Mathematica Mechanics Lab write-up – Experiment 2
Student: David Long (03457885)
Theoretical Background:
The aim of this second experiment was to study the effect of the rotation of the
earth on the motion of a pendulum. The pendulum considered was assumed to be of mass
m, and hanging on the end of a mass less string of length l. The point of suspension was
assumed to be at a great height, such that the angle of displacement was quite small. In
this case, the equation of motion is given by:

 
  T
a   g   2  v 
m

In this case, a is the acceleration of the particle, which can be split into x and y

components. g is the acceleration due to gravity, and acts downwards (in the –z

direction). T is the vector tension in the string due to the particle. This may be split into
x, y, and z components, with the x-component equal to –Tx/l, and the y-component equal
to –Ty/l. However, due to the fact that the string was assumed to be of very long length,
T ≈ mg, with the z-component of the tension approximately equal to T. Finally then, the

angular momentum  may be split into x and z-components, with the y-component equal
to 0. ωx = -ωcosλ (where λ is the displacement angle of the pendulum) and ωz = -ωsinλ.

The velocity v  of the pendulum may also be split into: vx  x , vy  y , vz  z  0 . By
working through the equation of motion for the pendulum, the following coupled
equations are achieved:
x   2 x  2 y sin 
y   2 x  2 x sin 
It was found that the pendulum is affected by the rotation of the earth, but that these
effects are too small to be seen by the naked eye. It was also found that the pendulum will
be affected in different ways in the Northern and Southern Hemispheres. i.e.: the
pendulum will move in opposite directions in the separate hemispheres, due to the
rotation of the earth (anticlockwise in the Northern Hemisphere, and clockwise in the
Southern Hemisphere.)
Part I:
r[t_]={x[t],y[t],z[t]};
[t_]={-cos,0,sin};
=10/50
=0.1;
=3Pi/4
(1)
sol=NDSolve[{x''[t]+*x[t]==2y'[t]**Sin[],y''[t]+*y[t]==2x'[t]**Sin[],x[0]==1,x'[0]==0,y[0]==0,y'[0]==0},{x,y},{t,0,300},MaxSteps5000]
(2)
pt[t_]={x[t],y[t]}/.sol
(3)
Show[{ParametricPlot[pt[t],{t,0,300},PlotPoints500],ParametricPlot[{Cos[],Sin[]},{
,0,2}]},AspectRatioAutomatic]
(4)
Above is the Mathematica code used in this experiment. In part (1), the various
parameters are defined. In this case, is equal to g/l, or the acceleration due to gravity
(constant) divided by the length of the string (variable). is the angular velocity of the
pendulum, and is variable. Finally, is the angle of displacement of the pendulum, and is
also variable. In part (2), these values are merely substituted into the two equations
shown above. The time limit for these experiments was set to be 300s, but it is a variable.
Part (3) redefines the equation, so that it may be graphed easily. Part (4) then, is the
actual code for graphing the equation. Again the time limit is set at 300s, with the
maximum number of plots set to 500 points.
Part II:
1
1
0.5
0.5
-1
-1
-0.5
0.5
-0.5
0.5
1
1
-0.5
-0.5
-1
-1
The graph on the left is the graph for ω = 0.1, λ = 30o, and α = 10/100. The graph on the
right however is the graph for ω = 0.1, λ = 30o, and α = 10/10. As can be seen from
comparison between the two graphs, in the case of the second case, where g and l are the
same, the pendulum tends to move with very small increments, and come closer to the
centre point, leading to a heavy dark line when the graph is drawn. Where the length of
the string is 10 times that of g however, the pendulum does not come as close to the
centre, and it moves in wider, curved steps. Therefore, by shortening the string, the
pendulum tends to almost retrace its steps with every swing. By lengthening the string,
the pendulum tends to move within a wider area, meaning that it will take longer to cover
a circle.
-1
1
1
0.5
0.5
-0.5
0.5
1
-1
-0.5
0.5
-0.5
-0.5
-1
-1
1
The graph on the left has ω = 0.5, λ = 30o, and α = 10/100, while the graph on the right
has ω = 0.1, λ = 30o, and α = 10/100. As can be seen, changing ω changes the size of the
inner circle of the graph, with a larger ω giving a larger circle, and vice versa. This is
because ω is the angular speed of the pendulum, and a larger ω means that the pendulum
is moving faster, and is hence unable to come as close to the centre point.
-1
1
1
0.5
0.5
-0.5
0.5
1
-1
-0.5
0.5
-0.5
-0.5
-1
-1
Here, the graph on the left has ω = 0.1, λ = 90o, and α = 10/100, while the graph on the
right has ω = 0.1, λ = 30o, and α = 10/100. As may be seen, a larger angle λ means that
the pendulum does not approach the centre point as much as for a lower λ.
1
1
In this case, ω = 0, λ = 60o, and α = 10/100. As can
be seen, nothing happens. This is because the particle
has no angular speed, and therefore has no speed to
swing.
0.5
-1
-0.5
0.5
1
-0.5
-1
Part III:
In this case, the pendulum was assumed to have a length of 50m, in a gravitational
field of g=10m/s2. The angles for which it was measured were 0o, 30o, and 60o, in the
Northern Hemisphere, for time t from 0s to 300s. This was then repeated for one angle in
the Southern Hemisphere.
1
For this graph, ω = 0.1, λ = 0o, and α = 10/50.
As can be seen, at 0o the pendulum is
unaffected by the rotation of the earth, and
hence does not tend to form a circle, instead
oscillating back and forth along the x-axis.
0.5
-1
-0.5
0.5
1
-0.5
1
-1
For the graph on the right, ω = 0.1, λ = 30o, and
α = 10/50. As can be seen, this results in a small
number of very sharp oscillations, which builds
up the circle.
0.5
-1
-0.5
0.5
-0.5
-1
1
In this case, ω = 0.1, λ = 60o, and α = 10/50.
This results in a large number of very sharp
oscillations. This shows that the number of
oscillations grows as the angle of Latitude
grows.
1
0.5
-1
-0.5
0.5
1
-0.5
-1
Finally, in this case, ω = 0.1, λ = 120o, and α =
10/50. This angle is in the Southern
Hemisphere. As can be seen, it is identical to
the similar angle in the Northern Hemisphere,
with the exception that this time the pendulum
is oscillating in the opposite direction.
1
0.5
-1
-0.5
0.5
-0.5
-1
1