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Nina del Ser - Mathematics
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The original position of the seat does not influence the shape of its path
In the introduction it was mentioned that the initial position of the seat was considered to be unimportant since
a different set of coordinate axes could be constructed for each individual seat. Now that we know the paramet-
ric equations for the seat, we can test this hypothesis by investigating the shapes of the curves produced by
seats which are not necessarily at (1.5,0) when their path starts to be tracked.
y
S fixed
Α
P t
t
Α t
S t
t
r
Α
O
P 0
S 0
x
figure 5
Figure 5 shows the setup of a seat which is located with an offset angle Α to the horizontal. It may be seen
from the diagram that the vector of the secondary rotor to the horizontal is 0.5icos(Α-t)+0.5jsin(Α-t). Here is a
Π
plot of the path of a seat when Α 7 :
figure 6
Α
2
As we predicted, the curve maintains its elliptical shape and the only effect of the offset angle Α is rotation
about the origin which, had we set up our coordinate system taking into account Α, would not have occurred.
Α
By investigating further we can show that the angle by which the ellipse is rotated is 2 . The most obvious
way to do this is to find an expression for the magnitude of the seat’s position vector, which will then enable
us to find the time t when |r| reaches its maximum value. Since we know that for the case when Α 0 the
ellipse’s semi-major axis coincides with the x-axis (i.e t 0), the value of the new time t at which |r| is at a
maximum will also correspond to the rotation of the original ellipse about the origin. From figure 5:
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