Computer Algebra Recipes

4.3. VARIATIONAL CALCULUS MODELS 201
to Vectoria. So Mr. X comes to you and asks what speed he would reach and
how long would his slide take if air resistance and friction are included, the same
wire shape being used. He tells you that his mass is about 60 kg. You may
assume that the force due to air resistance is 0:5 v 2 newtons, where v is Mr. X's
speed, and that the coe±cient of kinetic friction is about 0:6 (appropriate for
steel on steel). Mr. X would also like to know the maximum speed and trip time
if a braking action is applied. You may assume that the brakes exert a force of
400 newtons on the wire. Hint: You may ¯nd that using a do loop to compute
the tangential velocity and position of Mr. X at discrete time intervals is the
easiest way to attack this problem.
4.3.4 This Would Be a Great Amusement Park Ride
To gyre is to go around and round like a gyroscope.
To gimble is to make holes like a gimlet.
Lewis Carroll, English writer and mathematician (1832{1898)
Consider the following possible amusement park ride consisting of a small cage
of mass m (including the screaming victims inside) being swung around at the
end of a light, but strong, connecting arm of ¯xed radius r as in Figure 4.18.
The cage can trace out various spherical surface trajectories depending on the
y
z
θ
Figure 4.18: Con¯guration of the amusement park ride.
conditions imposed by the slightly sadistic ride operator. What kind of trajectories
can the victims be subjected to?
Our task is to investigate this question using Lagrange's equations of motion,
an alternative approach to the Newtonian formulation. The Lagrangian L is
de¯ned as L = T ¡ V ,whereT is the kinetic energy and V is the potential
energy of the system being studied. Formulating the Lagrangian is often much
easier than determining all the forces and their components that are required to
r
x
φ
m