Section 18–1 The center of mass

(Motion of center of mass / Center of mass vector / Stationary center of mass)

In this section, Feynman discusses the motion of center of mass, a definition of center of mass vector, and an example whereby the center of mass is stationary. This section only gives a short introduction to the concept of center of mass. The next chapter elaborates on this concept further with two sections: 19–1 Properties of the center of mass and 19–2 Locating the center of mass. We may understand this section from the perspectives of idealization, approximation, and exception.

1. Motion of center of mass:

“That is called the theorem of the center of the mass, and the proof of it is as follows… (Feynman et al., 1963, section 18–1 The center of mass).”

According to Feynman, the first theorem concerning the motion of complicated objects can be observed if we throw an object made of some blocks held together by strings, into the air. It demonstrates that there is a mean position (or effective center) which moves parabolically, but it is not definitely a point of the material itself. The theorem of the center of mass’s motion can be developed from a simpler model involving two particles first. The simplified theorem may be stated as “[t]he motion of the center of mass of a system of two particles is the same as the motion of a single particle of mass equal to the total mass of the system acted on by the resultant of all the external forces which act on the individual particles (French, 1971, p. 353).” Then, we can let students deduce a theorem involving a system of many particles.

It is not straightforward to develop an equation to describe the motion of complicated objects such as “flowing water” or whirling galaxies.” The equation describing the center of mass of an extended body involves idealizations and simplifications. Firstly, we idealize a rigid body as “a collection of particles whose relative distances are constrained to remain absolutely fixed (Thornton & Marion, 2004, p. 411).” The rigid body is an idealization that does not exist in nature because the atoms within the body are always in motion. In addition, we simplify the motion using the equation F = MR that describes only the translation of the body (or the motion of center of mass). This idealized equation does not provide information on the body’s orientation in space as well as the location of every atom within the body.

2. Center of mass vector:

“… M is the sum of all the masses, i.e., the total mass. Then if we define a certain vector R to be R = S_im_ir_i/M, Eq. (18.3) will be simply F = d²(MR)/dt² = M(d²R/dt²), since M is a constant (Feynman et al., 1963, section 18–1 The center of mass).”

Feynman defines the center of mass vector using R = S_im_ir_i/M and expresses the external force F on the total mass M of a body as F = d²(MR)/dt² = M(d²R/dt²). The point at R is called the center of mass of the body and it is a kind of average r in which the different r_i’s have weights that are proportional to the masses. In a sense, one may assume the weights of the masses are under a parallel gravitational field. Importantly, the center of gravity for an extended object that is relatively large does not necessarily coincide with the center of mass of the same object. By assuming the “same” gravity, we may say that the center of gravity of a body is approximately located at the same point as the body’s center of mass.

Another approximation is based on the assumption in which the parts of a body are moving at speeds very much slower than the speed of light. Using the nonrelativistic approximation for all quantities, we may deduce that the mass of the moving body is constant such that we can write F_i = d²(m_ir_i)/dt². We could include internal forces within the body using Newton’s second law by writing S_iF_i^ext + S_iF_i^int = M d²R/dt² and stating S_iF_i^int = 0 due to Newton’s third law. However, Newton’s third law is not always valid for moving charged particles because electromagnetic forces are velocity dependent. To be precise, magnetic forces exerted on a moving charge q in a magnetic field B obey the weak form of the third law (F_ab = -F_ba), but not the strong form (action and reaction must act along the line joining the particles).

3. Stationary center of mass:

“… then this little blob of gas goes one way as the rocket ship goes the other, but the center of mass is still exactly where it was before (Feynman et al., 1963, section 18–1 The center of mass).”

Feynman discusses a special case (or exception) of a center of mass’s motion. He asks whether it is absolutely impossible to have rocket propulsion because one cannot move the center of mass. Although the center of mass is stationary, he clarifies that we can propel an interesting part of the rocket by throwing away the uninteresting part (exhaust gases). On the other hand, there could be a discussion about the relativistic limitation to the concept of an absolutely rigid body. For instance, if we strike a blow at one end of the rocket, it is impossible for the effect to be felt instantaneously at the opposite end. This is related to the transmission of a signal with an infinite velocity that is forbidden by the theory of relativity.

In the real world, the lift-off of a rocket has to overcome the Earth’s gravitational forces while it accelerates to the outer space. Similarly, a system of (colliding) particles cannot be completely isolated from gravity. In the words of French (1971), “[t]he conservation of momentum in a collision process holds good only to the extent that the effect of any external forces can be ignored. If external forces are indeed present, the duration Dt of the collision must be so short that the product F Dt is negligible. A different way of stating this same condition is that the forces of interaction between the colliding particles must be much greater than any external forces which may be acting (p. 353).”

Questions for discussion:

1. What are the idealizations or simplifications in conceptualizing the center of mass’s motion?

2. What are the approximations needed in writing the equation of the center of mass’s motion?

3. What are the exceptions that are related to the center of mass’s motion?

The moral of the lesson: we may define the center of mass vector using R = S_im_ir_i/M and express the external force F on the total mass M of an idealized object as F = M(d²R/dt²).

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. Thornton, S. T. & Marion, J. B. (2004). Classical Dynamics of Particles and Systems (5th Edition). Belmont, CA: Thomson Learning-Brooks/Cole.

Section 17–5 Four-vector algebra

(Relativistic invariance / Scalar product / Momentum of a photon)

In this section, Feynman discusses relativistic invariance, scalar product of four-vectors, and momentum of a photon (an exception of four-vector).

1. Relativistic invariance:

“...complete the law of conservation of momentum by extending it to include the time component. This is absolutely necessary to go with the other three, or there cannot be relativistic invariance (Feynman et al., 1963, section 17–5 Four-vector algebra).”

According to Feynman, we must complete the law of conservation of momentum by extending it to include the conservation of energy (time component) to obtain a valid four-vector relationship in the space-time geometry. That is, it is absolutely necessary to have four components of momentum such that there is relativistic invariance. We can define relativistic invariance (or Lorentz invariance) in terms of a four-vector a_μ under a Lorentz transformation whereby a_t′²−a_x′²−a_y′²−a_z′² = a_t²−a_x²−a_y²−a_z². In chapter 16, Feynman explains that “the conservation of mass which we have deduced above is equivalent to the conservation of energy” in the context of collisions within the special theory of relativity. In a sense, this implies that we have a single principle involving the conservation of momentum, mass, and energy.

In Volume II, Feynman adds that “the quantity which is analogous to r² for three dimensions, in four dimensions is t²−x²−y²−z². It is an invariant under what is called the “complete Lorentz group”—which means for transformation of both translations at constant velocity and rotations (Feynman et al., 1964, section 25–2 The scalar product).” He also discusses the four-velocity vector with components v_x = dx/dt, v_y = dy/dt, v_z = dz/dt, and mentions that an incorrect guess of the time component is v_t = dt/dt = 1. It turns out that the four “velocity” components that we have written down also have an invariant quantity (the speed of light) if we include the proper time t in all denominators (cdt/dt, dx/dt, dy/dt, dz/dt). The invariant quantity of the four-velocity can be calculated easily using (cdt/dt)² – (dv/dt)² = g²(c² – v²) = c².

2. Scalar product:

“if a_μ is one four-vector and b_μ is another four-vector, then the scalar product is S′a_μb_μ = a_tb_t−a_xb_x−a_yb_y−a_zb_z. It is the same in all coordinate systems (Feynman et al., 1963, section 17–5 Four-vector algebra).”

Feynman states the notations that are used for a scalar product (or inner product) in terms of S′_μA_μA_μ = A_t²−A_x²−A_y²−A_z². The prime on S′ means that the first term, the “time” term, is positive, but the other three terms have minus signs. This invariant quantity is the same in any coordinate system (or inertial frame), and we may call it the square of the length (or norm) of a four-vector. In addition, Feynman (1964) mentions that “the only real complication is the notation.” However, some authors complicate the situation by changing the sign of all the terms and state the square of the length of the four-vector as +a_x²+a_y²+a_z²−a_t². Alternatively, one may state Sa_μ² = +a_x²+a_y²+a_z²+a_t² in which all have the same signs, and define a_t in terms of ict instead of ct (i is an imaginary number).

Feynman elaborates that the square of the length of a four-vector momentum of a single particle is equal to p_t²−p_x²−p_y²−p_z², or in short, E²−p². He says that the invariant quantity must be the same in every coordinate system (or inertial frame) and it is purely its energy, which is the same as its rest mass. In other words, the norm of the four-momentum vector is equal to m₀²c⁴ (i.e., E²−p²c² =m₀²c⁴) and the invariant quantity can be stated as rest energy instead of rest mass. In essence, we may save time in problem-solving if we use the four-momentum vector with a scalar product (relativistic invariant) that is the same in all inertial frames of reference. Simply put, the knowledge of an invariant quantity allows us to choose an inertial frame in which a problem can be solved more easily.

3. Momentum of a photon:

“Such a photon also carries a momentum, and the momentum of a photon (or of any other particle, in fact) is h divided by the wavelength: p = h/λ (Feynman et al., 1963, section 17–5 Four-vector algebra).”

Feynman ends the chapter by discussing the momentum of a photon. This is an interesting example because the rest mass of a photon is zero. Thus, the equation E²−p²c² = m₀²c⁴ can be simplified as E²−p²c² = 0 or E = pc. However, Feynman could mention that the momentum of any (massless) particle is equal to its total energy times its velocity (p = vE/c²) can be reduced to E = p if c = 1. More importantly, physics teachers should explain that the equations E = gm₀c² and p = gm₀v are not really useful for massless particles because m₀ = 0 and g approaches infinity for objects moving at the speed of light (i.e., E and p are not equal to zero). The energy of a photon would change and this is related to the relativistic Doppler effect that is elaborated later in the section 34–6 The Doppler effect (Feynman et al., 1963).

Feynman mentions that the rest mass of a photon is zero and ask whether the photon’s energy is zero using the formula m₀/√(1−v²) instead of E = gm₀c². He adds that the photon really can have energy even though it has no rest mass, but it possesses energy by always moving at the speed of light. In Volume II, Feynman states that “the energy E₀ has the relativistic mass E₀/c² the photon has a mass (not rest mass) ℏω₀/c², and is ‘attracted’ by the earth (Feynman et al., 1964, section 42–6 The speed of clocks in a gravitational field).” On the other hand, from an interpretation of Meissner effect, “[t]he photon mass inside a conventional superconductor is 10^–11 proton masses, or less (Wilczek, 2008, p. 213).” It is good to end the chapter by discussing a special case (or exception) of four-vector using the momentum of a photon.

Questions for discussion:

1. How would you define the concept of relativistic invariance?

2. Would you define the scalar product of a four-vector that has the same sign or opposite signs?

3. How would you explain the mass and momentum of a photon?

The moral of the lesson: the relativistic invariance of a four-vector a_μ is related to the principle of the conservation of momentum, mass, and energy.

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

3. Wilczek, F. (2008). The lightness of being: Mass, ether, and the unification of forces. New York: Basic Books.

Section 17–4 More about four-vectors

(Energy is mass / Composite velocity / Transformation of energy and momentum)

In this section, Feynman discusses four-momentum from the perspectives of mass-energy equivalence, composite velocity, and the transformation of energy and momentum. This section could be renamed as “four-momentum” instead of “more about four-vectors.”

1. Energy is mass:

“Energy and mass, for example, differ only by a factor c² which is merely a question of units, so we can say energy is the mass (Feynman et al., 1963, section 17–4 More about four-vectors).”

There are many examples of four-vectors such as four-velocity and four-force. Another example of four-vectors is four-momentum that has three spatial components (linear momentum) and a temporal component (energy). Because it is inconvenient to write c’s everywhere in the equations, Feynman introduces E = m by using the same trick concerning units of the energy. He explains that energy and mass differ only by a factor c² which is merely a question of units, and says that energy is the mass. On the contrary, Okun (1989) argues that mass is not equivalent to energy and emphasize that E = m₀c² is the correct equation instead of E = mc². Some physicists have also argued that the term relativistic mass is obsolete and suggested that it is no longer fashionable to teach this concept.

In The Evolution of Physics, Einstein and Infeld (1938) write that “according to the theory of relativity, there is no essential distinction between mass and energy. Energy has mass and mass represents energy. Instead of two conservation laws we have only one, that of mass-energy” (pp. 197-198).” Currently, some physicists explain that the definition of relativistic mass is redundant because mass and energy would refer to the same thing. In a sense, the crux of the problem is a matter of definition and one may argue whether the definition is useful or not. However, Galison (1997) observes that there are three sub-groups of physicists within the particle physics community (experimentalists, instrument developers, and theorists) that have a different specialized language (or definitions) for their internal communication.

2. Composite velocity:

“What is v′, the velocity as seen from the space ship? It is the composite velocity, the “difference” between v and u (Feynman et al., 1963, section 17–4 More about four-vectors).”

To find out the momentum and energy in another inertial reference frame, we have to know how the velocity transforms. If an object has a velocity v and an observer is in a space ship that is moving with a velocity u (with respect to the Earth), we can use v′ to designate the observed velocity of the object as seen from the space ship. Feynman adopts the concept of the composite velocity, the “difference” between v and u, and states it as v′ = (v−u)/(1−uv). Alternatively, we can explain that the four-momentum, P ≡ mV = (E/c, p) is obtained by multiplying the four-velocity by the rest (invariant) mass. More important, we need the Lorentz factor g in the expression of momentum and energy.

Feynman’s method to obtain the 4-momentum may seem unnatural by using a mathematical trick, v′² = (v²−2uv+u²)/(1−2uv+u²v²). As a suggestion, we should clarify that Feynman needs to derive the Lorentz factor that can be expressed as 1/√(1−v′²) = (1−uv)/(√1−v²)(√1−u²). For example, we can directly substitute the composite velocity, v′ = (v−u)/(1−uv) into the Lorentz factor, g = 1/√(1−v′²). Thus, we can get g = 1/√(1−[(v−u)/(1−uv)]²) = (1−uv)/√([1−uv]²–[v−u]²). The expression g_v′ could be quickly simplified as (1−uv)/√(1−u²)√(1–v²) and it is equal to g_ug_v(1−uv). In essence, we need the Lorentz factor of the composite velocity v′ (or observed velocity) that is from the perspective of a moving space ship (an inertial frame).

3. Transformation of energy and momentum:

“… transformations for the new energy and momentum in terms of the old energy and momentum are exactly the same as the transformations for t′ in terms of t and x, and x′ in terms of x and t… (Feynman et al., 1963, section 17–4 More about four-vectors).”

Feynman explains that the transformations for the new energy and momentum in terms of the old energy and momentum are exactly the same as the Lorentz transformations of space and time. That is, we simply replace t by E and replace x by p_x. We may elaborate that for a massive particle, the four-momentum (E, p_x, p_y, p_z) is derived using the particle’s invariant mass m multiplied by the particle’s four-velocity (cdt/dt, dx/dt, dy/dt, dz/dt). The four-velocity of a particle can be defined as the rate of change of its four-position (ct, x, y, z) with respect to the proper time, t. In short, the four-velocity is represented as (cdt/dt, dr/dt) and the four-momentum is simply (E, p), but it should be (E/c, p) if we remember to check the units.

Feynman concludes that we have discovered the four-vector momentum which can transform like x, y, z, and t. Furthermore, the “arrow” of the four-momentum has a temporal component equal to the energy and its spatial components are the three-vector momentum; but it may not be very clear how this arrow is more “real” than either the energy or the momentum. In a sense, the meaning of real is related to the concept of Lorentz invariance that does not depend on how we look at the space-time diagram. Specifically, the invariance of E² – (cp)² is really the rest energy m₀c² that remains unchanged under the Lorentz (energy-momentum) transformations. This invariance is also associated with the concept of rest mass (or invariance mass) that is the same in all inertial frames of reference.

Questions for discussion:

1. Would you explain that energy is mass in the transformation of four-momentum?

2. Would you adopt the concept of composite velocity (instead of observed velocity or Lorentz factor) in the derivation of four-momentum?

3. Why is the arrow of four-momentum more “real” than the energy or momentum?

The moral of the lesson: the “arrow” of the four-momentum has a time component equal to the energy (scalar) and three spatial components are equal to the three-vector momentum (vector); this arrow is more “real” than the energy or momentum.

References:

1. Einstein, A., & Infeld, L. (1938). The Evolution of Physics. New York: Simon and Schuster.

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. Galison, P. (1997). Image and Logic: A Material Culture of Microphysics. Chicago: University of Chicago Press.

4. Okun, L. B. (1989). The concept of mass. Physics Today, 42(6), 31-36.