[SOLVED] Physics 342 Homework 1

1: 1-9
2: 1-10
3: 1-13 (this is either really easy or really hard, maybe try crossing vector A with vector B)
4: 1-31 dont forget that r = p
x2 + y2 + z2 and @ f
@x = @ f
@r
@r
@x Also, you really only need to take
the first partial derivative with respect to x or the 2nd derivative with respect to x to figure this out
since all taking the derivatives with respect to y and z changes is to make x go to y or z. This can
be long or relatively short if you dont overdo the work.5: An equation often occurring in mechanics describes the behavior of a simple mass connected
to a spring where F~ = mx = kx. The motion of a pendulum displaced from equilibrium is
harmonic. Assume k = 4 and m =1 for simplicity. If you get complex exponentials recall Eulers
formula. You will also need to apply the initial conditions to position and velocity to solve for your
two unknown constants. Two useful identities are eit+eit
2 = cos(t), where is a number, and eiteit
2i = sin(t) . Hint, divide through by m.A) Using the initial conditions x(0) = 1 and x(0) = 0 put this equation into the general SOLDE
form shown in class and solve it and then graph it using any method you like. Approximate the
square root portion to two digits and remember, you can not combine the real and imaginary components. Comment on the behavior.B)If one allows for friction that is velocity dependent this equation may be rewritten as mx =
kxbx Leave everything as variables in this case and do not apply initial conditions to determine a
numerical solution. Your solution should look like C1e[ b+
p
b24mk
2m ]t +C2e[ b
p
b24mk
2m ]t. Assume C1 = C2
and recall et+t = et
et Simplify your solution. Three possible cases exist. What happens if
b2 = 4mk? What happens if b2 > 4mk? What happens if b2 < 4mk?
6:Integrate R 1
0
1ex
px dx using the 4 terms of the Taylor expansion of the integrand. Compare this
result to one done numerically. You only have to find the taylor expansion of one thing here then
do some algebra and simple integrals. Exams about x = 0. This should give you an idea of how to
write a simple computer program to evaluate tricky integrals. You get a number larger than the real
one. What does this tell you about your choice of x=0 for the expansion?7: Derive a series representation for the ln(1 + x) about x = 0. Can you do the same for ln(x)?Extra Credit
1-25 The unit vectors in the spherical coordinate system are given by
e = sin i + cos j
e = cos cos i + cos sin j sin
k
er = sin cos i + sin sin j + cos
kFollowing the procedure used in class for plane cylindrical coordinates do this problem assuming your initial vector is rer. Hint e depends on er and e in a way that is not as straightforward to
see as the time derivatives of the other unit vectors.See what you can do with factors of with
trig functions to make this work.