teodorakis said:
ok i got the f=ma but its rotational equivelant comes from where?From the 2nd law?Special case of linear momentum?2nd law doesn't say anything about rotation.Where's this vectoral cross come from?I nearly understand it as i tell in my previous message, but scientifically still I'm not satisfied.
Conservation of linear momentum for a single particle
is Newton's first law. Conservation of angular momentum for a single point mass particle is a direct consequence of Newton's first law. Point masses have a null inertia tensor, so the angular momentum of a point mass is simply {\boldsymbol r}\times {\boldsymbol p}. Since the cross product of a pair of parallel or anti-parallel vectors is zero, angular momentum of a constant mass point mass is trivially conserved whenever linear momentum is conserved.
To derive conservation of linear momentum for a system of particles you need to look at Newton's third law as well as Newton's second law. The weak form of Newton's third law -- the force exerted by particle B on particle A is equal in magnitude but opposite in direction to the force exerted by particle A on particle B -- suffices for deriving conservation of linear momentum, but does not suffice for deriving conservation of angular momentum for a system of particles. To derive the latter you need the strong form of Newton's third law -- forces are equal-but-opposite
and are directed along the line connecting the particles.
Domenicaccio said:
But I am sure you can derive \tau=I\alpha from F=ma by simply considering the extended object as the sum of tiny point-like particles each of which still obeys Newton's 2nd law.
You can't derive that because in general {\boldsymbol \tau}={\mathbf I}{\boldsymbol \alpha}
is not true. The rotational analog of Newton's second law is {\boldsymbol \tau} = {\mathbf I}{\boldsymbol \alpha}+ {\boldsymbol \omega}\times({\mathbf I}{\boldsymbol \omega}).
Strictly speaking, Newton's laws are only valid in an inertial reference frame. In particular, {\boldsymbol F}= d{\boldsymbol p}/dt is only valid in an inertial frame. Fictitious forces (e.g., centrifugal force, coriolis force) must be introduced to make Newton's laws appear to be valid in a rotating frame. Fictitious torques must similarly be introduced to make the rotational analog of Newton's second law, {\boldsymbol \tau}= d{\boldsymbol L}/dt appear to be valid in a rotating frame.
The angular momentum due to some composite body rotating around some axis expressed as {\boldsymbol L}={\mathbf I}{\boldsymbol \omega}. Taking the time derivative, d{\boldsymbol L}/dt=(d{\mathbf I}/dt){\boldsymbol \omega}+{\mathbf I}(d{\boldsymbol \omega}/dt)d{\boldsymbol L}/dt. The problem: While the inertia tensor
I of some constant mass, solid object is constant in a body frame defined by the object, it is anything but constant from the perspective of an inertial frame.
To have the time derivative of the inertia tensor be zero, which is exactly what is going on when one claims d{\boldsymbol L}/dt={\mathbf I}d{\boldsymbol \omega}/dt, one must express the angular momentum, inertia tensor, and angular velocity in a rotating frame. A fictitious torque term, {\boldsymbol \omega}\times({\mathbf I}{\boldsymbol \omega}), must be introduced to make the rotational analog of Newton's second law appear to be valid in this rotating frame.
Of course this is not the way but in the way most physics books this rotational analog comes up like this.I think he first analyse the rotational motion and as i mentioned in my previous message he saw the conserved quantity and after some manipulatons he evaluates this rotational analog.But books don't even mention about this, ok may be people don't need to learn where ,it comes form they just use it, but i can not accept the dogmatic formulas.May be just because i don't get it:).Excuse my sillyness again.But anyone shares this point of view and wants to fully understand the real meaning of this rather than some cold mathematical formulas, please contact with me or advise some books(i already got feynman's).
