Section 10–2 Conservation of momentum

(External forces / Point of view / Definition of equal masses)

In this section, the three interesting points are external forces, point of view, and definition of equal masses. Alternatively, this section could be revised and titled as “three conditions of conservation of linear momentum”: (1) no external forces; (2) frame of reference; and (3) conservation of mass.

1. External forces:

“…if there are no forces from the outside (external forces), there are no forces that can change the total momentum; hence the total momentum is a constant (Feynman et al., 1963, section 10–2 Conservation of momentum).”

Mathematically, Feynman shows that the total linear momentum of two particles remains constant even though there are mutual interactions between them. Generally speaking, all the internal forces (due to mutual interactions) between many particles will cancel out each other such that they do not change the total linear momentum of the particles. In essence, there is a law of linear momentum: the total linear momentum of a system of particles remains unchanged provided there is an absence of external forces. However, one may expect Feynman to elaborate that this law is an idealization because there is no isolated system of particles in the real world. For practical purposes, we assume gravitational forces and electromagnetic forces acting on the particles to be negligible.

In a sense, it is about an idealized condition necessary for the conservation of linear momentum of a system: the net force on the particles is zero. The concept of net force may be labeled as an external force or internal force depending on how we define a system of particle(s). If we focus on a system of two particles, we may consider the opposite forces acting on the two particles as internal forces that are equal and opposite in direction. In this case, the linear momentum of the system of two particles remains conserved. Alternatively, one may treat the two particles separately as two (one-particle) systems and label the opposite forces as external forces that can accelerate the two particles in opposite directions. In other words, the momenta of the two systems can be changed as a result of the so-called external forces.

2. Point of view:

“…we shall discuss the laws of impacts and collisions from a completely different point of view (Feynman et al., 1963, section 10–2 Conservation of momentum).”

Feynman explains that he will discuss the laws of impacts and collisions from a completely different point of view. It is based on Galilean principle (or principle of relativity) in which the laws of physics remain unchanged whether we are stationary or moving with a uniform speed in a straight line. Feynman illustrates this principle by giving an example where a child bouncing a ball in an airplane finds that the ball bounces in the same way as if it is bouncing on the ground. However, one may use the term inertial frame of reference instead of a point of view. That is, we are not simply viewing the bouncing of a ball in different directions (e.g., x-axis or y-axis).

An inertial frame of reference is also an idealized condition such that the linear momentum of a system of particles is conserved. This condition is idealized in the sense that there is no external force acting on any particles in an inertial reference frame. In the words of French (1971), “[t]here exist certain frames of reference with respect to which the motion of an object, free of all external forces, is a motion in a straight line at constant velocity (p. 162).” To be more accurate, there are external forces such as gravitational forces in a laboratory frame of reference (or simply Earth’s frame). The choice of a particular reference frame is usually for the purpose of problem-solving or a matter of convenience.

3. Definition of equal masses:

“…there are some physical laws involved, and if we accept this definition of equal masses, we immediately find one of the laws, as follows (Feynman et al., 1963, section 10–2 Conservation of momentum).”

Feynman suggests that we can use collision experiments to define equal masses. Firstly, two objects made of different materials, say copper and aluminum, may acquire equal (separation) speeds in a collision experiment if they have the same mass. Additionally, if one of two objects has the same masses as a third object (by measuring the velocities based on the same experiment), then we may infer that they all have the same mass. As a suggestion, one may explain that this is a thought experiment that cannot be performed easily and exactly. Interestingly, Feynman explained earlier that an object may gain mass when it moves at a faster speed. More important, the object may gain internal energy (e.g., the molecules in a ball move faster) through a collision or explosion such that its mass is increased.

Alternatively, we can suggest another idealized condition for the law of conservation of momentum: conservation of mass. For example, Sartori (1996) states that conservation of mass is a necessary condition in order for conservation of momentum to be covariant in the context of Galilean relativity. Furthermore, in Wilczek’s (2004) words, “the energies and momenta of such particles are given in terms of their masses and velocities, by well-known formulas, and we constrain the motion by imposing conservation of energy and momentum. In general, it is simply not true that the sum of the masses of what goes in is the same as the sum of the masses of what goes out (p. 10).” That is, we assume the masses of objects to be constant such that we can apply the equation, m₁v₁ + m₂v₂ = constant, in simple collision situations.

Questions for discussion:

1. Why is the absence of external forces needed as an idealized condition of the law of conservation of linear momentum?

2. Would you include an inertial frame of reference (or point of view) as another idealized condition of the law of conservation of linear momentum?

3. Would you consider conservation of mass as an important condition of the law of conservation of linear momentum?

The moral of the lesson: the law of conservation of linear momentum is dependent on the validity of idealization conditions.

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. Sartori, L. (1996). Understanding relativity: a simplified approach to Einstein's theories. California: University of California Press.

4. Wilczek, F. (2004). Whence the Force of F = ma? II: Rationalizations. Physics Today, 57(12), 10-11.