Radiocarbon and radiometric dating.

Radio Carbon Dating – Example for Exponential Decay

As cosmic radiation bombards earth’s atmosphere, nitrogen gets split into Carbon 14, an unstable isotope (C14 is called ‘radiocarbon’ because it is radioactive). Now because of lightening and other natural phenomenon, carbon 14 gets into the atmosphere that is taken up readily by the plants through photosynthesis. Now when we eat the plants, it gets deposited in our system. As long as we ingest carbon 14 till we die, the ratio of carbon 14 to carbon 12 remains stable in our body too, just like in the atmosphere. But the minute we die, whatever carbon 14 left in our system starts decaying. The half-life of carbon 14 is 5730 years. I.e., after 5730 years, the carbon 14 would have decayed half the amount when compared to the stable atmospheric ratio of carbon 14 to 12.

OK ….here is a simple problem to solve. In your backyard, when you are digging for your gardening work, you are coming across a mummified body . You are curious and determined that the ‘mummy’ belongs to B.C period. How will you find out the age of that ‘mummy’?

You can use the help of carbon dating to find out the ratio of carbon 14 to carbon 12 that is currently present in the mummified body. Let us presume it was 1.2 х 10 ^–12 grams. Then compare it with the atmospheric ratio of carbon 14 to carbon 12, that is 1.3 х 10 ^–12 grams.

Use the exponential decay formula. Check out the previous blog of exponential function.

A = C e ^kt

Where C is the initial amount, k is the rate of growth, t is the time, and A is the amount after time t.

We know half-life of carbon 14 is 5730 years. We need to find the rate of decay.

0.5 = e ^5730k

5730k = ln (0.5)

k =ln (0.5)/5730

Now substituting in our exponential decay formula, we have,

1.2 х 10 ^–12 = 1.3 х 10 ^–12 e ^{ln (0.5)/5730 t}

1.2 х 10 ^–12/1.3 х 10 ^–12 = e ^{ln (0.5)/5730 t}

ln (1.2 х 10 ^–12/1.3 х 10 ^–12) = ln (0.5)/5730 t

5730 ln (1.2/1.3) / ln 0.5 = t

t = 661. 69 years. So, this mummified body does not belong to B.C period. Nevertheless, it is an interesting 600 years old ‘mummy’.

Here, is a question for you. How long do u think that this radio-carbon dating be used to calculate the age? We know that every 5570 years, the radio-carbon in any given material is halved. So, if we keep computing, by the end of 60,000 years almost all the radio-carbon would have disappeared. So , radio-carbon dating would not be used to date anything that is beyond 60,000 years old!

Argon-argon dating (which has a half-life of  1250 million years) and other isotopes  are used to find the age of the rocks that are millions of years old and from that the age of the dinosaurs are calculated. This process is called as radiometric dating.

Constant Error, Variable Error, Absolute Error and Root Mean Square Error – Labview

Imagine a target where archer A and archer B are trying their luck for a good shot. The following figure is the outcome. Now let us try to figure out who has done the best. Again it depends on what you want to see. The common error measurements are as follows:

 

 

 

 

 

 

 

 

 

 

Constant Error:

 

Constant error measures the deviation from the target. The formula for it is: Σ (xi-T)/N, where T is the target and N is the number of shots. It comes with a positive or negative sign which points out the direction of the error. Absolute constant error will give the absolute value of CE alone without mentioning the direction.

 

Variable Error :

 

Variable error measures the consistency of the shots. It calculates the standard deviation of the total shots. Its formula is sq.root (Σ (xi-M)^2/N, where M is the average shot.

 

Absolute Error:

Absolute error is the overall deviation without considering the direction. In constant error there is a danger of cancellations of error because of direction. However, the error due to bidirection gets eliminated in absolute error. Its formula is Σ absolute ((xi-T)/N).

 

 

 

 

 

 

 

 

So here is a question for you? Among A and B, who is the better archer? Are you bothered about the deviationper se or concerned more about the consistency? I would say bet on archer B than A because he has lesser variablity.

Root Mean Square Error:

 

This measures both the deviation and the consistency of the shots. Its formula is sq.root (Σ (xi-T)2/N).

 

 

 

 

Coefficient of Variation:

It is nothing but standard deviation divided by mean. It gives a clear picture about the deviation. If there are lots of tremors, then the standard deviation increases and so the coefficient of variation.

Check:
Constant Error, Variable Error, Absolute Error & Root Mean Square Error

Ref:

Richard A. Schmidt, Timothy D Lee, (1999. Motor Control and Learning (Third Edition). Human Kinetics.

Click on the image to get a close-up view!

The explanations about all the errors are here. Click the link below.
Measurements of Error

Calculating Earth’s Circumference: Eratosthenus (276-195 B.C)

Eratosthenus was the first person to calculate the earth’s circumference with a fair accurate measurement. It took us another 2500 years to confirm his calculation!

His calculation is very simple and logical. He was intrigued that at Syene, a southern city of Egypt the sun was directly overhead at noon time (which means there were no shadows) on June 21 (summer solstice). At the same time, there was a considerable amount of shadow in Alexandria, a city 800 km from Syene. If earth were to be parallel this cannot be the case. At both places there won’t be a shadow from any pillar or building.

But in reality, in Syene there is not a shadow, but in Alexandria there is a shadow. He thought logically about this problem. He could  conclude that  only if the two places are not parallel that can happen, which means that earth is not flat but is curved or rounded!!!

He applied simple geometry and found the angle of shadow from the pillar to be 7⁰. He extended the angle to the center of the earth which corresponded to 800 kms, the distance between Syene and Alexandria. He then easily calculated the circumference of the earth to be 41, 000 kms which is very true with less than 1% of error!

Imagine he did this feat 2500 years back and it took almost 2000 years for the mankind to re-discover his calculation!

Generalized Motor Program – Schmidt

Motor program concept says that our movements are pre-programmed in an open-loop fashion. Imagine a seed which has a blueprint of what it wants to become in its core. Especially this motor program concept is applied to explain fast movements that are around 100-150 ms. If such were the case, would it then be possible to have programs/engrams stored in our brain for each and every action? Could this immense storage of the motor program in brain feasible? Another question that can come will be for novel movements. What about certain movements that we have never done before? Do you think that program is also imprinted on our brain?

Generalized motor program concept gives a solution for both the storage and the novelty problems. It says that by modifying the parameters such as force and timing of the movement you could have a single motor program that could be customized for different patterns.

A common example would be the walking pattern: you could walk slowly or fast, trot or run slowly or fast. Shapiro (1981) in his experiment showed the ratio of timing of a single step (100%): See the diagram. Now this ratio was unperturbed when the subject was walking or running. What does it mean? In slow walking the timing parameter was slower and for running the timing parameter was faster in the common motor program that oversees both walking & running! Another example to think about is a familiar song one could sing quickly-slowly or loudly-in whispers. You can make out the original song due to its fixed ratio of the timing and note of the song. Here all you have done is changed the parameters of that song with respect to time and audibility. The same thing applies for the handwriting, whether you write in your right or left hand or with foot or manipulating a pen between your teeth. One could make out the unique pattern of an individual’s writing style!

Ref:

Schmidt, R.A. (1988). Motor control and learning: A behavioral emphasis. Champaign: Human Kinetics.
Shapiro, D.C., Zernicke, R.F., Gregor, R.J., & Diestel, J.D. (1981). Evidence for generalized motor programs using gait-pattern analysis. Journal of Motor Behavior, 13, 33-47.

Posture Biomechanics

Supposing you weigh 50 kg in your weighing machine, then what is your force when you are standing on the weighing machine?

We know, f= m*a, so your force is 50 * 10 m/s2 (approx. of gravity which is 9.8m/s2) = 500 N.

So, even though you weigh 50 kg on earth or moon, your force on earth is 500 N and your force on moon would be different as the gravity of moon is lesser than earth!

Another point to ponder is you are exerting 500 N on earth and the earth is exerting that equal amount of force (500 N) as a ground reaction force on you. Both the forces cancel out and you are standing blissfully unaware of whatever is going on. If the ground is not exerting an equal amount of force on you then you would pierce against the ground with your force like a metal coin in a glass of water.

So, what exactly is a force?

Remember Newton’s I law? Force is that which alters the state of rest or of uniform motion of an object.

Balanced forces result in equilibrium.

Here comes another interesting question. What is then a moment?

Whenever a force acts at a distance from a joint/pivot, then it creates a torque or moment that tends to rotate the object around that joint/pivot.

Moment = force * distance

The classical example would be a seesaw. You tend to rotate with respect to the pivot up or down. Going back to our previous example, when you are standing on the ground, you are exerting your force (in this case 500 N) to the ground. Where exactly are you exerting? Is the force goes throughout the whole feet or hypothetically at some point? Remember center of mass of an object? Brush up your center of gravity and line of gravity before you read the following paragraphs. Supposing the center of mass or center of pressure of your foot is mid-point between the foot and your nearest joint/pivot is your ankle joint, then you do have a moment generated here, isn’t it? The force is say, 3-4 cms away from your ankle joint, and then the moment that is created would be

M=500 N * .03m = 15 N.m.

i.e., a moment of 15 N.m is trying to rotate your ankle joint up (dorsiflexion). But again there you are, standing blissfully unaware of what is happening at your foot. So, what is preventing your rotation? The action of your plantarflexors! Yes, that’s right; they produce an opposing plantar flexion moment that opposes your ankle rotation up.

Next time when you are standing for a long time and wondering why you get calf pain, you know the answer at least…

Now, coming out of the foot segment, let us concentrate on the whole body. Where do you think you have your center of gravity and line of gravity? The center of gravity is approximately located anterior to S2. And following are the moments that result due to the position of the LOG either in front or behind the joint, which are opposed by the moments developed by the opposite groups of muscles/ligaments.

Newton’s Laws of Motion

1. Law of Inertia:

An object tends to remain in its state of rest (or of uniform motion) unless it is disturbed by an external force.

Therefore, al objects tend to resist their change of state.

Example, if the driver applies a sudden brake, you tend to fall forward with a jerk because your body still wants to move in the same speed and direction of the car.

2. Law of Acceleration:

The acceleration of an object is directly proportional to the force of the object and inversely proportional to the mass of the object.

Accleration = force/mass or

F=m*a

F is expressed in Newton. One Newton is defined as the amount of force required for a mass of 1 kg to accelerate 1 m/s2.

3. Third Law of motion:

For every action there is an equal and opposite reaction.

For example, if you slap the ground with your hand, you get pain in your hand. With whatever force u exert on the ground, the ground hits u back equally

Another example is when you are swimming or paddling, you are using your arms to push the water behind. What does the water do to you? It propels you forward.

Another common example is when you are walking. You place your foot on the ground and push it backward. The ground exerts an equal and opposite force to push you forward!

Index of all my blogs

Click on ‘read full post’ at the end. You can see the links!!!!!

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Basic Statistics

Basic Statistics
Variability – Standard Deviation and Error
Z Scores
Pearson’s correlation
Common Statistical Tests
Linear Relationship and Linear Regression
Regression Basics
T-Test Basics
ANOVA Basics
post hoc
Two Way Independent ANOVA
One Way Repeated Measures ANOVA
Degrees of Freedom (df) in Statistics”

Everyday Math
Logarithmic Function
Exponential Function &amp
Doubling Time
Radio-Carbon & Radiometric Dating
arithmetic-geometric series
Linear Relationship and Linear Regression

Intellectuals of the past
Zeno’s paradox: 495 b.c
Calculating Earth Circumference: Eratosthenus: 276-195 b.c

Labview
Save_Data_with_Header
Labview-To read column from 2-dimensional array
Constant Error, Variable Error, Absolute Error & Root Mean Square Error

MotorControl
Measurements of Error
Speed Accuracy (Spatial) Trade Off: Fitts Law
Linear Speed (Spatial) Accuracy Trade Off-Schimdt
Speed Accuracy (Temporal) Trade Off
Generalized Motor program – Schmidt

Personal
My encounter with elephants
Migration of birds

Physics
Electromagnetic Radiation
Newton’s Laws
Velocity and Acceleration – Basics
diffusion-osmosis
Friction-Basics
Posture Biomechanics

Zeno’s paradox: 495 b.c

Zeno’s paradox origins from the concept that a single line segment will have an infinite number of points. The paradox is the following.

If Achilles tries to catch a tortoise that is slightly ahead of him, he cannot reach the tortoise at a finite time. By the time Achilles reaches the initial position of the tortoise, the tortoise might have already moved ahead. He needs to move infinitely, because as Achilles moves closer to tortoise, the tortoise would have moved slightly ahead, though at a very very miniscule level. So, where does this infinity end? Does it end at all? Where do we keep a pull stop?

Modern Implication: Think about calculating the area of a finite circle using triangles. You keep adding triangles inside the circle and fill most of the space of the circle. As you keep increasing the triangles, you might decrease the extra space but could never fill the whole circle with triangles. Infinite triangles in a finite circle! That is why we stick to approximations.

This approximation is exactly what we use in integration (calculus), which Newton and Leibniz discovered.

Velocity and Acceleration – Basics

Supposing your hand is on the table, which we term as position a. Then you move the hand forward to reach a book. We will term this new position of your hand as position b.

The amount of time it takes to move the distance from position a to b is the speed of your arm. If you add a direction component to it, then it is the velocity, that is, dx/dt. If you plot position vs. time, then the slope will be your velocity.

v=x/t

Constant velocity: If the rate of change of position is uniform, then you are moving at a constant velocity. Example, if you are going in a car from New York to New Jersey at a constant velocity of 70 miles per hour for two hours, can you calculate the distance traveled after 2 hours?

Distance = constant velocity * time

x= v * t

 

 

= 70 * 2 = 140 miles.

Now after 140 miles you take a break. Then you continue for half an hour with the same constant velocity. Now, how far have you traveled?

x= x0 + v * t

 

 

The total distance (x) will be 140 (x0 or starting distance) + (70 * ½) = 175 miles.

But is it this easy to calculate the distance? Do we drive in constant velocity all the time? What about the traffic and the pick up speed of the car? In reality the velocity is never constant. After a stop sign we increase the velocity from zero to 70 kms (70,000 m) per hour (3600 s) in 5 seconds. Here the rate of change of velocity is not constant, it is changing and is very fast and that is called as the acceleration.

a= v/t

 

 

Acceleration = 70,000/3600 * 1/5 = 3.89 m/s2. That is, after 1 sec the car is going from 0 to 3.89 m/s (velocity). After 2 sec, it is traveling at 7.78 m/s (velocity) and the acceleration is 3.89 m/s2 because acceleration is velocity divided by time (7.78/2 = 3.89). After 3 sec, car is traveling at 11.67 m/s (velocity = acceleration or 3.89 * 3), after 4 sec 15.56 m/s, after 5 sec 19.45 m/s.

Click on the figure to get a clear view:

Therefore, the velocity is ranging from 3.89 to 7.78 to 11.67 to 15.56 to 19.45 m/s. That means it is uniformly (constantly) accelerating at a pace of 3.89 m/s². If the velocity is constant, i.e., 3.89 to 3.89 to 3.89 to 3.89 m/s then the acceleration will be zero.

This can be applied for acceleration due to gravity. As a body freely falls, in each second the velocity doubles at a constant rate of 9.8 m/s².

Supposing a car is going from 0 to 20 m/s at a uniform acceleration of 5 m/s². How long will it take to reach 20 m/s? At 1 sec, the velocity will be 5, at 2 sec the velocity will be 10 m/s, at 3 sec it will be 15 m/s and at 4 seconds the velocity will be 20 m/s. Time = velocity/acceleration=20/5=4 seconds.

The increase in velocity from zero to 40 miles per hour, could take about 2 to 30 seconds depending on the capacity of the car. Greater the acceleration in a short period of time, higher will be the horsepower or pick up speed of your car.

But once u hit the highway and set your car to a cruise control of 70 miles per hour, then your velocity becomes constant and you have no acceleration (zero acceleration) at all.

Now you are facing a ramp and without releasing the cruise control if you turn around, are you in zero acceleration? NO. Even though you are in constant speed, you are changing direction constantly here, which means that you are changing velocity and therefore, accelerating.

The common formula to measure the final position of a freely falling object is

x=x0+v0+1/2at2

 

 

Common example: You are dropping an object from a height of 100m, after what time, will it reach the ground.

Here x₀ = 0, x = 100 m, v₀ = 0 (initial velocity), a = -9.8 m/s².

100 = 0+0-1/2*9.8t²

t² =100/4.9=20.40

t = 4.5 seconds.

If an object falls from a greater height, then it will achieve terminal velocity. What happens in these conditions is the air is playing a greater role by means of friction. If only the gravitational force alone were to act, then all objects obey acceleration of gravity, irrespective of the weight. If this frictional force is equal and opposite in magnitude to that of gravity, then the object is in terminal velocity (the rate change or velocity is constant) or in zero acceleration.

Here is a common question to you. If we are ought to fall towards the ground at zero acceleration (terminal velocity) then how can a guy in a parachute is coming down slowly? Or how a leaf is slowly coming down to rest?

By the help of parachute we greatly enhance our surface area (up to 100 times) and therefore lower our terminal velocity.

Check this website:

http://www.pacifier.com/~ppenn/accelgage.html

Electromagnetic Radiation

Have you ever wondered about a world without light? It brings to my mind about the sinister science fiction movies that always show an alien land where it is so dark with occasional flashes of light. Yeah, we all know that earth at least gets its light/electromagnetic radiation from sun. How does this light act? Scientists say light acts like waves as well as particles. Imagine a lamp that emits light and when you block them with a cup, you get a shadow of cup which explains that light is nothing but a stream of particles (massless). Now how will you explain a rainbow? Light falls into millions of tiny droplets of rain to scatter a wave like rainbow. Here light acts like a rippling wave. To compromise, we could say that light carries the streams of particles (photons) in waves.

The light what I am talking about not only includes the visible light that we see, but also the other ‘lights’ like infrared, microwave, x-rays etc…We collectively call them as electromagnetic radiation. Visible light (what our eyes see) forms a tiny part in that spectrum.

Now, a single source of electromagnetic radiation (eg. from sun) has different wavelengths of photons. The smaller the wavelength the more particles (energy) it carries, that is, the greater the ability of the light to act on some physical substance. Example being, ultraviolet rays that could burn your skin. Alternatively, the longer wavelength carries fewer particles (energy) in them that don’t harm you. You can stand next to a high powered radio transmitter without any potential problems.