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Particle Dynamics

 
 

Momentum

We define #~{linear momentum} of a particle, mass ~m, position #~r:

#~M _ #:= _ ~m deriv{#~r} _ = _ ~m #~v

where _ #~v _ = _ deriv{#~r} _ = _ d#~r ./ d~t

If a force #~F is acting on the particle then we have:

#~F _ = _ ~m deriv2{#~r} _ = _ deriv{#~M}

So if _ #~F _ == _ 0 , _ then _ #~M _ is constant - the #~{principal of conservation of momentum}.

Impulse

Now _ _ _ #~F ( ~t ) _ = _ deriv{#~M} ( ~t )

which, by the fundamental law of calculus, means

int{#~F ( ~t ),~t_0,~t_1,d~t} _ = _ #~M ( ~t_1 ) - #~M ( ~t_1 )

The quantity _ ~#I #:= ~{∫} #~F d~t _ is called the ~#{impulse} delivered by the force in the given time interval. So impulse is change in momentum, and is useful to describe situations where large forces act for short times.

Angular Momenum

The #~{angular momentum} #~H_O, about the origin O, of a particle with momenum #~M, and position #~r, is defined:

#~H_O ( ~t ) _ = _ #~r # #~M

The #~{moment} #~G_O ( ~t ), about the origin O, of the force #~F at position #~r is defined:

#~G_O ( ~t ) _ = _ #~r # #~F