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Re: What is wring here?



Regarding Ludwik's plea:

I must be blinded by something; please help.

Consider a modified Atwood machine, a horizontal
frictionless table with two massless and frictionless
pulleys. The vertically suspended mass on the left, m1,
is smaller than the vertically suspended mass on the
right, m3.
...
But what is wrong with the following reasoning?
I am interested in what happens to m2. Therefore I
draw the free body diagram for m2 with four forces.
Vertical forces cancel. The remaining two forces are
F1=m1*g (to the leftt) and F3=m3*g (to the right).
The net force on m2 must thus be F=(m3-m1)*g; it is
directed to the right. ...

No, this is not correct. The tension in the rightmost string is *not*
m1*g and the tension in the leftmost string is *not* m3*g as long as
the system is free to respond and accelerate in the absence of any
frictional or experimenter-imposed force on m2 preventing the system
from moving. But when m2 *is* appropriately anchored, *then* the
string tensions are what you claim above.

If you want to analyze the system via free-body diagrams with force
accountings then you will need to consider each of these string
tensions as two extra unknowns for the problem. But then you will
need to do analogous free-body analyses for masses m1 and m3 as well.
The dynamical equations from these other analyses will provide the
extra equations necessary to solve for the motion of the system as a
whole and to find the unknown string tensions.

If you want to analyze the system via Lagrangian mechanics you will
only need to figure out L = T - V including the kinetic and potential
energies of all 3 masses and use the constraints that each string has
a fixed length allowing you to express L entirely in terms of a single
dynamical variable (such as the horizontal location of m2) and its
time derivative. Once L is found the acceleration of the dyanmical
variable is easily found by differentiating L in the usual way to get
the Euler-Lagrange equation.

David Bowman
David_Bowman@georgetowncollege.edu