religious matters.  
Galileo did not have such luxury - he was short of money during the first half of his life with a big family to support.  His work was in direct conflict with the powerful Catholic Church and was hounded for 30 years by the religious mafia.  It is to Galileo's credit that in such an environment he was able to produce scientific work of such great merit and change the way science was to be practised in future years.