Section 9–5 Meaning of the dynamical equations

(Initial conditions / Good approximation / Dynamical equations)

In this section, the three interesting concepts are initial conditions of a simple harmonic motion, a good approximation of a differential equation, and the meaning of dynamical equations.

1. Initial conditions:

“…for then we can start with the given condition and compute how it changes for the first instant, the next instant, the next instant, and so on… (Feynman et al., 1963, section 9–5 Meaning of the dynamical equations).”

Feynman states the “given condition” of a simple harmonic motion as time t = 0, x = 1, and v_x = 0 that allows us to compute the subsequent motion of an object. He explains that the object continues to move because there is a force on it when it is at any position except at the equilibrium position x = 0. However, the object does not start to move at x = 0, but it would move if it is slightly disturbed from a stable equilibrium state. As an alternative, physics teachers can explain that the object would continue to move at the equilibrium position because it possesses an amount of kinetic energy or momentum. Specifically, the presence of a restoring force in simple harmonic motion may reduce the kinetic energy of the object.

Some physicists may prefer the term “initial conditions” instead of “given condition.” In classical mechanics, “initial conditions” may include initial positions and velocities of all particles that determine a system’s future. On the other hand, there are two “necessary (mathematical) conditions” of simple harmonic motion: (1) the direction of the force is towards the equilibrium position, and (2) the magnitude of the force is proportional to the displacement from the equilibrium position and it can be expressed as F = -kx where k is a constant. Moreover, physics teachers should remember that simple harmonic motion is an idealization. One may include two “ideal (physical) conditions”: (1) there is no air resistance or frictional force, and (2) the spring is perfectly elastic if the extension or compression of the spring is relatively short.

2. Good approximation:

“…if ϵ is very small, we may express the position at time t + ϵ in terms of the position at time t and the velocity at time t to a very good approximation as x(t+ϵ) = x(t) + ϵv_x(t) (Feynman et al., 1963, section 9–5 Meaning of the dynamical equations).”

Feynman expresses the position of an object at time t + ϵ in terms of its position at time t and its velocity at time t to a good approximation as x(t+ϵ) = x(t) + ϵv_x(t). He clarifies that this expression is more accurate if the time interval ϵ is shorter. In addition, the acceleration can be determined by Newton’s second law and the velocity at a short time interval later can be approximately expressed as v_x(t+ϵ) = v_x(t) + ϵa_x(t) = v_x(t) − ϵx(t). Physics students that do not like numerical methods should realize that Feynman enjoys calculations that involve approximations. In Surely You’re Joking, Mr. Feynman, he says that “I had a lot of fun trying to do arithmetic fast, by tricks, with Hans. It was very rare that I’d see something he didn’t see and beat him to the answer, and he’d laugh his hearty laugh when I’d get one... (Feynman, 1997, p. 194).”

Motions of objects can be modeled by using ordinary differential equations that may not be solved exactly. Thus, we can design numerical algorithms for differential equations and use a computer to simulate the motions of objects and form an approximate view of the motions. In general, the use of a numerical method may achieve an accurate approximate solution to a differential equation. Currently, there are many programs (or software packages) that help to solve differential equations. With today’s computers, an accurate solution can be obtained within seconds. More important, the simple harmonic motion is based on Hooke’s law that is a first-order linear approximation. To be more accurate, a real spring could be modeled by using F(x) = kx + γx² for some constant γ as a second-order approximation.

3. Dynamical equations:

“…Eq. (9.15) is dynamics, because it relates the acceleration to the force; it says that at this particular time for this particular problem, you can replace the acceleration by −x(t) (Feynman et al., 1963, section 9–5 Meaning of the dynamical equations).”

According to Feynman, the equation v_x(t+ϵ) = v_x(t) + ϵa_x(t) is merely kinematics because it determines how the velocity of an object changes depending on the magnitude of acceleration. On the other hand, the equation v_x(t+ϵ) = v_x(t) − ϵx(t) is dynamics because it is related to a force that is in terms of −x(t). To be precise, the acceleration could be expressed as –(k/m)x(t) instead of −x(t). The motion of the object that is attached to a gadget is dependent on the spring constant (k) of the gadget and the mass of the object (m) in motion. To avoid possible confusions, Feynman could set k/m equal to 1 later.

Kinematics refers to the nature and characteristic of the motions. Simply phrased, it focuses on the motions of objects without concern with the forces that cause motions. On the contrary, dynamics is concerned with the motions of objects due to the influence of forces. As a comparison, velocity is a kinematical quantity, whereas momentum is a dynamical quantity. Interestingly, the dynamical property of simple harmonic motion (SHM) can be related to circular motion. For example, French (1971) explains that “[i]n order to display the dynamical identity of this component motion with SHM, we can take the expressions for F_x and a_x separately, introducing the angular velocity ω and putting v = ωA (p. 234).”

Questions for discussion:

1. What are the conditions that should be specified in the simple harmonic motion?

2. What are the approximations that are necessary for the simple harmonic motion?

3. What is the meaning of the dynamical equations?

The moral of the lesson: if we know the position and velocity of an object that is attached to a spring at a given time, we would know its acceleration, which tells us the new velocity, and we would know a new position approximately — this is how the machinery works.

References:

1. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

Section 9–4 What is the force?

(A program for forces / Gravitational force / Restoring force)

1. A program for forces:

“…we have to have some formula for the force; these laws say pay attention to the forces. Our program for the future of dynamics must be to find the laws for the force (Feynman et al., 1963, 9–4 What is the force?)”

The title of this section “What is the force?” is potentially misleading. Instead of explaining the concept of a force, it is about some formulae for the force to deduce the motion of objects. More important, Feynman mentions that it is about a program for the future of dynamics to determine the laws for the force. In essence, the section is an introduction to the use of numerical methods to deduce the motion of objects. In general, numerical methods may involve the use of algorithms to solve mathematical problems and it is built on the foundation of approximation theory. Furthermore, numerical analysis is revolutionized by the invention of computers as well as the development of mathematical theories and algorithms.

Physics teachers may elaborate the program for forces by using Wilczek (2005) words, “…the law F = ma, which is sometimes presented as the epitome of an algorithm describing nature, is actually not an algorithm that can be applied mechanically (pun intended). It is more like a language in which we can easily express important facts about the world. That’s not to imply it is without content. The content is supplied, first of all, by some powerful general statements in that language such as the zeroth law, the momentum conservation laws, the gravitational force law, the necessary association of forces with nearby sources and then by the way in which phenomenological observations, including many (though not all) of the laws of material science can be expressed in it easily… (p. 11).” According to Wilczek, the zeroth law may refer to the law of conservation of mass.

2. Gravitational force:

“Thus the law of gravity tells us that weight is proportional to mass; the force is in the vertical direction and is the mass times g (Feynman et al., 1963, 9–4 What is the force?).”

The gravitational (radial) force near the earth’s surface is proportional to the mass of the object and is nearly independent of height for heights that are small compared with the earth’s radius R. According to Feynman, Newton’s law of gravitation tells us that weight is proportional to mass and is equal to the mass times g: W = mg = GmM/R², in which g = GM/R² and it is known as the acceleration of gravity. Essentially, the free fall of an object under gravity, which leads to the equations v_x = v₀+gt and x = x₀+v₀t + ½gt², is related to a theoretical definition of weight, W = mg. However, there is no consensus on the definition of weight. For example, one may prefer an operational definition of weight that is based on a weighing scale.

The debate on the definition of weight started during the man-in-space project in the 1960s (King, 1962). Some physicists explain that an astronaut would feel weightless and “float in the air” in a satellite coasting around the Earth. Thus, physics educators review the definition of weight and debate whether “weight is a gravitational force” (gravitational weight) or “weight is measured by a weighing scale” should be adopted for science education. The issue is not simply between the “theoretical definition” of weight as a gravitational force and the “operational definition” of weight. The controversy is also about whether weight should be theoretically defined as a “gravitational force” or “force on the support” (alternatively, a support force).

3. Restoring force:

“…the force is greater, the more we pull it up, in exact proportion to the displacement from the balanced condition, and the force upward is similarly proportional to how far we pull down (Feynman et al., 1963, 9–4 What is the force?).”

Feynman gives an example of a gadget (or a spring) which exerts a force proportional to the distance and the force is directed in an opposite direction. Based on Newton’s second law, we can describe the motion using the equation m(dv_x/dt) = –kx. In short, it means that the velocity of an object (connected to the gadget) changes at a rate proportional to x and it is in the x-direction. For reasons of simplicity, we can choose k/m = 1 and solve the equation as dv_x/dt = −x. Feynman needs not say that nothing will be gained by retaining numerous constants or imagine there is an accident in the units. We can interpret this as a sense of humor and explain that physicists have the freedom to arbitrarily (instead of accidentally) define the units involved.

What seems missing here is an explanation of the equation involving Hooke’s law (F = –kx). This equation is explained later in chapter 12: “[t]his principle is known as Hooke’s law, or the law of elasticity, which says that the force in a body which tries to restore the body to its original condition when it is distorted is proportional to the distortion. This law, of course, holds true only if the distortion is relatively small (Feynman et al., 1963, section 12–3 Molecular forces).” In De potentia restitutiva, Robert Hooke (1678) succinctly phrases the law of elastic bodies as “Ut tensio, sic vis,” which means, “the extension is proportional to the force.” Interestingly, Hooke claims his discovery of the law by stating a Latin anagram “ceiiinosssttuu” (French, 1971) without clarifying the “elastic bodies.”

Questions for discussion:

1. Why do physicists need a program of forces?

2. Do you agree with Feynman’s definition of weight?

3. What does Feynman mean when he says that “nothing will be gained by retaining numerous constants” and “imagine there is an accident in the units”?

The moral of the lesson: physicists need a program for the future of dynamics to determine the laws for the force that involves the use of numerical methods.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

3. Hooke, R. (1678). De potentia restitutiva. In R. T. Gunther (1931). Early Science in Oxford, Vol. VIII: The Cutler Lectures of Robert Hooke. Oxford: Oxford University Press.

4. King, A. L. (1962). Weight and Weightlessness. American Journal of Physics, 30(4), 387.

5. Wilczek, F. (2005). Whence the force of F= ma? III: Cultural diversity. Physics Today, 58(7), 10-11.

Section 9–3 Components of velocity, acceleration, and force

(Components of velocity / Components of acceleration / Components of a force)

In this section, the three interesting concepts are components of velocity, components of acceleration, and components of a force.

1. Components of velocity:

“…we have resolved the velocity into components by telling how fast the object is moving in the x-direction, the y-direction, and the z-direction (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force)

Feynman mentions that the velocity of an object is completely specified if we give the numerical values of its three perpendicular components: v_x = dx/dt, v_y = dy/dt, v_z = dz/dt. Furthermore, the magnitude of the velocity of the object can be calculated by using the equation, ds/dt = √(v_x²+ v_y² + v_z²). Essentially, the components of velocity refer to the speed of the object in the x-direction, the y-direction, and the z-direction. We can demonstrate these components of velocity by using a light source or projector. If we shine light vertically downward on a moving object, we can observe a shadow (or projection) moves in a specific direction. Physics teachers may explain that a component of velocity is projected onto the x-axis or y-axis depending on the direction of light rays.

There are gaps in Feynman’s explanation of components of velocity because this is a relatively easy topic. In Tips on Physics, Feynman adds that “the velocity in terms of x, y, and z components is very easy, because, for example, the rate of change of the x component of the position is equal to the x component of velocity, and so on. This is simply because the derivative is really a difference, and since the components of a difference vector equal the differences of the corresponding components (Feynman et al., 2006, p. 30).” In other words, the derivative of a position vector is related to a difference in positions of an object. Mathematically, the components (or shadows of an object) of a vector in the three-dimensional world also obey Newton’s laws of motion.

2. Components of acceleration:

The change in the component of the velocity in the x-direction in a time Δt is Δv_x = a_xΔt, where a_x is what we call the x-component of the acceleration (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force)

The action of a force can cause the velocity of an object changes to another direction and a different magnitude. Feynman explains that this apparently complex situation can be simply analyzed by evaluating the changes in the x-, y-, and z-components of velocity. Mathematically, the change in the component of the velocity in the x-direction in a short time Δt is Δv_x = a_xΔt, in which a_x is the x-component of the acceleration. Without loss of generality, we have Δv_y = a_yΔt and Δv_z = a_zΔt. Essentially, we can resolve the displacement, velocity, and acceleration of an object into components by projecting a line segment to represent these quantities.

In The Evolution of Physics, Einstein and Infeld (1938) write that “[b]y following the right clue, we achieve a deeper understanding of the problem of motion. The connection between force and the change of velocity and not, as we should think according to our intuition, the connection between force and the velocity itself is the basis of classical mechanics as formulated by Newton (p. 10).” In short, force is connected to a change in velocity instead of simply velocity. We should recall Feynman’s explanation that “the derivative is really a difference (Feynman et al., 2006, p. 30).” Thus, one may explain the connection by using the concept of “change in velocity” instead of only acceleration.

3. Components of a force:

“If we know the forces on an object and resolve them into x-, y-, and z-components, then we can find the motion of the object from these equations (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force).”

Feynman suggests that there are really “three” laws in the sense that the component of the force in the x-, y-, or z-direction is equal to the mass of an object times the rate of change of the corresponding component of velocity: F_x = m(dv_x/dt) = m(d²x/dt²) = ma_x, F_y = m(dv_y/dt) = m(d²y/dt²) = ma_y, F_z = m(dv_z/dt) = m(d²z/dt²) = ma_z. One may infer that Newton’s Second Law can also be represented by infinite possible combinations of x-, y-, or z-direction and hence there is an infinite number of laws governing the force in various directions. However, it is possible to simplify the motion of an object by using only two equations or even one equation depending on how we choose the x-, y-, or z-direction. Thus, Feynman does not need to identify each equation as a theoretical law.

Feynman states that motions in the x-, y-, and z-direction are independent if the forces are not connected. Historically, in his investigations of motion, Galileo is the first person to conceptualize the forces acting upon objects could be resolved into independent components. In Dialogues Concerning Two New Sciences, he writes that “the resulting motion which I call projection is compounded of one which is uniform and horizontal and of another which is vertical and naturally accelerated (Galilei, 1638, p. 244).” Galileo’s insights are remarkable because the ideal motion of projectile motion could not be directly observed due to the presence of air resistance. Importantly, physicists have assumed Euclidean geometry of space in the analysis of motions.

Questions for discussion:

1. Why are we allowed to resolve velocity into perpendicular components?

2. Why is a force connected to a change in velocity instead of velocity?

3. Why are we allowed to resolve forces into perpendicular components?

The moral of the lesson: force is connected to a change in velocity instead of velocity.

References:

1. Einstein, A. & Leopold, I. (1938). The Evolution of Physics. New York: Simon & Schuster.

2. Feynman, R. P., Gottlieb, M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Galilei, G. (1638/1914). Dialogues Concerning Two New Sciences. New York: Dover.

Section 9–2 Speed and velocity

(Redefining speed & velocity / Comparing speed & velocity / Formulating velocity)

In this section, the three interesting concepts are redefining speed and velocity, comparing speed and velocity, and formulating velocity.

1. Redefining speed and velocity:

“Ordinarily we think of speed and velocity as being the same, and in ordinary language they are the same (Feynman et al., 1963, section 9–2 Speed and velocity).”

In a lecture on quantum electrodynamics, Feynman (1985) explains that physicists use ordinary words such as work, action, energy, or light, in a funny way. Physicists also redefine speed and velocity that have the same meaning in daily life. In Regulae solvendi sophismata, Heytesbury defines the “velocity at any instant in non-uniform motion as the ratio of the distance traveled to the time that would have elapsed if the motion had been uniform at that velocity (Weinberg, 2015, p. 138).” Weinberg (2015) mentions that this definition is circular and hence useless. Grant (1996) explains that it defines “instantaneous velocity” by a uniform speed that is equal to the instantaneous velocity (it is yet to be defined). However, Heytesbury also derives the mean speed theorem that may be expressed as s = ½(v_i + v_f)t.

In 1928, Einstein posed the following questions to Jean Piaget: “Is our intuitive grasp of time primitive or derived? Is it identical with our intuitive grasp of velocity? (Piaget, 1969, p. xiii).” Einstein wanted to know whether children’s understanding of these concepts was intuitive or derived, and how their understanding of one concept influenced subsequent understanding of the other. Based on his findings, Piaget (1972) explains that “[t]he relationship v = d/t implies that v is a relationship and that both d and t are straightforward intuitions. The truth, however, is that some intuitions of speed, such as those of overtaking, actually precede those of time (p. 78).” In other words, children do not necessarily think of velocity in terms of the distance-time relationship and their concept of time could be derived from velocity.

2. Comparing speed and velocity:

“We carefully distinguish velocity, which has both magnitude and direction, from speed, which we choose to mean the magnitude of the velocity, but which does not include the direction (Feynman et al., 1963, section 9–2 Speed and velocity).”

Dictionary definitions of speed and velocity have essentially the same meaning. Currently, we can compare the concepts of speed and velocity from the perspectives of theoretical definition, classification, and equation. Firstly, speed is commonly defined as the rate of change of distance traveled by an object and velocity is the rate of change of displacement of an object. Secondly, the speed of an object can be classified as a scalar quantity and velocity is a vector quantity. Thirdly, speed can be mathematically represented by v = d/t which means a ratio of distance moved (d) over an interval of time (t) whereas velocity can be represented in terms of three components: v = v_x i + v_y j + v_z k. These three differences can be simply explained by the fact that speed is directionless in contrast to velocity that has a specific direction.

Some may prefer Einstein and Infeld’s (1938) comparison of speed and velocity: “consider two spheres moving in different directions on a smooth table. So as to have a definite picture, we may assume the two directions perpendicular to each other. Since there are no external forces acting, the motions are perfectly uniform. Suppose, further, that the speeds are equal, that is, both cover the same distance in the same interval of time. But is it correct to say that the two spheres have the same velocity? The answer can be yes or no! If the speedometers of two cars both show forty miles per hour, it is usual to say that they have the same speed or velocity, no matter in which direction they are traveling. But science must create its own language, its own concepts, for its own use. Scientific concepts often begin with those used in ordinary language for the affairs, of everyday life, but they develop quite differently. They are transformed and lose the ambiguity associated with them in ordinary language, gaining in rigorousness so that they may be applied to scientific thought (p. 12).”

3. Formulating velocity:

“We can formulate this more precisely by describing how the x-, y-, and z-coordinates of an object change with time (Feynman et al., 1963, section 9–2 Speed and velocity).”

In general, the motion of a particle in a specific direction can be resolved into three components that are independent of each other. Therefore, the position of the particle can be mathematically represented by three independent equations in terms of x, y, and z. Feynman explains that we can formulate the particle’s motion by describing how the x-, y-, and z-coordinates change with time. In a short interval of time Δt, we can assume the particle moves in a straight line and the total distance moved (Δs) can be resolved as a certain distance Δx in the x-direction, Δy in the y-direction, and Δz in the z-direction. Mathematically, the displacement Δx is equal to the x-component of the velocity times Δt, that is, Δx = v_xΔt. Similarly, we have Δy = v_yΔt and Δz = v_zΔt.

This concept of velocity is formulated based on the assumption of Euclidean geometry. In Feynman’s Tips on physics, he elaborates that “[i]n this case, where A is position, its derivative is a velocity vector; the velocity vector is in a direction tangent to the curve, because that's the direction of the displacements; its magnitude you can’t get by looking at this picture, because it depends on how fast the thing is going along the curve. The magnitude of the velocity vector is the speed; it tells you how far the thing moves per unit time. So, that's a definition of the velocity vector: it’s tangent to the path, and its magnitude is equal to the speed of motion on the path (Feynman, 2006, p. 29).” Because velocity is defined as a vector, it also needs to follow mathematical rules with regard to vector differentiation.

Questions for discussion:

1. How would you redefine the speed and velocity of an object?

2. How would you compare the differences between speed and velocity?

3. How would you formulate velocity in terms of vector quantities?

The moral of the lesson: physicists redefine ordinary words such as speed and velocity that have the same meaning in daily life.

References:

1. Einstein, A. & Leopold, I. (1938). The Evolution of Physics. New York: Simon & Schuster.

2. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

3. Feynman, R. P., Gottlieb, M. A., & Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

4. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

5. Grant, E. (1996). The Foundations of Modern Science in the Middle Ages: Their Religious, Institutional and Intellectual Contexts. Cambridge: Cambridge University Press.

6. Piaget, J. (1969). The Child’s Conception of Time. New York: Basic Books.

7. Piaget, J. (1972). Psychology and Epistemology: Towards a Theory of Knowledge. Middlesex: Penguin.

8. Weinberg, S. (2015). To Explain the World: The Discovery of Modern Science. London: Allen Lane.

Section 9–1 Momentum and force

(Newton’s First Law / Momentum / Force)

In this section, the three interesting points discussed are Newton’s First law of dynamics, momentum, and force.

1. Newton’s First Law:

“The First Law was a mere restatement of the Galilean principle of inertia just described (Feynman et al., 1963, section 9–1 Momentum and force).”

Feynman states the principle of inertia as “if an object is left alone, is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still.” He explains that this law never appears in nature because a sliding block will eventually stop. In essence, Newton’s First Law of dynamics is developed by Galileo’s imagination. Simply put, Newton’s First Law is based on idealizations and it cannot be directly (or exactly) observed in nature. Similarly, Eddington (1928) rephrases Newton’s First Law as “[e]very body continues in its state of rest or uniform motion in a straight line, except in so far as it doesn’t (p. 124).” Curiously, Feynman earlier (Volume I, Chapter 7) says that we do not know why an object coasting at a uniform speed in a straight line.

One may not agree with Feynman that the First Law was a mere restatement of the Galilean principle of inertia. Strictly speaking, Galileo did not explicitly state a general principle of linear inertia. On the contrary, Galileo suggests a concept of circular inertia: “a ship … would move continually around our globe without ever stopping and placed at rest it would perpetually remain at rest, if in the first case all extrinsic impediments could be removed, and in the second case no external cause of motion were added (Galilei, 1613, pp. 113–114.)” In other words, an object may continue in its state of circular motion unless there is an (external) resultant force. Perhaps Galileo would prefer this modern version of the law of inertia: “A free object continues in its state of rest or moves along a geodesic in spacetime.”

2. Momentum:

“Now the momentum of an object is a product of two parts: its mass and its velocity (Feynman et al., 1963, section 9–1 Momentum and force).”

Feynman mentions that a lot of words in physics have precise meanings in physics. He defines the momentum of an object as a product of its mass and its velocity. However, this is not a general definition of momentum. In the special theory of relativity, the momentum of a fast moving particle (p = γmv) includes a Lorentz factor, γ. In quantum physics, the momentum of a photon (p = h/λ) is equal to Planck’s constant divided by its wavelength. Alternatively, the momentum of electromagnetic radiations (p = E/c) can be calculated by the total energy of electromagnetic radiations divided by the speed of light. To be more precise, we should adopt the term linear momentum that is distinguished from angular momentum.

According to Feynman, the Second Law gives a specific way of determining how the velocity changes under different forces and the Third Law is essentially action equals reaction. However, Newton’s three laws of dynamics (or motion) can be consistently related to the linear momentum. We can rephrase the First Law as “a free particle always moves with a constant linear momentum relative to an inertial frame of reference. The Second Law can be more precisely stated as “the rate of change of linear momentum of a particle with respect to time is proportional to the force acting on it”. The Third Law can be related to the principle of conservation of linear momentum: the linear momentum of a system is constant if there is no external resultant force acting on the system.

3. Force:

“As a rough approximation, we think of force as a kind of push or pull that we make with our muscles, but we can define it more accurately now that we have this law of motion (Feynman et al., 1963, section 9–1 Momentum and force).”

Feynman elaborates that Newton’s Second Law may be written mathematically as F=d(mv)/dt and if the mass of an object is constant, it can be simplified as F = ma. This relationship does not only stipulate changes in the magnitude of the momentum and velocity but also in the direction. That is, the direction of the change in the momentum and velocity is the same as the direction of the force. Students should realize that acceleration, or a change in a velocity, has a wider meaning than its use in daily language: when an object slows down, we say it accelerates with a negative acceleration. However, Feynman in chapter 12 adds that if we insist upon a precise definition of force, we will never get it! This is because the Second Law is not exact and it involves approximations and idealizations.

Note that Newton did not specifically write the equation F = ma. In fact, Newton’s second law may be known as Euler’s First Law because Euler (1736) first develops the “F = ma” scheme and extends it to the motion of rigid bodies. Interestingly, Wilczek (2004) expresses his difficulties in learning F = ma and writes that “Newton’s second law of motion, F = ma, is the soul of classical mechanics. Like other souls, it is insubstantial. The right−hand side is the product of two terms with profound meanings. Acceleration is a purely kinematical concept, defined in terms of space and time. Mass quite directly reflects basic measurable properties of bodies (weights, recoil velocities). The left−hand side, on the other hand, has no independent meaning. Yet clearly Newton’s second law is full of meaning… (p. 11).”

Questions for discussion:

1. Is Newton’s First Law of dynamics a mere restatement of the Galileo’s principle of inertia?

2. Is there a general definition of linear momentum? (The linear momentum of an object is the ability to generate an impulse over a period of time?)

3. What are the meanings of Newton’s Second Law of dynamics as expressed by F = ma?

The moral of the lesson: Newton’s First Law of dynamics is related to Galileo’s method of idealization and this law cannot be strictly observed in nature.

References:

1. Eddington, A. (1928). The Nature of the Physical World. New York: Cambridge University Press.

2. Euler, L. (1736). Mechanica sive motus scientia analytice exposita. Saint Petersburg: Press of the Academy of Sciences.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Galilei, G. (1913). Letters on Sunspots (translated by S. Drake). In G. Galilei (1957). Discoveries and Opinions of Galileo. New York: Doubleday.

5. Wilczek, F. (2004). Whence the force of F= ma? I: culture shock. Physics Today, 57(10), 11-12.