Work out the time derivative of the unit vectors r^ and
theta^ in cylindrical coordinates
Show that the time derivative of A.B is dA/dt
. B + A. dB/dt
Due Monday Sept 13
Text problem 1-32
Problem B (Sun shadow angle)
Problem B1(parallel and perpendicular components of a
vector)
Show that A x (B x C) = B (A.C)
-C (A.B) [ the 'bac-cab' rule ] in rectangular coordinates
Due Monday Sept 20
A 2-kg mass is subject to a force F = 10 N/s t - 1N/s^2
t^2 . It is at rest at the origin at t=0.
a) Find the velocity and displacement as functions of
the time
b) What is the maximum positive displacement of the particle?
Write down lagrange's equations for a simple pendulum
of length L, and solve for the equation of motion of the mass M hanging
at the end of the pendulum.
text problem 2-12
text problem 2-42
Due Monday Sept 27
For a damped oscillator, work out the overdamped case.
Let omega = i gamma, so beta >sqrt(k/m). Determine the equation of motion
in terms of two arbitrary constants, and the initial conditions, just as
we did for underdamped motion.
Problem E (black hole and mass oscillating)
Potential problems (A, B,C) from Maple session.
Osage orange problem from Maple session
( Maple files can be sent to Saturn\Class\PH\MJM\Homework
Turn-In )
Due Monday Oct 4
Problem F
3-10 (approximate freq with light damping, n periods
to 1/e)
3-29 (find the capacitance and determine current at t=0.2
ms)
A 2500 kg mass is to be dropped from 0.4 m onto a (massless)
platform sitting on a spring whose force constant is 125000 N/m. You are
to install a dashpot (damping force proportional to velocity) so that the
platform will come to rest without overshoot in the shortest possible time.
Determine the damping constant for the dashpot. ( Parts a)-c) are 5 points,
part d) is 5 points )
a) Determine the dashpot damping for critical damping
b) Determine the time with critical damping for the mass
to be within 1% of its final resting value without overshoot
c) Double the dashpot damping and find the time for the
overdamped system to come within 1% of its final resting value without
overshoot (this should be larger than part a)
d) Determine the 'optimum' value for the damping (it's
not critical damping!) and determine the time for the mass to come within
1% of its final resting value without overshoot. (This will be smaller
than for part a).
Due Monday Oct 25
Show that for a circular orbit 2El^2/(mk^2) = -1, so
that e = 0 {e=sqrt(1 +2El^2/(mk^2) }
E = K+U = 1/2 mv^2 -k/r
l = mvr
use F=ma for circular orbit
A spherical galaxy is found to have a uniform velocity
V of stars about the center for all radii. (The tangential velocity of
any star in the galaxy is independent of radius). Determine the density
of matter as a function of radius from the center of the galaxy, assumed
to be spherically symmetric. The density should be expressed in terms of
V, G, distance r from the center, and other mathematical constants. [Keep
in mind that the net force on a body in a spherically symmetric distribution
of mass involves only the mass closer to the center than the body is.]
Text problem 5-16 about a sphere of mass M and radius
R a height h above an infinite sheet. (If you can't deal with the sphere,
let the sphere be a point mass, for slightly reduced credit).
Text problem 5-15
Construct the home oscillator with light string and thead.
Report the lengths used, the times measured, and the agreement or disagreement
with theory.
Due Monday November 1
8-5
8-31
10-9
10-12
Due Monday November 8
11-3 (moments of an ellipsoid, of semi-major axes a>b>c)
Problem P (started in class. Flat plate rotating about
a diagonal.)
Problem Q (symmetric rigid body with torque about symmetry
axis)
Problem S (frequency of 'wobble' of Earth North)
Problem AF (Obtain I(ij) for water molecule. Find principal
moments of inertia via similarity transform)