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Unformatted text preview: VII1 CYLINDRICAL COORDINATES;
DEFINING EQUATIONS COORDINATE SURFACES,
COORDINATE CURVES,
COORDINATE VECTORS, AND
COORDINATE FRAMES PATHS, VELOCITY, AND
ACCELERATION IN CYLINDRICAL
COORDINATES CENTRALFORCE MOTION,
AN GULAR MOMENTUM, AND
KEPLER’S SECOND LAW VII2 Cyl. Coords and Deﬁning Equations: See ﬁgure [1—1] for
geometric relationship between the cyl. coordinates {r,0,z} and
their “associated” Cartesian system of {x,y,z}. Note that for any
given point P, the usual position vector R1» in Cartesian coordinates
is related to the associated cyl coordinates by the vector equation: Rp(x,y,z)=rcosei + rsian + zk.
This vector equation can also be expressed as three scalar
equations, see (2.1). Coordinate Surfaces and Coordinates Curves: Given a point P,
we can hold the values ﬁxed for any chosen pair of cyl coordinates
and let the third cyl coordinate vary. This gives us the coordinate
curve through P for that third coordinate. Thus the rcurve through
P is a ray which starts on the zaxis and runs through P parallel to
the plane z = 0; if we use the deﬁning equation with r as parameter
and with the ﬁxed values for 6 and 2 given by P, all the concepts
and deﬁnitions of Chapter 6 can all be applied to the rcurve
through P. Similarly, the 0curve through P is a circle of radius r
with center on the zaxis and in a plane perpendicular to that axis;
it uses 9 as parameter. Similarly, the z—curve through P is a line
through P parallel to the z—axis, and can use 2 as parameter. Unit coordinate vectors. For any point P not on the zaxis, the
unit tangent vector at P to the rcurve through P is called the unit
vector rp, or simply, the unit vector 1' at P. The unit vectors 0p and
21) are deﬁned similarly. For any point P not on the zaxis, these
three unit vectors at P are mutually orthogonal and form the
coordinate frame at P. For any point P, if we use the frame
identity, the components of any given vector with respect to the
coordinate frame at P can be found. It is important to note that the
directions of the unit coordinate vectors r and 0 may change as we
shift ﬁ'om one given point P to another. VII3 Paths. In certain problems, the path R(t) of a given “moving
point” P may have a simpler expression in cyl coordinates than in
Cartesian coordinates. In such cases we view the position of P as
given by the dependence of its cyl coordinates upon our parameter
t. Thus we will begin with a parametric system of the form r 2 r(t),
9 = 9(t), and z = (t). Then, using the cyl coordinate frame, we can
express R(t) as R(t) = rr + 22, where r and z are unit coordinate
vectors and r and z are the given scalar functions r(t) and z(t), see
ﬁgure [51]. We must immediately note however, that the unit
vector 1' will change in direction as our point P(t) moves along the
path. This means that 1' must itself be treated as a vectorvalued
function of t. The dependence of r on t can be expressed in the
associated Cartesian system by: r(t) = r cos 6 i + r sin 9 j , where r and 9 are the given scalar
functions r(t) and 6(t). Similarly we have: 0(t)=—rsin6i + rcost. Velocity and acceleration. We can now proceed to apply the
methods of calculus to ﬁnd expressions for velocity and
acceleration in terms of the cyl ﬂame and the functions r(t). 0(t),
and z(t). This development is derived in the text, beginning at the
bottom of page 149. We can then use these results, for example, to
ﬁnd the curvature at any point on a curve which has been presented
in cyl coordinates. ...
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 Two '04
 Duorg
 Math

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