Re: Work/Energy Theorem

[SCIAMANDA@edinboro.edu]

Jim Green commented:
> Bob Sciamanda said -
> > The word "work" is defined as the left hand side of the Work-Energy 
> > theorem (at least for me).

> Yes, Bob, but John says "Which forces?"  and  "Which path?"  If the system
> is a particle, life is straightforward, but, if the system is a gob of goo,
> it ain't obvious which forces should be used so John considers all the
> possibilities he can think of.

Hi Jim!
It is very clear from the derivation of the W.E. theorem "which forces 
and which path":

It begins with the vector statment F=ma (N#2). If m is a particle, then 
F is the vector sum of all of the forces influencing that particle; if m
is the total mass of a system of particles (solid, liquid, gas or plasma) 
then F is the vector sum of all of the external forces influencing the 
particles making up the system (internal forces can also be included - 
they will cancel in pairs in the vector sum, by N#3). In both cases then, 
F is the vector sum of all of the external forces influencing m (particle 
or system of particles).

Then one forms the dot product of each side of this vector equality with 
the vector dr, which is the vector displacement of m (in the single particle 
case) or the C.M. of the system (in the case where m is the total mass of 
a system of particles).  One then integrates each side over the actual 
trajectory of the particle or (in the system case) over the trajectory of
 the C.M. of the system.

The result of the RHS is the change in the quantity .5 mv^2, where v is 
the speed of the particle (when m is a particle), or v is the speed of 
the C.M. of the system (when m is the total mass of a system of particles).

The theorem follows because *if the above is followed* the above RHS 
result is independent of the details of the motion and is just the 
difference in the quantity .5mv^2.  The argument works only if at every 
point in the integration dr is the vector displacement of m (or the C.M),
and F is the current vector sum of all the external forces on the 
particle (or the system of particles).

If dr represents something else, all bets are off, and you are not 
generating the W.E. theorem. (I hope it goes without saying that all of this 
is done in an inertial frame, ie., in which F=ma.)

Bob Sciamanda           sciamanda@edinboro.edu
Dept of Physics
Edinboro Univ of PA     http://www.edinboro.edu/~sciamanda/home.html