Download Lesson 27 notes – Oscillation Graphs - science

Lesson 24 notes – Oscillation Graphs
Objectives
Be able to select and apply the equation vmax = (2πf)A for the maximum speed of
a simple harmonic oscillator.
Be able to describe, with graphical illustrations, the changes in displacement,
velocity and acceleration during simple harmonic motion.
Outcomes
Be able to use the equation vmax = (2πf)A for the maximum speed of a simple
harmonic oscillator correctly for different situations.
Be able to rearrange and then use the equation vmax = (2πf)A for the maximum
speed of a simple harmonic oscillator correctly for different situations.
Be able to interpret graphical illustrations of the changes in displacement,
velocity and acceleration during simple harmonic motion;
Be able to derive the equation vmax = (2πf)A for the maximum speed of a simple
harmonic oscillator
Be able to draw graphical illustrations of the changes in displacement, velocity
and acceleration during simple harmonic motion;
Displacement – Time graphs
Recap from lesson 6 and 5 Mechanics G481.1 on displacement time graphs and
velocity time graphs.
Imagine a mass on a
spring being displaced
and then dropping
through the equilibrium
position to as far down
as it can go and back
up again:Error!
positive
SHM Graphs
If we let the mass hang it will stay in the equilibrium point. The strain of the spring
equals the weight of the mass.
If we pull the mass up, the strain is less and so it will accelerate towards the
equilibrium point.
If we pull it down the strain will be greater than the weight and so it will accelerate
towards the centre again.
At the equilibrium point the forces cancel leaving no acceleration and a maximum
velocity.
If we consider up positive then any displacement, velocity or acceleration
directed downwards will be negative.
The top graph shows displacement against time. The second graph shows the velocity of
the oscillator against time and the third graph shows the acceleration against time.
Maximum Velocity
You can see that when the mass goes through the equilibrium point, x=0, and
velocity is maximum. Maximum velocity of the oscillating system will happen
when vmax = (2πf)A. When x is maximum, acceleration is negative maximum –
the mass is being pushed back the opposite way (acceleration is directly
proportional to negative displacement). This happens when the velocity is zero –
since it is at its maximum amplitude. This isn’t easy so take some time going
through it and drawing your own diagrams of an object oscillating about.
The dotted green lines show how the graphs line up easily. You can see that the
velocity graph is led by the displacement graph and is π/2 out of phase with it
and the acceleration graph is led by the acceleration graph and is π/2 out of
phase with that one.
Explanation
If we go back to the diagram we used to define some terms and look at the
phasor diagram we can start to see clearly where the equations for SHM come
from:
Language to describe oscillations
Sinusoidal oscillation
+A
Phasor picture
s = A sin t
amplitude A
A
angle t
0
time t
–A
periodic time T
phase changes by 2
f turns per 2 radian
second
per turn
 = 2f radian per second
Periodic time T, frequency f, angular frequency :
f = 1/T unit of frequency Hz
 = 2f
Equation of sinusoidal oscillation:
s = A sin 2ft
s = A sin t
Phase difference /2
s = A sin 2ft
s = 0 when t = 0
sand falling from a swinging pendulum leaves
a trace of its motion on a moving track
s = A cos 2ft
s = A when t = 0
t=0
Motion of harmonic oscillator
velocity
force
displacement
against time against time against time
large displacement to right
right
zero velocity
mass m
large force to left
left
small displacement to right
right
small velocity
to left
mass m
small force to left
left
right
large velocity
to left
mass m
zero net force
left
small displacement to left
right
small velocity
to left
mass m
left
small force to right
large displacement to left
right
zero velocity
mass m
large force to right
left
Dynamics of harmonic oscillator
How the graph continues
How the graph starts
zero initial velocity would stay
zero if no force
velocity
force changes
velocity
force of springs accelerates mass towards
centre, but less and less as the mass nears the
centre
change of velocity
decreases as
force decreases
new velocity
= initial velocity
+ change of
velocity
trace curves
inwards here
because of
inwards
change of
velocity
t
0
0
trace straight
here because no
change of
velocity
no force at centre:
no change of velocity
time
time
Extension: More SHM Graphs
You should learn to be able to understand and draw the following graphs:
+2
r
a
a
x
+r
-r
t
a = -r sin
t

-2r
v
x
v = rcos
t
+r
>
1
=
1
t
v
-r
x
t
x = rsin t
r is the amplitude in these graphs.
And if you haven’t done it go back to the extension last lesson and try it. Use the
diagrams below to help:
Force, acceleration, velocity and displacement
Phase differences
Time traces
varies with time like:
displacement s
/2 = 90
If this is how the displacement varies
with time...
cos 2ft
... the velocity is the rate of change
of displacement...
–sin 2ft
... the acceleration is the rate of
change of velocity...
–cos 2ft
...and the acceleration tracks the force
exactly...
–cos 2ft
velocity v
/2 = 90
acceleration = F/m
same thing
zero
force F = –ks
 = 180
displacement s
... the force is exactly opposite to
the displacement...
cos 2ft
Maximum Velocity
Maximum velocity of the oscillating system will happen when vmax = (2πf)A. Let’s
look at the equations.
v= 2πfAcos(2πft) (a)
and
v=-A2πf sin(2πft) (b)
are both solutions.
Now, cos(2πft) or sin(2πft) have values of between 0 and 1,
So to get the maximum for each of these we need cos(2πft) to be equal to 1 in
(a) and sin(2πft) to be equal to 1 in (b).
cos0 = 1 and sin(π/2) = 1
(if you’re not sure of this try it on your calculator)
And if you look back at the graphs for displacement, velocity and acceleration
you will see that velocity is a maximum when this occurs.
If you can see the graphs in your head when you are tackling problems on SHM
you will find them a whole lot easier.