[SOLVED] Physics 342 Homework 5

1: 7-6 Hint The hoop has kinetic energy 1
2 I2. The wedge also has kinetic energy. See the
figure below. Take the center of the hoop as the generalized coordinates. Also, look at what pieces
are changeable and what are not, example, R and l are fixed.You can also relate S and .
For this problem decouple the two equations and present the result for S and . Interpret your
answer
2: A long, straight, frictionless wire is attached to the z axis a distance h above the origin. This
wire rotates at a constant angular velocity ! about the z axis. A bead of mass m is threaded onto
the wire such that it can move up and down it. The wire makes a constant angle with the z axis.
Denote the length along the wire that the mass is as r. Essentially and ! t takes the place of
& in the spherical coordinate system. Make sure you reduce the form for kinetic energy to
something I will check. It gets down to two terms.A) Set up the Lagrangian for this system. B) Write out Lagranges equations. C) Determine the
motion of the particle in time essentially r(t) where r is the distance along the wire from the z axis.
Ask me if you fail to understand the geometry and Ill draw it for you.For part C, remember, this is a non-homogenous second order linear dierential equation. First
solve for the homogenous portion then solve for the particular solution using something like the
method of undetermined coecients. Pick up your dierential equations book if you need to.
3: (LONG) Look at example 7.7 (done in class). Relax the assumption that r=R and that = !t.
Instead, allow the bead to have some initial angular velocity so that (t) = b+!t and (t) = (
b+!t).Now z still = cr2 pick c to be 1 for simplicity.
A) Construct your Lagrangian. it should be L = m
2 [r2 + 4r2r2 + r2(
b + !)
2] mgr2. Solve for
Lagranges equations in r and .B) NOW, pick three sets of initial conditions, r(t = 0),r(t = 0), (t = 0), and (t = 0). (I should
be letting the bead fall with no rotation or initial angular velocity. Youll also need to pick a value
for !. Plot z(t),r(t), (t). Is your (t) reasonable? Why not (if not)?C) Does your solution agree with equation 7.50? Do you get a constant radius for the right value
of !? See what happens if you vary, by a tiny amount, the condition set for ! in equation 7.50. Is
your solution stable? What happens if you pick a small time step? Do you believe your solution?
Answer this question with more plots and a brief discussion.NOTE: I would prefer you actually program this but you can use Maple. If you use Maple you
will need to create a set of 4 dierential equations (or maybe 2 depending on how you do it).
See the Pendel worksheet on the course site and this link for the double pendulum.http://www.maplesoft.com/applications/view.aspx?SID=4873
or this link for the spring pendulum http://www.maplesoft.com/applications/view.aspx?SID=4897
or look through the Maplesoft applications site.Essentially you will need to define
Vr = dr
dt
V = d
dt
Ar = F(r, ,r, ,!)
A = G(r, ,r, ,!)
Then set initial conditions and use desolve and odeplot.
4: (LONG )7-15 Hint b is the unextended length, l is the extension or contraction it can be in
either direction.A) Construct dS~ = d(l + b)r + (l + b)d. Also, put the origin of your spring mass system at
(0,0) so y is negative. remember l = l(t) in the radial direction. Using the solutions manual for
this book will show me you dont understand how to do these problems. Also, dont forget where
explicit time dependence comes in.B) Plot the solutions for this one as well for a set of five dierent initial conditions. One
initial condition should be that the mass is hanging vertically with no initial velocity but some
initial stretch. Another initial condition should be no stretch, no initial velocity, with the pendulum
released from 90 degrees. Pick 3 more initial conditions and plot plot both (t) vs time AND plot vs . These are phase plane diagrams. Do the same for l and l. What do they tell you? Play around
a bit to get a sense of the behaviour of this system. Is this system chaotic?Tip, one check on the veracity of your solver should be to check whether energy is conserved or
not. Using a simple spring example with the initial angle being zero you should find that your total
energy is conserved to better than 100th of a percent over several cycles. Another check is to turn
o k and see if you get the behaviour of a simple pendulum.