Homework Assignment 2 (due 2/2)
Simple harmonic motion, coupled oscillators
- A 0.20 kg thin rod of length 1.0 m swings freely from one end.
At time , it is at an angle of
rad with respect
to vertical, and its angular speed is 0.10 rad/s.
(a) The general solution to the equation of motion of the system can
be written in the forms
Find the constants , , , , and describing the
situation described above.
(b) Sketch a circle in the complex plane representing the trajectory
traced by the function
. Label and
in the diagram. Also place a dot on the circle corresponding to
the time .
- Two particles of mass are connected to each other and to two fixed
points by three identical, massless springs of strength as shown
in the diagram. Find the frequencies and amplitude ratios of the
normal modes of the system. Consider only longitudinal motion, as we
already considered the transverse modes in lecture.
- The general solutions to the double pendulum equations of motion can
be written as a linear combination of the two normal modes,
A double pendulum with strings of length 0.50 m is released from
rest at with
(a) What values of , , , and describe this
(b) Use Maple or a spreadsheet to produce a plot of ,
vs. time over the first few
periods of the slow mode. What is the physical meaning of the average?
(c) What are the angular positions of the masses after four periods of
the slower normal mode?
Lab 1: The Double Pendulum
Copyright © 2003-2009, Lewis A. Riley
Updated Tue Apr 14 02:04:45 2009