Problem set 5 Due: Tuesday, Nov 17, 1998
1.
Calculate the area of the "rounded square"
x4 + y4 = R 4 to
obscene accuracy.
2.
Calculate the integral from 0 to pi/2 of x(-2/3) cos(x) .
3.
This velocity data is the velocity (in m/sec) of a
satellite orbiting around the earth as a function of time (1st column, in
secs). What was the distance traveled by the satellite in that day?
4.
The differential equation governing the oscillations of a pendulum is
A'' + sin A =0
where A is the angle between the pendulum and the vertical.
In introductory mechanics courses, this equation is often approximated
by the equation for the harmonic oscillator : A'' + A =0,
but this is only valid for small A.
Use Matlab's ode23
to calculate the solution of the pendulum equation when the pendulum is
released with zero angular velocity A'=0 from the angles A0=pi/4 and
then A0=pi/2. Compare to the solutions of the harmonic oscillator problem.
What is the period of oscillation of the pendulum for both initial conditions?
Compare your solution to the function A(t) defined by
sin (A/2) = sin(A0/2) sn(t+d, m)
where sn(t,m) is the Jacobi elliptic function of the first kind,
m= sin 2 (A0/2), t is time
and
d =ellipke(m) =
the integral from 0 to pi/2 of 1/ sqrt(1- m sin2x) dx.
(Hint: "help specfun").
Fabian Waleffe
Mon Nov 09 16:23:47 CDT 1998