Nina del Ser - Mathematics
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5
Solving the original problem
In the following simplified diagram (also made using Mathematica) all of the seats except for seat S have been
ignored in order to make the explanation of its motion in time clearer:
y
S fixed
P t
t
2t
S t
t
r
S 0
O
P 0
x
figure 2
An x-y system of coordinates has also been established. The motion of seat can be worked out from the
following steps:
1.
We know that the speed of the primary rotor is 1 rad/s anticlockwise, therefore after time t, point P 0
will have moved t radians anticlockwise relative to the origin to point P t , as can be seen on the diagram.
2.
If the secondary rotor were fixed, point S 0 would move to point S fixed after time t. The angle between
the secondary rotor and a line going through P t that is parallel to the x-axis would be t, because it corresponds
with angle P t OP 0 . However, we know that it moves clockwise at 2 rad/s , which is twice the magnitude of
the angular velocity of the primary rotor. Therefore the seat will actually be at point S t , 2t radians clockwise
(or -2t anticlockwise) from point S fixed , relative to the point P t . The final angle between the seat at S t and the
parallel line mentioned earlier will therefore be -2t+t=-t.
3.
The position vectors of point P t relative to the origin, O, and point S t relative to point P t can now be
summed to give the position vector r of the seat as a function of time.
Thus,
the x-component of P t cost,
the y-component of P t sint.
Similarly,
the x-component of S t 0.5 cos(-t)
the y-component of S t 0.5 sin(-t).
4.
By adding the vectors P t and S t together, we obtain a position vector, r, for the passenger seat as a
function of time:
r x cos t 0.5 cos t 1.5 cost
r y sin t 0.5 sin t 0.5 sint
In the next section we will look at the graphical representation of these vectors.
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