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Newton's Laws and Force
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0:02
Force is a very important concept in physics.
0:07
In fact, the Newtonian mechanics can be considered as a study of forces. Do you agree? Perhaps you can remember Newton's three laws in mechanics, all of them involve the force: the first law of inertia talking about what would happen in the absence of external force; the second law provides a quantitative relation between the force and acceleration, based on acceleration the the motion of the object can be solved, can be predicted; the third law tells us that forces
0:47
always appear as a pair, so when we have one force, you have to consider another force which is opposite interaction but equal in magnitude, so that three laws are all
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concerned about the force. Actually the three law of Newton's mechanics provides a foundation
1:16
for dealing with different mechanical problems over past hundreds years, and many problems have been solved, even including today's uh
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activities, including today. We still do use Newton's mechanics, the Newtonian mechanics, to predict how a rocket be launched, and how to go to from the Earth to the Mars.
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Unfortunately, the importance of this tradition, or the achievement of the Newtonian mechanics had been underestimated by many Youtubers, who are trying to advocate that the future for physics must be based on quantum mechanics and Einstein's relativity.
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Of course, this series is focused on the field theory, both field and field theory, in which we'll eventually discuss various abstract ideas and the approaches
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introduced in the quantum field theory, including the "Standard Model" , or string theory,
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but before that I would like to give a short introduction to the Newtonian mechanics to demonstrate how force was originally treated as a key to solve the problems in physics.
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Welcome to Yong Tuition. This is a unique channel to explore and share the basic knowledges about the issues in physics and beyond. If you like this channel, please subscribe, and also please inform other people to subscribe, so that I can provide more interesting and in-depth
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short videos on variety of the subjects. Okay, let's go to our blackboard.
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Where shall we start? yes. What is the Newton's mechanics? The objectives of Newtonian mechanics is to describe and predict the motion of the object, either the flying objects near the surface, or the remote objects in the sky, or in the heaven. That is a grand objective for Isaac Newton before his journey started. To do so, Newton had invented Calculus in the first place to quantitatively describe these three quantities of motion, namely, the position, the velocity, and the
4:40
acceleration. If you know one of them, the other twos can be calculated, can be derived by using calculus, either use differentiation or integration. This was the first time that how the motion can be so accurately described. That is a great achievement made by Isaac Newton, in which the calculus invented played a unique role. To be specific, if we consider the motion of the object in space, we first have to choose the frame of reference
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in which we can set up the three coordinates, three dimensional coordinates, x, y, and z. In this coordinate then we can describe the position of an object.
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If we just focus on one dimensional motion, we can just write the position as the x as function of time. Of course in the 3D scenario we have to add another two components, y and z. But the the method is the same. Once we know the position, then we can define the velocity as the first derivative of the position, denoted by this derivative of x dot. In addition, Newton's introduced how to describe the change of velocity. That was a very new idea in his days
6:32
that acceleration must be taken into account. In order to describe the the motion observed,
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so that is the three equations you can also consider this is differential equations.
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That three quantities have been, can be used to describe any motion of any object in space.
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If you know one of them, then we can know all. Which one did Newton choose?
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He chose acceleration, because he first realized that acceleration is caused, is determined by the force acting on the object, which is consistent with his first law. Actually his first low is based on Galileo's experimental observation.
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Also that is also consistent with ... It's called the paradigm shift that Aristotle's pure philosophical reasoning was thrown away, abandoned.
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Because, according to Aristotle, the moving object must be sustained, or must have the force exerted on the object. Both Galileo and Newton realized
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that is not the case. Therefore, from point of view of physics, the simple relation between acceleration and the force was established in his second law.
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In other words, instead of going to observe and just purely observe the phenomena of the motion, we can actually focus on or identify how many forces acting on this object. For example, these two forces represent the gravity and the air resistance force, two forces. Once we know these two forces, then the trajectory of this projectile motion can be mathematically determined. That is essence of the Newtonian mechanics.
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A good example in using Newton's mechanics is to study the mass-on-spring,
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which provides, which is eventually known as the harmonic oscillator. This is a classical one. Late on I'm going to discuss the quantum harmonic oscillator, which, although the description is different, the idea, the system, the model, is the same. In this simple system, we have the object connected with a spring and we put this object into the liquid, so that when the object is moving, resistance or drag force is generated.
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and in these issues the acceleration the the these. On the left hand side is a product of mass and acceleration on the uh right hand side that were
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the the sum the vector sum of the three forces involved the last one will be external force in addition to the spring elastic force and air resistance if you just just simply rearrange these equations we can obtain a standard second degree inhomogeneous differential equations
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therefore the rest of the work is just how to solve the the differential equation mathematically
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so that is a good example of how to apply newton's mechanics to solve the variety of problems in which force is the key so every effort
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should be made to identify and to quantify the force acting on object
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another aspect of the newton's mechanics is its integration form here we in the original newton's second law we see the differential form that's that is the first second derivative represents the acceleration what would happen if we do integration in space and time? That is a very important issue in the Newton's, Newtonian mechanics. Actually, that also provides a starting point for other mathematicians, such as Lagrange and Harmilton, to introduce their analytical mechanics.
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Integration can be done either in space or in time. Let's do the integration in space first. If we multiply the dx on both sides and do integration from position 1 to position 2, we can solve this integral by first rewriting the acceleration as dv/dt, and then, if we shift this dt under the dx, we can simplify the integral as v dv from position 1 to position 2. Every student who has studied
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calculus know how to do this integral. Then, we will get this famous and familiar form: half m v_square.
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If you just use the delta sign to represent the change of this quantity. By integrating in space, we can obtain this simple relation. By introducing the two quantities: work and the kinetic energy,... Notice that this is the first time the idea of energy is invoked.
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That integral can be called as Work-Energy Theorem. The work done by the external force is equal to the change of the energy of the system. Tthat is a very important and basic theorem in Newtonian Mechanics. In a special case, of course, if the work down by the external force is zero, then we will obtain the Energy Conservation Law, in which delta E is equal to zero. In other words, in the absence of external,.. in the absence of work done by the external force,
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the total energy of the system will become a constant. That is another idea. This is another motive for Lagrange and Harmilton to build their analytical mechanics. We're going to discuss this issue next time. You can argue that this Work-Energy Theorem can be
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considered as the cumulative effect in space of the force. So that we are not dealing with the force at a particular state. Instead, we are considering a process in which the work is done so what would happen? Well, the energy will change.
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similarly we can do the integration in time starting from newton's second and o and then if we multiply dt on both sides and do the integration definite integration from from from position one to position two let's focus on uh right hand side
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if we cancel this dt with this integral became so simple and we can obtain the result right away
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so that is a change delta mv so mv if you still remember the newton's mechanics you know that mv better financially is called momentum is different from kinetic energy because momentum is a vector the carried out the in the masjid you must describe not just the quantity magnitude but also you have described the direction so this i put these two integral side by side so you can appreciate that how versatile the newton's second law is
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we can just move from his differential form to the integration for integral four
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Similarly, the simple relation between the impulse, this integral is defined as impulse, and the change in the momentum is called Impulse-momentum Theorem, similar to the work energy theorem.
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Again you could argue that the impulse momentum theorem can be considered as a cumulative effect
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in time of the force. From here you can see that space and time are related.
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Consequences of this integral have different outcomes, but they still remain connected. It could be argued that the space has no sense of direction, because we generate the scanners which is a change in energy, energy doesn't have direction, and it is just a scalar, the quantities of something, or matrix of something, but time does. It would appear the the cumulative effect in time. that still can be traced in relation to the process of motion,
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That is a difference between these two theorems.
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In passing, I would like to point out that the Newtonian energy defined in this integral form of the second law differs from that introduced by Leibniz, so-called living force. Nowaday we know that the so-called living force used by Leibniz is atheproduct of mass and v squared. Actually Tesler also used this idea, Leibniz's idea of energy. Of course, this is just a relevant issue we can discuss further, because Leibniz's idea about
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living energy, or living force.... Be careful. living force has nothing to do with the force
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defined in the Newtonian mechanics. We'll talk about the living force in relation to the more recent idea about the potential energy and also the energy conservation or energy transformation, the essence of Newtonian mechanics. In addition, Newton not only provided the foundation, he laid the foundation for mechanics, but also Newton was first person to discover that there is a universal gravitational law
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about the force formula. This particular force is called universal gravity, or gravitational interaction, and can be described, just by trial-and-error, by this simple formula, so that is a force between the two objects. Again, this is another demonstration that
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ex the new to the certain uh is relevant so the interaction between the auto object or gravitational interaction uh involves two forces, a pair of forces, one is the force on object i, comes from object j, another is the force on object j, comes from object i.
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This pair of forces are equal in magnitude but opposite in their directions. So, the formula is for their magnetudes are the same.
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That is uh that's a very important discovery so not only the provide newton needs a foundation for the mechanics but also he actually discovered the gravitational force
21:48
by using if you if this eventually and newton has applied his new mechanics to describe quantitatively the capitalists removed about the orbit of the planet if sun is considered as a center so that is a great breakthrough in human history you could argue that without Newton's, Newtonian mechanic, human would be still in the Platonic, Plato's Cave.
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Today, even high school students can actually appreciate the Newtonian mechanics by approximating the orbit of the planet as as circle. If we in doing so we can simply combine newton's second law with his formula for gravitational force
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by using very simple algebra we can predict the velocity of planets such as the velocity of earth
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about the sun so that is uh some of the greatest achievement
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made by isaac newton of course if the number of the of of this object increase
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and you can imagine that in principle we should we still can apply newton's mechanics to describe and predict however there's no guarantee we still can get the accurate or analytical formula in fact when the number of the object in the system is more than three including three and you one have one must apply one must be obliged to uh using appropriate approximation method such as
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different orders of perturbations otherwise we cannot find any quantitative solutions so that is a reality perhaps that is why that is one reason after eyes conducted mathematicians such as Lagrange and Hamilton had decided to rewrite Newton's
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basic law into different forms, so that we can still work, to some extent, without considering the force, that is our topics in the future., because that is the motive to introduce so-called analytical mechanics.
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Of course, this is something inevitable. For example, if you want to deal with say the gas,
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if you want to describe the ideal gas by Newtonian mechanics, then you have to derive three equations, because it's a 3D motion, you have derived three differential equations for each particle, or each atom. You can multiply the total number of these particles by three. so you can see the the coupled equation actually are not independent. They are coupled differential equations you can see that the the the the task the mathematical challenge
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is enormous. Even you can invoke some supe-computers. I don't think you can solve this problem. That is why, in my opinion, that Newton's law can primarily apply to study two or three, or finite number of the objects. otherwise we have to follow the analytical mechanics, that would be our next subject.
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I hope that this discussion will give you some overall pictures about Newtonian mechanics and why force or forces are treated as the very important concept
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in Newtonian mechanics. Of course, gradually you will notice that and there is one question Newton didn't answer, that is: how fast this action, namely the gravity, can propagate in space?
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I think that is only question Newton didn't answer, but in many a practical problem solving, who care? It is not an issue. It could be faster, could be very close to speed of light, as people advocated nowaday in the context of so-called gravitational wave. But,
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is that the only reason we have to abandon Newton's approach? Instead, we have to adopt Einstein's general relativity? Think about it. I will stop here and enjoy your viewing. Please make your suggestions as a comment, so that I can improve my presentations, and also to answer your new questions. Thank you for watching. See you next time, goodbye.
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