2) page 18 # 24, 26 and 36 plus page 43 # 1-4 Some remarks: On problem 26, tMAX
is approximately20.4 seconds.
problem 36, you are given that xMAX =
2.25, from which you want to compute v0. First compute tMAX. You’ll find that tMAX =v0 / gE ( Earth’s
gravitational constant) Then using that, you find that xMAX
=v02 /2gE. and
from that you can figure out v0.
3) page 43 # 21, 22, 25
4) Using the value of k that we found in
class, predict the U.S. population in 1920 ( compare
with the actual value of about 106 million) Plus, on page 43, do numbers 33-38
5) page 84 #32 ( just
do the derivation as discussed in class…i.e., go from equation (*) to this form
of the solution) and # 29 ( on part c , maybe just see how accurately the Verhulst model predicts P(2000)or P(1990)
6) page 44-45 # 43,
7) page 54 # 2, 3, 4, 8, 15, 17, 22, and 24
8) page 55 # 33, 35
9) A handout on exactness. I’ll post a couple of extra copies on
pg 73 # 33, 35, 37, 39 Do
these by the systematic way introduced in class, please.
Using the numbers provided in class ( copies on my
door) compute escape velocity for the Sun, Earth, Moon and Antares. Then, for
the Sun, Moon and Antares,
compute what radius will produce a black hole…as we did in class
for the Earth.
pg 111 # 1, 2, 5, 6
A handout with 8 problems on solving second order equations..copies on my door ( The equation in problem 4
should end in 25y, not just 25. Sorry for the misprint.) Comments added
Saturday morning….if you have complex roots a+bi and
a-bi and the a is equal to 0, the corresponding real solutions are just cos(bx) and sin(bx), since eax=1.
Also, in the first problem, the roots are real; they are approximately -3.62
Solve the spring/mass equations with the following data: (1) m =4 k= 16
x(0)= 2 x’(0) = -3 and (2) m = 1 k= 9
x(0) = -1 x’(0) =0
Announcement about late homework….I’m afraid I have been encouraging
bad habits among a very few of us. From now on, unless you have spoken with me
in advance, as soon as I start grading an assignment any previous assignments
that have not been handed in will be regarded as late. They may incur a late
penalty or, if they’re REALLY late, they may not get graded at all.
page 147 # 13 a, # 15,16 In each case
simply find the solution using initial
data, and, in #13, find the time to reach max displacement: tmax
page 147 # 18 and 20
page 161 # 1, 2, 3
I also passed out a list of topics from which
exam questions will be selected. I’ll post copies on my door.
October 13 th
page 161 # 31, 32, 3320)
Calculate the response amplitude for the system x” + 16x = sin (
omega)t when omega = 4.1, 4.08, 4.05 , respectively. Suppose the system will
collapse if the response amplitude reaches 5 units. What omega will produce
this? Then, let’s do an experiment.
Suppose we look at x” + 9x = cos 3t. Show that the method of undetermined
coefficients fails if we try a trial solution of the form xT
= A cos 3t. ( We’ll see what to do about this
#1) for the equation x” + 9x = cos3t, substitute the trial solution xT = t (A cos3t + Bsin3t) and, after much
computation, verify that the particular solution is xp
=(1/6) ( t sin3t) #2) Then, using the
initial data x(0) =1 and x’(0) = 5, find the complementary solution xc
= sqrt (34)/3 cos ( 3t – phi).
point here is to illustrate the dominance of xp
when you have pure resonance. #3) This problem gets us started on resonance in damped systems.
Take the equation mx” + cx’ +kx = F0cos
(omega)t. We want to work out the general form of the
steady state solution, xp, so substitute
the trial solution, xt = A cos
( omega)t + B sin(omega)t and calculate what A and B are in terms of
K,m,omega, c and F0. We’ll go from there
Starting with the values of A and B that we found in the last problem, show
that the response amplitude, denoted C( omega), is equal to F0
divided by the square root of ( k – m(omega)2)2 +
(c(omega))2. We want to
rewrite this in terms of the RLC constants. Show that when we use current I as
the dependent variable, C(omega) becomes E0 divided by the square root of
( 1/(omega)C –(omega)L)2 +R2.
And, from there, we saw in class how to
tune a radio….
page 206 # 2, 3, 4 plus complete the solution of the equation: y” + 9y =0
23) No assignment