Re: third law forces cancel? Momentum and Angular Momentum vs. Energy

[Hugh Logan <hloganPHY@CFL.RR.COM>]

John Clement wrote:

> Now Knight's objective may be OK, in the context of momentum, but it is
> difficult to tell in a snippet.  The statement should be judged in context.
> What page is it on, Chapter&section?
The following was written last night except for editing. I don't know if
it will answer Jim's question.

Quite a few years ago, I heard a Harvard physics professor tell his
general physics evening students something to the effect that, although
they had previously been cautioned not to add the forces in an
action-reaction pair, that they would now have the opportunity to do so.
He was talking about a system of particles -- probably in the context of
momentum conservation or the principle of momentum, which implies the
former when the system is isolated (from forces external to the system).

I do not have Randy Knight's book, but I think the idea of  canceling
the equal and opposite forces of
action-reaction pairs in the context of the momentum of a system of
particles may be found in almost any upper level undergraduate mechanics
text.  I have _Principles of Mechanics, 2nd ed._ by Synge and Griffith,
McGraw-Hill, 1949 in front of me, having referred to it to answer a
question on another forum. Like most texts, they start out with a system
of n particles. The momentum of the system _P_ is the vector sum of the
momenta of the individual particles, _P_= (sum from i=1 to n) m_i * v_i.
S&G state the principle of
momemtum as: "The rate of change of momentum of  a system is equal to
the vector sum of the external forces." [They use the qualifier,
"linear," for momentum -- which I have deleted -- to distinguish it from
angular momentum. I have also changed their notation to more familiar
notation. Also, I am using lower case to refer to individual particles
and upper case for the entire system.] Differentiating the above
expression for the net momentum of the system with respect to time, one
obtains

d_P_/dt = (sum from i=1 to n) m_i * a_i

where a_i  is the acceleration of the ith particle, and  thus, by
Newton's second law, m_i*a_i is the net force on the ith particle.
Letting f_i  be the external force acting on the ith particle and f '_i
the internal force on it, we have

d_P_/dt= (sum from i=1 to n) (f_i + f '_i )= (sum from i=1 to n) (f_i) +
(sum from i=1 to n) (f '_i ).

Since the internal forces occur as members of equal and opposite
action-reaction pairs,
(sum from i=1 to n) (f '_i )= 0 (the promised summation and cancellation
of the forces in action-reaction pairs), so that

d_P_/dt=(sum from i=1 to n) (f_i) or  d_P_/dt=_F_, where _F_ is the net
external force on the system.

In the case that the system is isolated, _F_=0, and thus d_P_/dt=0. In
other words, momentum is conserved for the system.

I think that, when interpreted, this agrees with what John, Michael, and
Bob wrote. However, I am not convinced that Newton's third law and
conservation of momentum are completely interchangeable as John D.
wrote. What about the case of a particle that emits radiation? The
particle recoils, and the radiation carries equal and opposite momentum
so that momentum is conserved, but considered classically, there is no
reaction force to the force associated with the recoil of the emitting
particle. I was of the impression that momentum conservation is more
general than Newton's third law. I recall that this was pointed out in
the college version of _PSSC Physics_.

Michael mentioned that the cancellation of forces in an action-reaction
pair was used in arriving at the equation for the motion of the center
of mass of a system of particles. It is an easy consequence of
differentiating the equation defining the center of mass,

_R_(cm)=[(sum from i=1 to n)(m_i*r_i)]/[(sum from i=1 to n)(m_i)],

_r_i being the position vector of the ith particle, to get  the velocity
_V_(cm) of the center of mass:

_V_(cm)=[(sum from i=1 to n)(m_i*v_i)]/M, where _v_i is the velocity of
the ith particle and M is the total mass of the system. Now

M*_V_(cm)=(sum from i=1 to n)(m_i*v_i)=(sum from i=1 to n)(p_i)=_P_.

Differentiating _P_=M*_V_(cm), we get  d_P_/dt=M*_A_(cm). But the
principle of momentum says that d_P_/dt=_F_, where _F_ is the net
external force on the system. Thus we get the required equation for the
motion of the center of mass of the system,

_F_=M*_A_(cm) .

The above was based on the presentation in S&G, but I condensed his
component equations for the CM into a single vector equation.

S&G go on to show that moments (torques) produced by internal
action-reaction pair forces similarly cancel in the summation used in
the derivation of the principle of angular momentum, d_J_/dt=_L_, where
_J_ is the angular momentum of the system, and _L_ is the net torque
acting on the system. This principle holds with respect to either a
fixed point or the CM. (fixed line or line through the CM in plane
mechanics). Again, S&G's notation for angular momentum and torque is
different.

S&G compare (p. 137) the principles of  momentum (d_P_/dt=_F_) and
angular momentum (d_J_/dt=_L_) with the principle of energy, which they
state as, "The rate of change of kinetic energy of a system is equal to
the rate of working of all the forces, external and internal." As an
equation this would be dT/dt=dW/dt, where T is the total kinetic energy
of the system and W is work done on the system by all the forces,
external and internal. Condensing S&G's component notation into vector
notation and using notation that is easier to type,

T=(1/2)(sum from i=1 to n)[m_i*(_v_i dot _v_i)],

dT/dt=(sum from i=1 to n)[m_i*(_v_i dot  d_v_i/dt)]=(sum from i=1 to
n)[m_i*(_v_i dot  a_i)],

dT/dt=(sum from i=1 to n)(_v_i dot  m_i*a_i)=(sum from i=1 to n)(_f_i
dot  v_i) where f_i is the total force, both external and internal,
acting on the ith particle. Note that
_f_i dot  v_i =_f_i dot dr_i/dt = (_f_i dot dr_i)/dt =dW_i/dt, the rate
of working (power) of all forces, both external and internal, on the ith
particle. Thus, summing the rate of working on all n particles,

dT/dt=dW/dt .

S&G had earlier derived this equation for a single particle where any
work done was necessarily done by external forces. S&G state
insightfully (p. 137),  "There is a sharp difference between the
principles of linear and angular momentum on the one hand and the
principle of energy on the other. In the principles of momentum the
internal forces are eliminated; in the principle of energy they are not
eliminated, except in the special case where they do no work and so
contribute nothing to dW/dt.  In our idealized mathematical models,
consisting of rigid bodies with smooth contacts, no work is done by the
internal forces, and so they disappear from the principle of energy. In
cases of collision, however, work may be done by the internal forces ...
; that is because, in such cases, it is impossible to regard the bodies
as absolutely rigid."

I couldn't resist extending the  discussion to energy, since there has
been so much talk about the inadequacy of the work-energy theorem as
usually presented (considering only work done by external forces). S&G
has been on my bookshelf for over half a century, but I am only now
starting to appreciate the conceptual clarity which it is reputed to
have. It emphasizes the role of models including mathematical models.
Now, I can hardly open the book without finding something insightful. I
didn't appreciate Synge and Griffith in the fall of 1951, because
sophomore engineering students were taking Applied Mechanics from it
concurrently with non-calculus general physics from _College Physics_ by
Sears and Zemansky, and we were just starting integral calculus.  The
traditional high school physics courses of that era did not provide much
of a conceptual foundation in mechanics, and I am not convinced that the
present-day conceptual physics courses do much better. Maybe there is
some hope for Modeling.

To complete the discussion, the principle of conservation of energy for
a conservative system is obtained by recognizing for such a system,
dW/dt= -dV/dt, where V is the potential energy of the system.
Substituting in dT/dt=dW/dt , one gets dT/dt=-dV/dt or d(T + V)/dt =0,
showing that T+V is constant. Calling the constant E, one gets T+V=E, E
being the constant total energy.

Hugh Logan

> > Many authors say that N's 3rd law pair forces do not cancel because they
> > are acting on different (i.e. opposite) objects.  Randy Knight, in his new
> > PER-based textbook, says, in his discussion of conservation of momentum,
> > that
> > \vec{F}_{k on j} + \vec{F}_{j on k} = \vec{0}.  Is the discrepancy between
> > these two statements real or imagined?  Would you tell your students one
> > or
> > both or neither of these statements?  I have told students that we don't
> > even add forces on different objects together; should I not have?  What's
> > the best pedagogical approach to this?  Further comments?
> >
> > Thanks in advance,
> > Larry