Due: 10:00pm on Friday, October 29, 2010
You will receive no credit for late submissions.
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You should make a serious attempt at these problems before your October 29 workshop. You only have a couple of hours after the work shop to make final adjustments before the due time !!
You will do poorly in the work shop if you are unprepared. !! If you need help consult the Help Blog and/or visit the help room.
Energy of Harmonic Oscillators
To learn to apply the law of conservation of energy to the analysis of harmonic oscillators.
Systems in simple harmonic motion, or
, obey the law of conservation of energy just like all other systems do. Using energy considerations, one can analyze many
aspects of motion of the oscillator. Such an analysis can be simplified if one assumes that mechanical energy is not dissipated. In other words,
is the total mechanical energy of the system,
is the kinetic energy, and
is the potential energy.
As you know, a common example of a harmonic oscillator is a mass attached to a spring. In this problem, we will consider a
moving block attached to a spring. Note that,
since the gravitational potential energy is not changing in this case, it can be excluded from the calculations.
For such a system, the potential energy is stored in the spring and is given by
is the force constant of the spring and
is the distance from the equilibrium position.
The kinetic energy of the system is, as always,
is the mass of the block and
is the speed of the block.
We will also assume that there are no resistive forces; that is,
Consider a harmonic oscillator at four different moments, labeled A, B, C, and D, as shown in the figure . Assume that the force
constant , the mass of the block,
, and the amplitude of vibrations,
, are given. Answer the following questions.
Which moment corresponds to the maximum potential energy of the system?