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.
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.
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