In-class exercise 7.1
Modify the predictor-corrector program
(C++
or Python versions) studied
previously to study the effect of friction on pendulum motion. Assume
that a particle moves under the influence of a restoring acceleration
-k sin(x), with k = 1, and
that friction also acts, producing a force -ALPHA
v that always opposes the motion. Note that the
variable x is angle, replacing θ in our earlier
discussion. The particle starts with position x =
0 and velocity v = 1 at time
t = 0.
- Using time steps of dt = 0.01, answer the
following questions for ALPHA = 0, 0.1, 0.2,
0.5, and 1.0.
- By what factor does the amplitude decrease per oscillation
-- that is, what is the ratio of x
from one maximum to the next?
- determine the maximum by looking for a change in
velocity from positive to negative, and take the maximum
of the current and previous positions as a good estimate
of the maximum value
- do this twice, storing the maximum values you obtain
- By increasing ALPHA in small
increments until the particle never quite returns to
x = 0, estimate the value of the critical
damping factor ALPHA_CRIT, beyond which
oscillations cease.