For SHM to occur we require stable equilibrium, about a point. For example, at
the origin we could have:
which
would describe a system in equilibrium. This however is not necessarily stableequilibrium.
The lower part of the figure shows the case of unstable equilibrium. The upper
part shows the case of stable equilibrium. These situations often occur inmechanical systems.
For example, consider a mass attached to a spring:
In general, in a case of stable equilibrium we can write the force as a
polynomial expansion:
where
the
are positive constants. There is always a region of
small enough that we can write:
This is satisfied by an equation of the form
where
and
are constants that are determined by the initial conditions.
Draw a diagram of a sinusoid and mark on it the period T and Amplitude A
Is an arbitrary phase which shifts the sinusoid.This is also
satisfied by an equation of the form
Lets
show this:
Again
there are two constants determined by the initial conditions
and
The equation can be rewritten
Thus
if
then
the equation is identical to the SHM equation.
So another way to write the equation of Simple Harmonic Motion is
or
It is also important to remember the relationships between freqency, angular
frequency and period:
Another solution to the SHM equation is
Recall Taylor's expansions of sine and cosine
Then
(an alternative way to show this is the following)
Thus we can write
as
We will use the complex representation a lot, so you need to
become familiar with it. It is used a lot in Optics, Classical and QuantumMechanics and Electrical Engineering so it is a good thing to know.
Now for physical systems we are interested in just the realpart so
This
will be implicitly understood. In physics we just write
One thing that will seem to be confusing is that there are all these different
solutions. They are all just different forms of the same thing. Which form isused in a particular circumstance is simply a matter of convenience. Some
forms lend themselves to to solutions of certain problems more easily thanothers. Also the most convenient form can depend upon the initial conditions.
For example if
is at its maximum displacement at time
then a
form may be the most convenient. As a general rule I like using the complex
representation because natural logarithms are so easy to work with. Forexample
which
is all pretty simple to remember
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
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