Power: discrete or continuous

Article status:                              Draft
Time Estimate for Reading:        30 min
Learning Objectives:                  Summarizing the concept of Power
Effort Required:                          Medium
Pedagogy Model:                       Evolution, Formula Analysis, Inter-disciplinary
Prior Physics Concepts:             displacement, velocity, acceleration, mass, power, force
Prior Math Tools:                       Secondary school level Arithmetic, geometry and algebra

We have been discussing about power of motion and power of rest without considering how do we provide this power. providing power would require an interaction. May be we can visualize power applied as discrete (per second) impacts. For an interaction it takes some amount of displacement and time. But before we get into impacts, we need to understand a little more of rest. Even rest is a result of invisible discrete impacts. we call it continuous push but in reality, they are discrete.    

Let us continue with "rest". We discussed about falling bodies and got to understand that independent of mass, all of them fall with constant acceleration. we seem to repeat this quiet often. This seems to be the starting point for every thing. Starting with this we were able to define and measure mass. we were able to measure constant velocity and constant acceleration motion. we also deduced the concept of force and ended up with equal and opposite reactions.

So, we will be looking at two situations. one, a ball at rest on a table. Second, how do we keep a ping-pong ball at rest without a table.

Situation 1- Ball on table:
Let us resume our discussion from "power of rest". An object  should not accelerate and we need to provide an acceleration in the opposite direction. This can be provided by placing the ball on a reasonably rigid table; table providing the necessary opposite acceleration.

Notice the use of word reasonably rigid. If the table is not rigid enough, it would undergo extensive deformation (imagine placing this ball on a cloth) and in extreme cases, it might even break.  

This brings us to another question. How can a non-moving table provide acceleration in the opposite direction? To answer this question, let us consider a simple example of pressing a ball with our fingers. Once we release our fingers, the ball gets back to its original shape. We can roughly imagine the ball to be made of springs inside (relate it to the newtons mass determination experiment). So springs when released, have the ability to provide an acceleration. The amount of push (or acceleration) the spring can provide is given as kx. K is the stiffness of the spring and x representing the displacement. A wooden table in this case acts like a spring (the deformation will be so minuscule and may not be visible to the naked eye).

The concept of spring may be used as a means of visualization. Later in the 18th century, once we understood that matter is made of atoms and atoms in turn are made of charged particles, the effect of spring comes from repulsion between electric charges. The invisible springs.

So, the invisible springs in the table will be acting on the ball continuously. the word continuously is important here (you may refer to "power of motion" as a refresher). We may visualize it like this. The table, seeming at rest, pushes (accelerates) the ball, a very little, upward, for a very short duration and comes back to its initial position. the ball falls down along with the table. this process goes on again and again. this may not be visible to our eye but this is what must be happening (based on our understanding so far). You might have come across sagging rack's. racks's left unattended for a long period, might sag due it's self weight. There is fatigue. Even inanimate objects seems to get tired.

You might have experienced this if have tried to retain a trolley at rest in an inclined plane. you could feel the trolley pushing (or pulling) back on you. this would not be the case if you left the trolley on a horizontal plane. there is no motion. that is no visible acceleration or deceleration of the trolley. How do we handle this?

With advancement in science, we came to know that atoms and molecules are in continuous motion. A measure of this motion is termed as heat energy (total kinetic energy of all molecules. but the concept of kinetic energy is yet to be discovered). So, power for acceleration might be taken from the atmosphere.  Image something like this. The table pushes on the ball and loses some power and there would be a drop in temperature (average kinetic energy of molecules).  It is natural for all bodies to be at the same temperature. So, atmosphere provides this to the table. So, looks like every object, seeming at rest is an engine working at microscopic level.

We are still in the days of newton. atomic structure, engines, understanding of heat and temperature and yet to be discovered.

So, newton might have come to a conclusion that, for objects at rest, we do not require any power. All that we need is an acceleration in the opposite direction. Some how, the earth along with the table should provide this acceleration. Action = reaction again.

Situation 2: Ping-pong ball at rest without table

                                                              
The concept of rest can be better understood if we try to balance a ping-pong ball by blowing air with the help of a compressor driven by electric motor (instead of seeking support from earth by resting it on a table). With out getting into the actual numbers, this is how it works. The pin-pong ball has some mass 'm'. it is naturally accelerated towards the earth by a quantity 'g'. for this ball to remain at rest we need to provide an acceleration opposite to 'g'. How can air that we blow offer this equal acceleration?

We know how to calculate the power required for uniform velocity motion.
power = mass of ball * acceleration x uniform velocity

This power should be provided by electric motor and is equal to V*I (more on this later)

Motor should give this power to air. So for air,

Acceleration required should be equal to 'g'. equal and opposite reactions.

the mass of air displaced should be equal to mass of ping-pong ball. mass of air can be calculated from the volume of ball . for the sake of simplicity, consider the ball to be a cylinder of diameter d and height h. knowing the density of air, we can calculate the mass of air that will occupy the volume of 1 ball. from this information, we can calculate the can calculate the height of the cylinder required for an equal amount of air.

This height of air has to be displaced in 1 sec. that brings us to calculate the velocity.

Now power of air = mass of air * acceleration of air * velocity.

Since this air passes over the ball, the velocity of air will not carry the ball away and it will just keep the ball at rest. The concept of continuous interaction.

So, the power of motor should can be calculated by an appropriate choice of voltage and current.

There are some factors which we have not considered, but this would gives us an understanding that it takes some power to keep an object at rest. another example would be the hill start feature in automobiles.

We just used a simple method. There are other accurate methods using Bernoulli's principle which were discovered later.

Even rest requires power.

Having discussed on rest, we can move on to deformation and Impact.

What happens to objects when they hit the ground after a free fall? A solid ball does not break nor bounce back, a rubber ball bounces back to more or less the same height, an egg or tomato loses its shape beyond recognition.

When an egg hits the ground, we call this Interaction an impact, egg undergoes deformation. The atoms and molecules that form the shell of the egg can only undergo a certain amount of deformation and beyond this they will break. A macroscopic study has a relationship between stress and strain. Within a specific limit of strain, the objects would not break or yield. We have introduced a few words like stress and strain to our vocabulary. To understand these words, we need to add one more word pressure, which we shall take it up later.

Objects that bounce back are considered to be elastic. They can deform without breaking.

We can summarize the discussions as follows.
- For an object to be at rest we use m1a1 = m2a2  (supported by table or earth)
- For an object to be at rest without support from earth we need power (ping-pong ball example).
- When applying power, one object (cause) interacts with another object (effect) and during an interaction, there would be deformations at macroscopic and microscopic level
- If either object does not withstand this deformation, objects may break and we would not get the intended motion


In the discussions on power of motion we had the engine and object moving together. The engine providing the necessary power for the object to move with a required velocity or acceleration.

In the discussions on power of rest of ping-pong ball we had air moving past the ball.



But in applications like bat hitting a ball (gun firing a bullet, bow and arrow etc), the bat does not travel along with the ball. We shall get into the regime of special class of interactions called "collisions" in the next article.

Power of Rest and concept of force

Article status:                              Draft
Time Estimate for Reading:        30 min
Learning Objectives:                  Understanding the concept of Rest and introduction to Force
Effort Required:                          Medium
Pedagogy Model:                        Evolution, Formula Analysis, Inter-disciplinary
Prior Physics Concepts:             displacement, velocity, acceleration, mass, power
Prior Math Tools:                       Secondary school level Arithmetic, geometry and algebra


A ball allowed to fall, will travel towards the ground with a constant acceleration. So, if we do not want it to fall, we need to cancel this acceleration by providing a sufficient acceleration in the opposite direction.

In real life, we may choose to hold the ball in our hand or leave it on a table. So, the hand or table should be providing a sufficient opposite acceleration.

We should also be aware of the point that, all objects fall towards the ground with the same acceleration, independent of their masses. So, earth should somehow vary the pull depending on the mass of the object.

Instead of saying earth is pulling, we may also say something is pushing. there is the atmospheric air that might push. Can this be? if we consider two bodies of same shape but of different masses, air would have to push them at the same rate (why this is not possible to be addressed here) . Gilbert had already established that earth has a magnetic field. and probably, the bodies are attracted due to magnetic effects. but this can be discounted as there are both magnetic and non-magnetic materials. The concept of electricity (static electricity and magnetism were discovered as early as 650 BC. may be before that.) can also be discounted in a similar fashion as there are electrically neutral materials.

giving a little thought, we arrive at two parameters. the mass and acceleration. advancing a little and assuming that there are no other parameters, mathematically LHS = RHS and considering the fact that it might be due to earth (just assuming. not sure i would have done so in the days of newton), we may say

mass of object * acceleration of object should be equal to mass of earth * acceleration of earth. (just using arithmetic)

how do we measure the mass of earth. we have defined the mass and we have also defined density. so if we know the radius of diameter of earth, we can arrive at a rough estimate of mass of earth.

Eratosthenes of 100 BC comes to our help. He came up with an experiment and calculated the radius of earth to be close to 6350 km. You may watch this video for Eratosthenes experiment.

here mass of object and mass of earth are constant. acceleration of object is also constant. the only thing that could change is the acceleration of earth.

M1* a = Me * Ae

M1/Me = a/Ae1  ---- (a)

M2/Me = a/Ae2 ----- (b)

divide a/b

M1/M2 = Ae2/Ae1   - (c)

we know, M1 and M2.  from here we can calculate Ae2/Ae1.

with a very rough estimate of mass of earth and substituting in this formula, we can calculate the acceleration of earth; a very,very small quantity. remember, this is only possible if m1a2 = m2a2.

So, in the process of doing all this mathematics and calculations, we may also come to a hypothesis that m1a1 might be equal to m2a2 between any two objects. the quantity m*a assumes significance and can be given a word; force (as per Aristotle force = mv). force cannot equal to mv because, in the course of interaction the velocity will change and change in velocity is acceleration.

Starting from Galileo's experiment, getting on to define mass and also coming up with law of action and reaction. F1 = F2 (in the opposite direction). it is Newton again.

Can we apply this to our mass experiment using a spring.
mass of ball * acceleration of ball = Force of Spring.
                                                       = stiffness of spring * displacement.

Note: It is important to note that the force of spring varies with displacement.

if the spring is not fixed, there cannot be any displacement of the spring and the ball cannot accelerate.

To answer the question of keeping the ball at rest, the supporting table should provide an equal and opposite force which should be equal to mass of ball * acceleration of the ball.

Having analyzed the vertical motion, let us consider the horizontal motion. there is no acceleration acting on the ball. But think of a situation where the ball is moving with a constant velocity.  we need to bring it to rest. in order to bring it to rest, we need another object. Now, we have to understand what happens during an interaction between two objects.

We could leave at that. but let us go a little further and analyze the situation. If we happened to hold the ball for a sufficiently longer time, we would feel a little pain. for that matter, we cant even stand for a sufficiently long time. we feel tired. so, we choose to sit. given sufficiently long time in a seat, we would start moving a little. for that matter, we cant even lie down for long. we would feel tired again. we cant sleep for a duration that is longer that required. we cant even rest for long!.

So, let us see if we connect the concept of power, the concept of force and the law of equal and opposite reactions.

Power: The cause of motion

Article status:                              Draft
Time Estimate for Reading:        30 min
Learning Objectives:                  Velocity, Accelration and Power
Effort Required:                          Medium
Pedagogy Model:                        Evolution, Formula Analysis, Inter-disciplinary
Prior Physics Concepts:             displacement, velocity, acceleration, mass
Prior Math Tools:                       Secondary school level Arithmetic, geometry and algebra

We wish to analyse and predict the motion of objects. So far we have identified that the amount of motion can be measured using the concepts of displacement, velocity and acceleration. We also understood the significance of acceleration; the bridge between space, time and mass.

Just to refresh our memory, measurement of velocity requires two points. Measure the distance between them and divide by the time interval.

Similarly, measurement of acceleration requires 3 points. measure the velocity between points 1 and 2, 2 and 3. also clock the time interval between points 1 and 3. now acceleration can be calculated by taking the difference in velocities and dividing by the total time taken. By getting the 3 point closer and closer, we would get more accurate measurement of acceleration. Mathematically speaking we say acceleration at a point (we shall see how geometry helped newton to settle this issue). but in practice we need 3 points.

Now that we know how to measure, We know the effect. Having known the effect, we are interested in causing the effect. Let us try to understand the cause of motion; motion in a straight line with constant velocity or constant acceleration. For that we will have to add the concept of power to our vocabulary.

What is power?

Let us begin with the definition, take a real life example and try to understand it step by step.

Definition:
Power = mass of the object x required acceleration  x velocity of the object. This might look incomprehensible in the beginning but it is rather a very simple concept. It would be all the more intriguing if you have come across the text book definition "power is rate of doing work or rate of spending energy". Similar to displacement, work and energy are abstract concepts. They cannot be experienced in real life. We wont need them for the time being. Let us defer a discussion on work and energy later. So all we need is power to cause motion.

Since we state that power is real, let  us take a real life example of using a two wheeler. Say a bike. the bike may be a by cycle (mechanical power), gasoline power bike (thermal power) or an electric bike (Electric power). Notice that power can be provided from different sources.
Electric Power = Voltage x current
Thermal power = mass x Specific heat capacity x average temperature difference/second
Mechanical power (linear motion) = mass x acceleration x average velocity
Mechanical power (rotational motion) = 2 x pi x revs per second x mass  x acceleration x radius

Do not bother if you have not understood the formulas. we will get to it shortly.

Real life example 1: Bike
State 1.Initially the bike is a rest (initial velocity is zero).

State 2: Mount on the bike and start the engine. Let us assume it has an automatic gear and the road is perfectly horizontal.

State 3:Turn the accelerator (throttle) handle and the bike starts moving. You experience a change in velocity as long as you keep turning the accelerator handle (within the limit of traffic and vehicle capability).
Note down the distance traveled and time taken to reach the required velocity

State 4: Once we reach the required velocity (we wont be using the word speed. speed is required only for academic purposes), notice that we will not be twisting the accelerator handle any more. may be if you have noticed carefully we will release it quite a bit once we reached the required velocity.

State 5: Once the destination is reached (we are considering only straight line motion for now), Notice the distance traveled and time taken at constant velocity. Now, we release the accelerator handle from current position, we will observe that the velocity automatically decreases. Again measure note the distance traveled and time taken to reach zero velocity. The rate of decreasing velocity is termed deceleration. Just the opposite of acceleration. 

If acceleration or deceleration are not constant, we will observe jerk; and jerk is not a comfortable experience.

Let us now go on to calculate the power required at various states.

Consider mass of the bike (including the person) = m kg

State 3: Power should be equal to mass x required acceleration of the object x average velocity
             therefore power required = m x a x d/t.  and is measured in watts.

m x a is quite intuitive. but, we may ask why d/t?. The answer is, any displacement will take a certain amount of time. So to define power in a sentence, Power is a quantity which is a product of mass to be accelerated through space in a given amount of time. the given amount of time is one second. Observe that during an accelerated motion, the velocity will keep changing every second. So, we need to keep changing the power every second (reason for gradually twisting the throttle). we may say that we should provide a continuously varying power. notice that, if you happen to accelerate for a period of 2 secs, the power required in 1 sec is less that the one in the next second (velocity would have increased by now)

There are other simpler methods to calculate continuously varying power using abstract concepts like work and kinetic energy. the concept of kinetic energy had to wait for calculus and calculus was invented (though there are many others who have contributed) by newton and leibnitz. So, we shall defer it for now.

State 4: Now we need a constant velocity motion. We would be tempted to apply the same reasoning above and would arrive at a quantity m x v. But this is not so. Why?
We have friction and air drag acting in the opposite direction. They offer deceleration (the opposite of acceleration) and so we cannot use v. How to calculate friction and air drag will be covered in a separate article. for now, it is sufficient to know that they offer deceleration.

Now the formula for power = mass x external deceleration x velocity

Notice that the required acceleration of the object has become equal to external deceleration. We just need to keep accelerating through the required distance every second. that is the interpretation of the formula for power. Notice that we don't have to use the concept of average velocity in this formula. in a constant velocity motion, the velocity will be same at any point we choose to measure.

We may also consider a zero friction and zero air drag scenario. There wont be any deceleration. Hence, we would not require any more power. We just need the power sufficient enough to take it to the required velocity and the bike will keep on moving. This is the condition is space. This is Galielo's law of inertia.

Stage 5: Now we need to decelerate. All that we need to do is to release the throttle completely and the bike will come to a stop. Friction and air drag will offer the required power. But if we have to stop it in a shorter distance and shorter time, we can apply brakes.

Having introduced the concept of external deceleration, we can modify the power calculation for stage 3.
Power = mass * (required acceleration + external deceleration) * average velocity.

So, in order to cause motion of constant velocity or constant acceleration all that we need is power. a continuous or varying power supply as long as (for the duration of time) we require motion.

Now it is clear why bike manufactures specify power (maximum power). Along with this they will also specify the mass and acceleration of the bike.

Going back to the formula p = m x a x d/t, we know p, m and a. we need to know either distance or time to calculate the remaining unknown. acceleration comes to our help again. a = (v-u)/t. v the final velocity, u the initial velocity (consider it as zero) and t the time taken for this change in velocity. from this we can find a. From the given power, we can now calculate how much displacement it will take to reach the required velocity. remember, the displacement in each second is not same.  so we got to add the distance traveled in each second to arrive at total displacement.

Real life example 2: Lift
The bike example considered only the horizontal motion and we need not have to bother about vertical acceleration (it is important to note that a part of vertical acceleration would be used in calculating the amount of friction. for sake of simplicity, we just said friction is a decelerating action).
Now, a lift would involve motion is vertical upward and downward directions. so, what is the difference. the difference is that we have to move against gravity during the upward motion and move with (assisted by) gravity.
It might look complicated. but all that we need is to use the same logic that we used for bike.

Upward motion:
power = mass x (natural downward acceleration + upward acceleration required) x velocity.

Downward motion:
power =  mass x (natural downward acceleration - downward acceleration required) x velocity.

You will observe that this is similar to horizontal motion with friction and air drag.  Instead we have considered downward acceleration. an interesting point here is gravity will assist downward motion and oppose upward motion. but friction and air drag will always oppose motion.

It is good to introduce the concept of counter weight here. use of counter weight will reduce the power requirement and hence the size and cost of the motor can be reduced. How does a counter weight help?. simple. It will cancel out the natural acceleration.

For the sake of convenience we may add some more terms to our vocabulary by analyzing the concept of power.
varying power = m * a * average velocity     -> instead of average use may use varying ever second.
continuous power =  m * a * velocity
force = m * a
work = f x d. Work is another abstract concept measured in joules
Hence power = w/t or f * v depending on our interpretation and is measured in watts.

You might have also observed that we need not have to use the concept of force till now for solving real world problems. Force is yet another abstract concept. once we say force, we need to say across what distance and in what time interval and that becomes power.

Having understood intuitively, the concept of power, we shall get to the concept of work, energy (potential and kinetic) and also the concept of power of rest and power of collisions

Acceleration: the bridge between Space, Time and Mass.

Article status:                              Draft
Time Estimate for Reading:        30 min
Learning Objectives:                  Importance of Acceleration in understanding physics
Effort Required:                          Medium
Pedagogy Model:                        Evolution, Formula Analysis, Interdisciplinary
Prior Physics Concepts:             displacement, velocity, acceleration, mass

Prior Math Tools:                       Secondary school level Arithmetic, geometry and algebra




Every student who continued beyond primary school would know this story. The story of Newton sitting under a tree, observing an apple fall and coming up with the theory of gravitation. We chose to remember the story and may be did not give enough attention to the concept behind this story. Even we went on to sending a little speck of the tree to Space (a part the same apple tree has been taken to space in 2010 )

The concept behind the falling apple is acceleration and Newton (1642 - 1727) would have got the idea of using acceleration as the core concept, to bridge space, time and mass; eventually leading to the three laws of motion (dynamics) to analyze and predict motion on earth (terrestrial mechanics) and Planetary dynamics.

Take at look at the seemingly simple mathematical relationships between space, time and mass.

                                       Displacement                 =  acceleration * time^2
                                       Uniform Velocity            = Displacement/time
                                       uniform acceleration     = Change in velocity/time
                                       mass                               = force/acceleration (m = f/a)
We should also make note of sinusoidal acceleration and Jerk. Jerk is abrupt change in acceleration (The change is neither continuous nor sinusoidal)

Apart from being a bridge, acceleration ss significant because, nature took close to 2000 years to reveal this story.

To understand acceleration, we need to begin with Euclidean point (geometry- 350 BC), The Galilean Falling ball experiment (1590), The Galilean Telescope (1620), The Kepler point - (1571 to 1630) and for the sake of completeness; the Planck Point - 1900 AD (the Measure of Minimum Possible Energy). With the introduction of Planck Point, there are no infinitesimals anymore. A Point is no more abstract and seems to have a definite, discrete value of Planck Point.

Let us time travel to 300 BC. Euclid has given a formal account of geometry (space) and it was made possible by the abstract concept of point. Abstract in the sense that something that is not natural, that cannot be physically experienced, that which is not intuitive or that which is beyond common sense. Now science gets beyond common sense.

Many years pass by. 2000 years. Quite a long time and we reach the times of Galileo. Galileo establishes the relationship between space and time with his vertical falling ball experiment (acceleration), the horizontal motion (rest and constant velocity motion does not require an external force), a combination of vertical and horizontal motion leading to parabolic motion (all of these are formalized using geometric mathematics and thanks to Euclid).

Galileo, with his experiments, more or less, helps define the concept of acceleration (and of course velocity) It is unbelievable that just linking up space and time took such a long time (and from there it took hardly 30 years for Newton to define mass and link up space, time and mass).

Why did it take so much time?

Let us begin our journey by first getting to terms with distance, displacement, velocity and acceleration.

To begin with, Descartes (1637) helps us with the concept of coordinate geometry. The euclidean non-existent point can now can be defined with well established co-ordinates (x, y, z) with respect to a fixed origin.

With the definition of Descartes "co-ordinate" point, Distance can be measured between two points. The measurement being made in stadia, ft or the SI units, The distance may be measured across any path. Notice that we are using the word path instead of straight line.

Now let us define and we need to introduce one more word "displacement". Displacement is the shortest distance between any two points. In real life, there cannot be any displacement without considering the time interval (ah. that's velocity). Hence displacement is again an abstract concept.

It is easy if we confine our analysis to straight lines in planar space. Archimedes (300 BC) has shown us that a circle can also be geometrically considered to be made of a polygon with sides of infinitesimal length. We can now extend the same concept to any other path and henceforth we can conveniently use displacement and do away with the concept of distance from our science vocabulary. The concept of spherical and hyperbolic spaces are yet to arrive (and geodesic is a common term to describe the shortest path)

Having defined displacement, Velocity can be defined as the displacement (measured in meters) in unit time (measured in seconds). Velocity is given the unit of meters/second. Simple. Notice here that, space (m) occupies the place of numerator (may be because, those days, we were yet to get to terms with time.)

Having defined velocity, Acceleration can be defined as the rate of change of velocity. Now there are two choices; rate of change with respect to space or with respect to time.  This was the fundamental problem in the past; the past before the days of Galileo. Those were the days when relationship between space and time were yet to be established.

Choice 1: Let us try to define acceleration as rate of change with respect to space. This is what we get. acceleration = velocity/space = (meters/second)/meters = m/(ms) = 1/s. Okay. Now, What is the inverse of time?. It is frequency; measured in Hertz. But frequency as a concept was established for analyzing oscillatory motion; and analysis of oscillatory motion had to wait for discovery of differential calculus. Nature chose to reveal the secrets of calculus to newton and Leibniz (and they are yet to be born).

Choice 2: The choice now is to use the rate of change with respect to time. Galileo experimentally proved that, for falling objects closer to the surface of earth, displacement varies with respect to square of time. d is proportional to t^2 (independent of the mass of objects).

Getting back to acceleration, how can the unit of displacement (m) be equated to square of time(s^2). Well, we may replace the proportionality constant with something that would give us just 'm'; a measure of displacement. So, this is the clue to acceleration. Acceleration can be given the units of m/s^2. On a similar note we can also apply this logic to velocity and define acceleration as the rate of change of velocity with respect to time (m/s^2). Fits Nicely.

So far, we have put the concepts of displacement, velocity and acceleration into perspective. Now the challenge is to measure them.
- Displacement (rather length) can be measured with a measuring tape. Simple.
- Velocity can be measured with a tape (length between two points) and a stop clock (end time - start time ). Simple again.
- To measure acceleration, we need final and initial velocities between two points (say point A and Point B). Sounds interesting. It is no more simple. How do we do it?  Somehow we need to arrive at a method to measure the velocity at every point. Feeling dizzy.

The way forward is to stretch point A into A1-A2 and Point B to B1-B2 to some known length; the smallest possible length, depending upon the resolution of our measuring equipment. Remember, we should also measure the time interval and the resolution of clock should also be taken into account. With this in place, we can calculate the initial and final velocities. Once we know the velocities and time interval, we can calculate acceleration.

Well. We may now understand (hopefully) and we may know how to measure displacement, velocity and acceleration. With that we can understand the significance of acceleration as the bridge between space and time. A bridge that has taken almost 2000 years to build.

Newton is set out to establish the bridge between space, time and mass.
From Galileo's falling ball experiment, it is fascinating to see that, for bodies of independent mass to fall all the same rate (constant acceleration), earth has to some how know the mass of each object and try to pull them with an appropriate force.

In order to validate this idea, Newton must have come up with an experiment to observe the behavior of mass on a horizontal plane and applying a constant force; the force being exerted by a spring (instead of gravity). This leads to an interesting and expected observation that, acceleration of the object varied according to mass of the objects.

By calibrating a spring with a known reference weight and conducting the horizontal block experiment and measuring the acceleration, mass can be determined.

An excellent account of this experiment can be found in this video by Prof Shankar Ramamurthy of Yale University  (it is a 1 hr video. This experiment is discussed between 20min and 40min)

There comes the equation m = Force exerted by the spring/acceleration (F/a). it is important to not here that, force from the spring will vary with displacement. since we are measuring the acceleration at the same point for any mass and also the forces cancel out from the equation, this experiment is considered valid.

We could have also used a hanging mass instead of spring. but we will consider that later (till we cover free body diagrams and interactions)

The definition of mass (inertial or gravitational mass) in this way settles the long standing dispute of mass,  density (mass/volume), weight (mass * acceleration due to gravity of 9.81 m/s^2) and force (1 kg mass accelerated at 1 m/s^2 acceleration).

By rearranging the formula, we can also arrive at f=ma. The formula that is most misunderstood.

With such a significance, we may say that as much A for apple is for babies, A for acceleration should be introduced to beginners of science.

Foot Note.
With this in place we may move to understand the concepts of instantaneous velocity and acceleration
Instantaneous velocity       = infinitesimal displacement over infinitesimal time interval.
Instantaneous acceleration = infinitesimal change in velocity over infinitesimal time interval.

Now, the interesting question is, how small is the smallest possible length?. We cant just say "depending on the resolution of measuring equipment".

Mr Kepler provides a clue. The Infinitesimal. The smallest possible length which is very close to zero and not exactly zero. For this we need to begin with Archimedes  (350 BC). This will be done in another article. Kepler is mostly known for establishing the laws of planetary motion. But he also established the more important concept of infinitesimal point (Archimedes in 300 BC is known to have advanced the concept if infinitesimals.). The infinitesimal point forms the basis for non intuitive concepts of instantaneous velocity, instantaneous acceleration and also to the concept of calculus (calculus deals with arithmetic with infinitesimal quantities).

Who said science is common sense? We are dealing with abstract points and infinitesimal lengths. Add to it the concept of Planck Point. A Planck point being the volume of a cube made of Planck Length (not sure of the reason for not using a sphere to calculate the Planck volume. for us a sphere is a more natural shape than a cube. May be nature uses cube as the building block at that microscopic level. Planck length is in the order of 10 raised to the power -35). May be it is time to redefine the concept of infinitesimals, instantaneous velocity and instantaneous acceleration.

As though, this is not enough, Einstein comes up the concept of curved space-time. Space and time are not independent anymore.

Apart from this, the story of mass does not end here. Once we understood that matter is made of atoms and molecules, we redefine chemical mass as "amount of matter" (we have to wait for Avogadro to relate gravitational and chemical mass). We also have Einstein to come up and say mass and energy are equivalent. E=mc^2, the most bizarre equation. Energy is another abstract concept.

So we have abstract concepts like point, displacement and energy, the concepts that are not real. We cannot feel them. But we need to use them to deal with nature.

Euclid has a Point

Article status:                              Draft
Time Estimate for Reading:        20 min
Learning Objectives:                  Significance of Abstract Concepts and Introduction to Geometry
Effort Required:                          Medium
Pedagogy Model:                        Evolution, Formula Analysis, Inter-disciplinary
Prior Physics Concepts:             displacement, velocity, acceleration, mass

Prior Math Tools:                       Secondary school level Arithmetic, geometry and algebra


Euclidean geometry (300 BC), could well share something in common with Galileo's law of falling bodies (1600),  Newtons laws (1667), Maxwell's equations 1861), Planck's Constant (1900) and Einsteins theory of relativity (1905). The common ground is that all of science is modeled around space, time, mass and energy (also include power) and it was a collaborative effort spanning more than 2000 years to connect these concepts.


-  Euclid gives an account of how to understand the concept of 3 dimensional space (points, lines .. etc )
















-  Galileo demonstrates the relationship between space and time for bodies in free fall (displacement is proportional to square of acceleration)

-  Newton goes on to combine the concepts of space, time and mass (f=ma) and treats each of them as independent entities

- Maxwell sets the limit for maximum velocity (displacement/time). Determines the maximum speed of light from his wave equations and also lays the foundation for wave-particle duality
- Planck's constant determining the minimum quanta of energy (E = hf).








- Einstein introduces the concept of 4 dimensional space-time (theory of relativity) where space and time are no more independent entities. He also goes on to establish that mass and energy are mutually convertible (mass energy equivalence and the most famous equation E=mc2)







From Euclid to Einstein, 2200 years of rich history. Rather we would call it a game that is played with concepts of Space, Time, Mass and Energy. Euclid begins this game and established the first set of rules. The first rule is about a point. "A point is that which has no part". Well. Well. 

Let us embark on a time travel to 300 BC. We had the number systems (cant believe zero and negative numbers are yet to be discovered), the Pythagoras theorem, astronomical studies and theory of music. During this period, During this period, for some of ancestors stars and planets were the gods and for some others it was numbers (considered to be perfect) that where the gods. The earth was still flat.

During this period, most of activity involved measurement of land, tracking stars and planets. There was no clear method of mathematically describing space.

Such a description had to wait till the definition of the concept called "point". 

A point in geometry is a location. It has no size i.e. no width, no length and no depth. A point is shown by a dot. This definition is a physical impossibility but an abstract concept. But once, the point is accepted as an abstract concept, we could go on and define a line.

A line is defined as a line of points that extends infinitely in two directions. It has one dimension, the length. So, a non-existent point leads to a non-existent line. On a similar note, rules for area, volume, parallel lines, angles etc could be derived.

Returning from the past, we observe this to be a case of extreme simplification (watch out! Complexity is simplicity in disguise) of geometry. It should have taken years for Euclid to arrive at the abstract concept called "point". Once "point" was defined and understood, everything else fell in place. The emphasis here is that, every subject has a core concept. once that is identified all other aspects will reveal itself.

Now we do have non-euclidean geometry for spherical and hyperbolic spaces. The Euclidean definitions still hold and just extended.

The evolution of rules for connect space, time, mass and energy are not over as yet. We are still in pursuit of a single unified theory which connects space, time, mass and energy. Planck's constant might take us one step closer.

We may redefine the point now based on Planck's constant. The Euclidean definition of point was for a reason that, we may mark a point with different sizes of pens. But now that we have Planck's constant and Planck Length, we may redefine the point in 3 dimensions as "A point is defined as a Sphere of radius equal to half Plank Length". That's the minimum of a point which we can mark. But i am sure, we cant mark this point on a paper made of wood. May be we could, on a paper made of dark matter or dark energy. Great. What sort of pen do we use to write on such a dark matter paper? Life gets interesting.

For those interested in numbers;
Planck's Constant = 6.62607004 × 10^-34 m² kg / s
Planck Length       = 1.6 x 10^-35 m or about 10^-20 times the size of a proton 
Planck Point          = 2.1436E-105 m^3 (volume of a sphere of half the radius - 4/3*pi*R^3)

Well. A number raised to the power of -105. That should be pretty small. And people say, at this scale, quantum gravity takes over (whatever that means?)  and is not Newtonian anymore. And with that, We pretty much feel like Euclid, trying to make a point. Starting allover again. Time for new game and new rules.

Foot note - Impact of Euclid and Euclidean Geometry

No wonder, Abraham Lincoln was fascinated by this and spent two years to study Euclid's book on Elements and this is why?. He wanted to know, What does it really mean to “prove” something? How do you “demonstrate” that a person is innocent or guilty? Lincoln wanted to understand proof in a deeper way. A case where mathematics helps a lawyer. Inspiration and knowledge may come from any any source. Just that we got to be open for ideas and exploration.

On a similar note, Galileo is supposed to have spent 2 years (when he was 22 years of age) learning and teaching Elements. Probably this could have made him to convince himself to pursue mathematics and incidentally, most of his proofs (parabolic motion, law of falling bodies etc) are based on geometry. Rather higher mathematical concepts like differential and integral calculus were yet to be invented.

As much as Galileo is considered the proponent of experimental science, Euclid is considered to be proponent of deductive and logical reasoning. Let us not leave newton and Einstein out, they are the proponents of Thought Experiments.

References:

An excel account of Euclid by Sal khan

The history of Newton’s apple tree

The significance of apple tree story

The knowledge before Newton

Maxwells Equations

Origins of theory of relativity

https://en.wikipedia.org/wiki/History_of_special_relativity

https://en.wikipedia.org/wiki/Planck_constant
https://en.wikipedia.org/wiki/Planck_units