Chapter 4 – Motion in 3 Dimensions


F  ma or
Fx  mx
Fy  my
Fz  mz
if the Fx  x  is the function of x only and
Fy  y  is the function of y only and Fz  z  is
the function of z only, then there is nothing
new, just 3 equations, one for each
direction. However, if variables other that x
and its derivative appear in Fx and likewise
for Fy and Fz then there is a significant
complication. For this case, we have
coupled D.E. that may ever be non-linear.
The work energy principle is still true:
 
F
 dr  T

path
that is the change in Kinetic energy of a
particle is equal to the line integral of the
resulting force acting on the particle along
the path the particle moves. If the resulting
forces are all conservative, the line integral
will be independent of path and depend
only on the end points.
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The test tosee if a force is conservative is
that xF  0 or
i

x
Fx
j

y
Fy
k

0
z
Fz
The derivation of this idea that is the
integrations of a vector along a path being
independent of path is a special case of
Stokes Theorem which states:

 
 
F  dr  area   F  n da

Stokes Theorem is true whether F is
conservative or not. But if the integral on
the LHS is to be zero, one must have
  F  0.
This is a very powerful theorem. It states
that a line integral around a perimeter is the
same as a surface integral over the area
inside the perimeter.
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Potential Energy
In one dimension, the potential energy is
dV
related to the force by F  
.
dx
In three dimensions, this becomes:

 V
V
V 
F   i
j
k
  V

x

y

z


The Del operator is defined in rectangular
coordinates to be:
i



 j k
x
y
z
Some terminology and properties of Del
are:
V is said to be the gradient of V
The gradient is a vector that gives the
maximum change in V. It is perpendicular
to lines of V = constant.
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


  F is called the curl of F . If   F  0 , then


F can be written as F  V because
  V  0 , that is the curl on any gradient is
always zero.


  F is called the divergence of F .
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Curvilinear Coordinates
(back of book, page A12)
To be able to use the equations in Appendix
F one needs to understand what the “h”s
are. The equation that defines them is:

dr  e1h1du  e2h2dv  e3h3dw
This is an orthogonal coordinate system
with coordinates u, v, and w that have unit
vectors associated with them of e1, e2 , e3.
Notice that to get h1 , one holds v and w
constant and generates a length element.
In an xyz coordinate system, the length
along the x axis is dx, so h1  1.
For cylindrical coordinates, one has
x  R cos , y  R sin  , z  Z
The u, v, and w are R, , and Z.
If  and Z are constant, the differential length
generated by letting R increases in just dR ,
so h1  1.
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If R and Z are constant and  is allowed to
increase, the length generated is
Rd , so h 2  R.
And if R and  are constant and Z
increases, one gets dz h3  1.
In other words, if two of the dimensions are
held constant and the third is allowed to
change, one has the
d distance generated  hu du , in the eu
direction.