PHYS 2210
UNIVERSITY OF COLORADO AT BOULDER
CLASSICAL MECHANICS AND MATH METHODS, SPRING, 2011
Homework 11
(Due Date: Start of class on Thurs. March 31 )
1.
Consider a simple pendulum of length
L
= 10
m
.
(a) In an ideal world (assuming no damping, and making the small angle approximation) determine the
period of oscillation.
Now imagine we take into account air friction and determine that it causes the period to change by
0
.
1%. Using Taylor’s notation introduced in Eq. 5.28, what is the damping factor
β
? By what factor
will the amplitude of oscillation decrease after 10 cycles?
(b) Which eFect of damping would be more noticeable  the change of the period or the decrease of the
amplitude? Explain.
2.
Imagine two concentric cylinders, centered on the vertical z axis, with radii
R
±
°
,where
°
is very small. A
small
frictionless
object of radius 2
°
is inserted between the two cylinders, so that it can be considered a
point mass that can move freely at a ±xed distance from the vertical axis. At time
t
= 0 the puck is released
at height
h
with a purely angular initial velocity
ω
0
.
²igure 1:
(a) Write down Newton’s second law ±rst in terms of cartesian coordinates and then in terms of cylindrical
polar coordinates. Which form would be easiest to use to solve for the motion of the ball described
above? Why? (
If you need a reminder about what cylindrical polar coordinates are, problem 1.47 on p.
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 Spring '11
 STEVEPOLLOCK
 mechanics, Energy, Simple Harmonic Motion, Work, Polar coordinate system

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