Parametric Equations Examples
Example 2-3: Use of Parametric Equations |
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A particle moves along a path described by the vector equation
with
R given in meters and
time in seconds. |
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(a) Eliminate time t from the parametric equations
and
to
obtain the path equation:
(b) Plot the path. Note that though the path shows both an upper and lower leg, the particle only moves along the upper leg. |
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(c)
Velocity and acceleration vector equations are found from time derivatives
of the vector position equation.
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(d)
At time t =
sec, the position, velocity and acceleration vectors are, in
both cartesian and polar form:
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(e)
The angle, b,
between the v and
a vectors, as shown above, is 9.30°. (f)
The normal and tangential components of acceleration (shown above)
are easily found once b
is known. Note:
The angle b
between v and
a
can also be found from the dot product:
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Example 2-4: x-y Plotter Parametric Equations Problem | ||
An
x-y plotter consists of an x slider and a y slider,
with a pen located at the intersection of the two sliders.
Both sliders are capable of moving independently in time in order
to plot virtually any kind of curve.
If for a particular plot the pen position is described by x =
2t and y = 5 sin 4t , where x and y are
in feet and t is in seconds, do the following:
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(a)
Eliminate time t from the parametric equations
and
to
obtain the path equation:
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(c)
Velocity and acceleration components are found from time derivatives of
the parametric equations: |
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At
x = 2 ft, time is t = 1 sec (since
x = 2t ). The
position, velocity and acceleration vectors are, in both cartesian and
polar form:
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(d)
The angle, b,
between the
v and
a vectors, as shown above, is 8.70°. (e)
The normal and tangential components of acceleration are easily
found once b
is known. These are shown in
the picture above. |
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