Most problems have moved to:
| http://electron9.phys.utk.edu/phys513/Modules/module_4.htm |
Additional Problems:
Problem:
An infinitely long rod is being rotated in a vertical plane at a constant angular velocity w about a fixed horizontal axis (the z-axis) passing through the origin. The angular velocity is maintained at the value w for all times by an external agent. At t = 0 the rod passes through zero-inclination, i.e., q = 0 at t = 0 where q is the angle the rod makes with the x-axis. There is a mass m on the rod. The mass' coordinates and velocity components at t=0 are
,
where g is the acceleration due to gravity. The mass m is
free to slide along the rod. Neglect friction. Hint: Recall that in plane polar
coordinates the unit vectors 
and
are
not constant.
(a) Find an expression for r(t), the radial coordinate of the mass, which holds as long as the mass remains on the rod.
(b) Show that r(t) > r(0) for small t (t >0).
(c) There is a component of the mass' weight acting down the inclined rod, but no force component acting up the rod. With this in mind, explain why the mass begins moving farther out along the rod instead of down the rod.
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Problem:
A massless rod of length R is caused to rotate about one end in the x-y
horizontal plane at constant angular frequency w.
A massless
string of length s is tied to the other end of the rod, and a point mass m
is attached to the far end of the string.
I. At time t = 0, both the rod and the string lie on the x-axis, and m
is given a velocity w(R + s) along the y-axis.
A. Which of the following are conserved in the motion that
follows, and why?
(a) Linear momentum p
(b) Energy E
(c) Angular momentum L
B. Find the trajectory as a function of time.
II. Suppose that the mass is given an initial velocity that is in the y-direction, but slightly different from w(R + s) in magnitude. Show that the mass will execute simple harmonic motion about a line, which is an extension of the rod. Find the frequency of the oscillation. Use the small angle approximation freely.
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Problem:
Consider a system consisting of a mass m, a spring, and a rigid, massless lever arm
of length L with one end fixed at the origin. The spring has unstretched length l0
with force constant k and joins mass m with the lever arm.
The entire
assembly rests on a frictionless surface.
(a) Calculate the Lagrangian function in terms of the lengths L and l and the angles q and f and their derivatives.
(b) Now impose the additional constraints that f = p/2. (One can do this by making m move in a track attached to L.) Obtain the equation of motion for this constrained system
(c) Consider the case where
. Describe the motion of m.
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Problem:
A massless wire rotates about the z-axis with constant angular speed w and a constant angle of inclination a.
A
particle of mass m is free to slide without friction along the wire.
Let r
be the distance of m from the fixed midpoint of the wire and let a uniform
gravitational field act in the negative z-direction.
(a) Obtain the Lagrangian of the system.
(b) Obtain the equations of motion.
(c) Show that the solution is
.
(d) Let 
and r=r0
at t=0 and show that the particle moves up or down along the wire
depending on whether r0 is greater or smaller than 
.
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Problem:
Consider a particle constrained to move on the surface of a cone of
half angle a. The particle is subject to a gravitational field.
Take the apex of the cone to be the origin, with its symmetry axis along the z-axis.
By choosing the cylindrical coordinates r and q as the
generalized coordinates, use Lagrange’s equations to
(a) show that the angular momentum about the z-axis is constant in magnitude, and
(b) derive the equation of motion for the r coordinate.
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Problem:
Two particles of mass m and M are connected by a light, inextensible string of length L, which passes through a smooth hole in a smooth horizontal table. The mass M is suspended below the table, and the mass m rests on the table with an initial distance r0 from the hole. A gravitational field g acts on this system. The mass m starts out with speed v0 on the table initially at right angles to the string.
(a) Find, but do not solve, the equations of motion for m and M, if M moves only in a vertical line.
(b) Assume initial conditions were such that equilibrium nearly held and solve
for the equilibrium distance re, when
Solve the equations of motion for small departures
from equilibrium. Describe this motion.
(c) What effect would inclusion of the Coriolis force on m have on the motion?
Include a discussion of the possible changes in the path of the motion and a comparison of the magnitude of the Coriolis force and the gravitational force, assuming that v0 is 1000m/s.
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Problem:
The bearing of a rigid pendulum of mass m is forced to rotate
uniformly with angular velocity w. The angle between the
rotation axis and the pendulum is called q.
Neglect the inertia
of the bearing and of the rod connecting it to the mass. Neglect friction. Include the
effects of the uniform force of gravity.
(a) Find the differential equation for q.
(b) At what rotation rate wc, does the stationary
point at q=0 become unstable?
(c) For w>wc what is
the stable equilibrium value of q?
(d) What is the frequency W of small oscillations about this
point?
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