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Linear motion
A Nexus mindmap
I much prefer a stack of numbers to represent the
components to using unit vectors. It's much easier to
see what is going on, and linear transformations on
vectors can be prepresented as matrices.
0
0
0
0
1
Position is simple
enough. In general this
will be a vector.
3
0
0
1
Since position is a vector, we are free to choose the
basis in which to express the vector. This does not
have to be an orthogonal basis, it just has to span the
space. However choosing an orthogonal basis means
that often the physics is independent in each
dimention.
2
2
0
0
By v^2 we mean the length of v squared.
i.e. the dot-product with itself.
0
0
1
1
0
1
2
0
First we'll take a look at motion
in a line just to remind
outselves of the form of various
relationships in kinematics and
dynamics.
1
0
0
0
0
1
1
Basic relationships that govern
the dynamics of objects
4
0
1
2
0
For a whole lof of particles each particle and
dimension contributes to the kinetic energy
independently.
0
0
1
0
0
0
1
Note that because the force
derives from the derivative of the
potential, any constant terms in
the potential have absolutely no
effect on the dynamics. i.e. we are
free to place the zero of the
potential wherever we like.
2
1
3
2
0
1
0
0
0
We are doing something crazy here - we are adding all
the force vectors together as if that means something.
3
With
n
particles there will be
n(n-1)
such forces (which is always even)
1
Because of Newton's 3rd law the total force vector is
zero for all the forces between the particles. An
interpretation of this remarkable result is that the
total momentum is not changing in time.
0
2
1
This is the same argument as before
0
0
1
0
0
0
1
Imagine some complicated object
with lots of internal interactions
as well as external forces
3
It would be really neat if we could write this
expression in the form of
F
=
ma
...
0
2
2
0
1
So we CAN write the expression in the form
F
=
ma
. It
describes the motion of a ficticious point particle with
the total mass, located somewhere within the
collections of particles.
0
2
3
5
1
Say we have an object composed
of two pieces
A
, and
B
.
Where is
the centre of mass
R
?
0
Divide and conquer
4
3
What about rocket propulsion?
0
4
(Because we will be concerned with
rotations not linear motion)
0
5
3
Linear motion