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.