Particle Dynamics

 

1 Momentum

We define linear momentum of a particle, mass m, position r:

M ...:= ...m

·
r
.
...= ...m v

where ...v ...= ...

·
r
.
...= ...dr ⁄ dt

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

F ...= ...m

··
r
.
...= ...
·
M
.

So if ...F ...≡ ...0 , ...then ...M ...is constant – the principal of conservation of momentum.

2 Impulse

Now .........F ( t ) ...= ...

·
M
.
( t )

which, by the fundamental law of calculus, means

⌠t1
│
F ( t )dt
⌡t0
...= ...M ( t1 ) – M ( t1 )

The quantity ...I := ∫ F dt ...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.

3 Angular Momenum

The angular momentum HO, about the origin O, of a particle with momenum M, and position r, is defined:

HO ( t ) ...= ...r × M

The moment GO ( t ), about the origin O, of the force F at position r is defined:

GO ( t ) ...= ...r × F