Re: moment of inertia
mmeron_at_cars3.uchicago.edu
Date: 02/08/05
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Date: Tue, 08 Feb 2005 23:29:27 GMT
In article <1107890193.810813.118290@z14g2000cwz.googlegroups.com>, "PD" <pdraper@yahoo.com> writes:
>
>Randy Poe wrote:
>> PD wrote:
>> > Randy Poe wrote:
>>
>> > > But for me personally, it helps me understand what's
>> > > going on if I can also think about it in terms of
>> > > mass, forces, and linear momentum because those are
>> > > the fundamental quantities. Conservation of angular
>> > > momentum isn't a new law, it's a consequence of conservation
>> > > of linear momentum. Rotational KE isn't a new thing,
>> > > it's the sum of the linear KE of all the particles.
>> >
>> > I don't think I agree with that statement. Linear momentum and
>> > rotational momentum are conserved separately, and the two cannot
>> > commingle, unlike linear and rotational kinetic energy.
>>
>> Yes, in a sense. Yet every particle in a rotating
>> body does in fact have a linear momentum at each
>> instant in time, a momentum which is constant in
>> amplitude because no work is done in maintaining
>> circular motion.
>>
>> It is a fact that rotational KE = 0.5*Iw^2
>> = 0.5*SUM(m_i*r_i^2)*w^2
>> = 0.5*SUM(m_i*v_i^2)
>> = SUM(0.5*m_i*v_i^2)
>>
>> i.e. the rotational KE of a rotating body is exactly
>> the total of the linear KE of all the particles in
>> the center-of-rotation system. This is no accident.
>>
>> - Randy
>
>
>There's an important difference. In the case of energy, you CAN
>decompose rotational energy to be the sum of linear kinetic energy of
>comprising particles. The same is true for thermal energy. This is
>precisely what allows a *single* energy conservation law and the
>transformation of one kind of energy to another.
>
>In the case of momentum, there are *connections* between angular
>momenta and linear momenta, but I would hesitate that there is identity
>between them. The laws are separately conserved and there is no
>transformation of one to the other in process in any closed system. If
>I have to, in order to defend this point, I'll cite quantum mechanical
>spin.
>
Before we even get to QM, one should mention that linear and angular
momenta are related to inherently different symmmetries, the first to
translational and the second to rotational. And while one can,
locally, represent the effect of an infinitsimal rotation as a
translation, one can never do this globally.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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