Physics 321 -- Spring 2003
Homework #11, due Wednesday
April 23
1. (7) Do problem 7-6 in Marion & Thornton without using
the Lagrangian method. Here is how to
do it:
a) (2) Write down the total kinetic energy T of
the hoop and incline plane, using the x and y coordinates of the hoop and the
angle f. Don’t forget that the hoop has both rotational and translational
kinetic energy. Use I=mR2
for the hoop.
b) (1) Now switch to the coordinates s and x, and eliminate x,y, and f from your expression for K. (What is the relationship between s and f?)
c) (1) Write down the potential energy U of the
system. (It doesn’t matter where you
take your origin – adding a constant to U does not change the physics.)
d) (1) Write down an equation describing conservation of energy. Assume the hoop and incline plane start from
rest when s=0.
e) (1) The horizontal component of linear momentum of the
(hoop + plane) system is also conserved, because there are no horizontal forces
acting on the system. (The plane slides
without friction along the table.)
Write down an equation expressing this conservation law.
f) (1) Now calculate the acceleration of the incline
plane, , and the acceleration of the hoop with respect to the plane,
. You will need to
combine everything and take some time derivatives to get these.
2. (5) Now do Marion & Thornton: problem 7-6 using the
Lagrangian method. Some of the work,
such as calculating T and U, you have already done in the previous
problem. So you just need to write down
the Lagrangian and Lagrange’s equations in terms of the coordinates s and x.
Then you can solve for and to uncouple the
equations.
(continued
on other side)
3. (6) Marion & Thornton, problem 7-3. You may use either the Lagrangian method or
the Conservation of Energy method. In
either case, the constraint of no slipping gives you a relationship between the
two angles q and f. Use
that to eliminate f, so you end up with a
differential equation for the variable q. (Note that the distance of the center of the sphere to the center
of the hollow cylinder is (R-r), rather than R.)
4. (7) Marion & Thornton, problem 7-12. Do this problem using the Lagrangian
method. (To use Newton’s 2nd
Law, you would need to consider a non-inertial reference frame, which we have
not discussed in this course.) Here are
the steps you should follow:
a) (2) Write down the Lagrangian, using the variables r
and q. Now
put in the explicit form for q and , so you are left with only the coordinate r.
b) (1) Lagrange’s equation should give you an inhomogeneous differential equation for r. (You may recognize the loathsome “centrifugal force” in this equation.)
c) (2) Now use all the tricks you learned about
differential equations in Chapter 3.
The general solution consists of the solution to the homogeneous
equation, plus a particular solution to the inhomogeneous equation. For the particular solution, try rp
= C sin(at), and plug in to the D.E.
to find the constant C.
d) (2) Now use the initial conditions to determine
the two unknown constants in the homogeneous solution. Simplify the answer using cosh and sinh, and
you are all done.