Simple
Harmonic Motion
Graphical
and analytical treatments
Understanding and use
of the following equations:
The characteristic
features of simple harmonic motion are summed up in the first equation
(NB you must always mention both characteristics!)
- Acceleration
(a) is proportional to the distance (x) from a fixed point
in its path
- Acceleration
is directed TOWARDS that point (hence the negative sign
in the equation below!)
a=
-w2x = - (2pf)2x
The constant of proportionality
is w2
A strange constant
you may think! .... But, in order to understand where it comes from we
have to do a bit of mathematical analysis.
w
is the angular velocity of the circular motion that corresponds to the
SHM. It is equal to 2pf where f is the frequency
of the sinusoidal waveform that is associated with SHM. (See circular
motion).
Now, at what point
should we begin timing? Does it matter? Well, no, not really, the motion
is still SHM! But the mathematical desription of the motion will alter
depending on where we begin the cycle!
If we time from the
centre of the oscillation (where x=zero at time t=0s) we get a sine curve
to describe the displacement as time varies, if we time from a point where
x=A (maximum displacement) we get a cosine curve, but either way, after
a bit of mathematical analysis, we get rid of the original trig function
for displacement and get 'a' in terms of 'x' and a constant!
Let me show you how:
Mathematical
treatment 1 (t=0 when x=0)
In a graph of displacement
against time for a particle exhibiting SHM the amplitude of the graph
would be A and
the equation would be:
x
= Asinwt
The differential (gradient
with respect to time) of this equation would give the velocity 'v'.
v
= Awcoswt
The differential (gradient
with respect to time) of this equation would give the acceleration 'a'.
a
= -Aw2sinwt
BUT
Asinwt
= x
so
a
= -Aw2sinwt
= -w2x
Mathematical
treatment 2(t=0 when x=A) -
this is the one AQA give you on the data sheet!
In a graph of displacement
against time for a particle exhibiting SHM the amplitude of the graph
would be A and
the equation would be:
x
= Acoswt
The differential (gradient
with respect to time) of this equation would give the velocity 'v'.
v
= -Awsinwt
The differential (gradient
with respect to time) of this equation would give the acceleration 'a'.
a
= -Aw2coswt
BUT
Acoswt
= x
so
a
= -Aw2coswt
= -w2x
BOTH
treatments have the same result! Care when tackling questions that you
choose the correct trig function (the one in the syllabus is the cosine
one... but in Lowe and Rounce you many need to use the sine one occasionally)
to describe the situation given in the question - look to what is happening
when t=0s!!
Lots of natural systems
exhibit SHM - bouncing, swinging, vibrating things are all usually moving
in SHM: from atomic vibrations to suspension bridges!
Note that the
equation has acceleration 'a' proportional to the negative value of the
displacement' x' (i.e. proportional to the distance (magnitude of the
displacement!) but opposite in direction - acceleration increases as it
gets closer to the point we are measuring from. The constant of proportionality
is linked to the frequency 'f' of the oscillation. Therefore the general
equation for SHM could be written:
a
= - (2pf)2x
Now
v
= - Awsinwt
So
v2
= A2w2sin2wt
But
sin2wt
+ cos2wt
= 1
so
v2
= A2w2 (1-cos2wt)
v2
= A2w2 -
A2w2cos2wt
However we know that:
Acoswt
= x
Squaring this we
get:
A2cos2wt
= x2
Therefore, by substituting
it into our equation we get:
v2
= A2w2 - w2x2
And taking the square
root of this we get:
v
= +w(A2 -
x2)0.5
You
are GIVEN this on the data sheet - you do not have to worry about deriving
it - this just shows where it comes from (same thing can be done with
the sine expression... try it!)
This
equation gives the velocity at any displacement x.
If
it is at the extremities x=A therefore the velocity is zero.
At
the centre of the oscillation x=0 so v= wA=2pfA
(maximum velocity) the +/- indicates the direction of travel.
- You should be able
to sketch the graphs of displacement, velocity and acceleration against
time for a system operating SHM. If you start at the point x at time
t=0 then the displacement graph will be a sine curve (amplitude A),
the velocity a cos curve (amplitude wA) and
the acceleration a negative sine curve (amplitude w2A)
- where w is 2pf.
If you start at a distance A from the point x at time t=0 then the displacement
graph will be a cosine curve (amplitude A), the velocity a negative
sine curve (amplitude wA) and the acceleration
a negative cosine curve (amplitude w2A)
- where w is 2pf
Exchange
of potential and kinetic energy in oscillatory motion
- The energy of the
system is taken to be constant - made up of kinetic (max at x, where
velocity is a max and zero at A because velocity at A is zero) plus
potential which has the opposite characteristics of the kinetic.
The link between
circular motion and SHM
If you observe a body
rotating uniformly in a circle from a side perspective (in one plane only)
it exhibits SHM - see
Java applet.
Graphical representations linking displacement, velocity, acceleration,
time and energy
- You should know
how to skech and interpret displacement/time and velocity time graphs
for SHM.
- You should know
that velocity is the gradient of displacement/time graph and
- Acceleration the
gradient of a velocity time graph.
- You should know
that the amplitudes of the graphs are related by a factor of 2pf
= w
- Kinetic energy
will follow the velocity squared graph - NEVER goes negative! (and Potential
energy plus KE = constant!)
- see this Java
applet
Simple
pendulum and mass-spring as examples and use
of the equations:
- The pendulum
as an example of SHM - 'g' is gravitational field strength and 'l' is
the length. The period will vary according to which planet you are on.
- The mass
on a spring - 'm' is the mass (kg) and 'k' is the spring constant
(stiffness of it (which relates to interatomic forces) - revise Hooke's
Law!). This will not vary according to the planet it is on!
- A light gate can
be used to detect the times that a swinging pendulum or bobbing mass
on a spring cast a shadow over it. This is a more accurate and less
tedious way of finding the period (and hence the frequeny) of an oscillation.
(The syllabus specifies that you should be aware of this!).