MadSci Network: Physics

Re: Please answer question i can't find answer ,sorry.

Date: Wed Feb 13 05:43:00 2002
Posted By: Radu Grigore, Undergraduate
Area of science: Physics
ID: 1012358789.Ph

Hi Roman,

It is always better to try to "prove something is wrong" than
to "prove someone wrong". This is because the former point of
view allows you to focus on IDEAS and not people.

In order to solve the problem I'll reformulate it more formally
and I will also make some assumptions not mentioned by you.

* all happens in an uniform gravitational field characterized
  by g constant acceleration towards "DOWN"
* the "T" body is constrained so that it can only rotate around
  an axis that passes thru the 'join' of its 'arms' (horizonatal
  and vertical); this axis is perpendicular to both arms
* this motion is accompanied by 'some' friction
* there are two masses Mleft and Mright attached to the heads of
  the horizontal arm; it does not matter how they are attached

? specify the angle of inclination which corespond to 
  the equilibrium

The fundamental law of mechanics (Newton) is:
"Translating" this to rotational motion (such as the motion
our T body is confined to):
   M = J (d/dt)K
       M = Force momentum
       J = Rotational inertia
 (d/dt)K = variation (in time) of kinetic moment

At equilibrium the body does not move so it has zero kinetic
moment. And this kinetic moment is constant: it does not vary
in time. So the above equation becomes:
   M = 0                       (1)

This is the equation used to solve your problem! Only one more thing...
How exactly is the force momentum calculated? This is defined
in the process of "translating" mentioned earlier. The result is
   M = F d                     (2)
        F = force magnitude
        d = distance from the force to the reference point
            (the 'join' in our case)

There are two cases:

1) The T body has no mass
In this case if Mleft > Mright the T body will end up in this position:


Because this is the only position where the momentum of the two
weights compensate each other: they are zero because d in equation
(2) is zero for both of them.

2) The T body has a mass Mt
In order to solve this we must use another important fact from mechanics.
The mass of a rigid body can be considered to lie exclusively in a
point named center of mass (which is not necesarily 'inside' the body).
Now I need to make another hypotesis:

* the center of mass of the T body is on the vertical arm, at a 
  distance Lcm from the 'join'
* the length of the horizontal arm is 2xLa
* the angle made by the 'vertical' arm with the vertical is alpha
* Mleft > Mright

Now I write the force momentum for the three weights involved here:
   M1 = + Mleft g La cos(alpha)                  (3.1)
   M2 = - Mright g La cos(alpha)                 (3.2)
   M3 = - Mt g Lcm sin(alpha)                    (3.3)
The signs are different because the forces tend to rotate the body
in different directions. By using equations (3) and (1) you can 
find that:

              Mleft - Mright   La
tan(alpha) = ---------------- -----               (4)
                    Mt         Lcm

In the first case the angle of inclination does not depend on the
masses but in the second case it depends accordingly to (4).

PS: Equation (4) pretty much describes how a balance work.

Now, I must tell that I'm not sure this is what you've asked. If
not then please be more specific.


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