Investigating Complex Pendulums

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Investigating Complex Pendulums
Student’s Name
Institutional Affiliation
Date

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Aim
To explore the motion of a pendulum moving in two directions.
Method
a)
i)
x=cos 2t
y=sint

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i)
We choose the following points,
( 1,0 ) , ( 0,1 ) , ( 0,0.71 ) , ( 0 ,−0.71 ) , ( −1 ,−1 ) , ( −1,1 ) , ( 0.5,0 .5 ) ,(0.5 ,−0.5)
These coordinates can be obtained by substituting the various values of t in the given parametric
equations,
For example consider point ( 1,0 )
For this point, x=1 , y=0. Therefore from x=cos 2t,
cos 2 t=1
2 t=cos−1 1=0
t=0
Also,
sint=0
t=sin−1 0=0
Therefore, by substituting the values of t in the given range of 0 ≤ t ≤ 2 π in the given parametric
equations for x and y, the coordinates on the graph can be obtained.
ii)
The given parametric equations are,
x=cos 2t
y=sint
Squaring both sides of y we have,
y2=sin2 t
We also know that cos 2 t can be written as,
cos 2 t=1−2 sin2 t
But cos 2 t=x
x=1−2sin2 t
And sin2 t= y2
x=1−2 y2
2 y2=1−x
y2=0.5−0.5 x
y= √0.5−0.5 x

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b)
i)
x=cost
y=sin2 t
ii)
x=cost
y=sin2 t

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We know that sin 2 t can be written as,
sin 2 t=2 sintcost
Therefore,
y=2 sintcost
But cost=x therefore sint= √1−x2
y=2 √ 1−x2 cost
And cost=x
y=2 x √1− x2
Squaring both sides we have,
y2=4 x2 (1−x2)
This Cartesian equation gives the graph below which is similar to the one plotted using parametric
equations.

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c)
x= Acos( at+c )
y=Bsin (bt +d)
0 ≤ t ≤ 2 π
With A=B=1 , c=d=0 ,a=b=1

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With A=B=2 , c=d=0 , a=b=1

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With A=B=1 , c=1 , d=0 , a=b=1

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With A=B=1 , c=0 , d=1 , a=b=1

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With A=B=1 , c=5 , d=5 , a=b=1

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With A=5 , B=4 , c=1, d=0 , a=2 , b=1

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With A=5 , B=4 , c=1, d=0 , a=3 , b=1

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With A=5 , B=4 , c=0 , d=1 , a=1 , 2=1

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With A=5 , B=4 , c=0 , d=0 , a=1, b=7

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With A=5 , B=4 , c=0 , d=0 , a=7 , b=1

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With A=5 , B=4 , c=0 , d=1 , a=5 , 7=1

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The parameters A and B define the amplitudes of the signals. Parameters a and b define the
frequencies of the signals while c and d define the phase shifts of the signals. It can be observed that
increasing A or B generally increases the amplitudes and vice versa. A limits the amplitude on the x-
axis while B limits the amplitude in the y-axis. Increasing the frequency of the functions increases the
number of meshing figures. If the frequency of the sine function is held constant while increasing the
frequency of the cosine function, the number of meshing figures increases along the y-axis.
Increasing the frequency of the sine function with the frequency of the cosine function constant
increases the number of meshing figures along the x-axis. If the values of c and d representing the
phase shift are changed, the symmetry of the figures disappear as one of the signals lags the other.
The equations of the Lissajous figures are just parametric equations which are both functions of time
t with the introduction of phase shift. These equations can be plotted parametrically by considering
different values of t. For instance, the two parametric equations given below still yield a figure
similar to that generated using the given equations for Lissajous figures,