Transcript SHM

Oscillations
© 2014 Pearson Education, Inc.
Periodic Motion
Periodic motion is that motion in which a body
moves back and forth over a fixed path,
returning to each position and velocity after a
definite interval of time.
1
f 
T
Amplitude
A
Period, T, is the time
for one complete
oscillation. (seconds,s)
Frequency, f, is the
number of complete
oscillations per
second. Hertz (s-1)
Simple Harmonic Motion, SHM
Simple harmonic motion is periodic motion in
the absence of friction and produced by a
restoring force that is directly proportional to
the displacement and oppositely directed.
x
F
A restoring force, F, acts in
the direction opposite the
displacement of the
oscillating body.
F = -kx
Simple Harmonic Motion—Spring
Oscillations
We assume that the surface is frictionless. There
is a point where the spring is neither stretched nor
compressed; this is the equilibrium position. We
measure displacement from that point (x = 0 on the
previous figure).
Simple Harmonic Motion—Spring
Oscillations
• The minus sign on the force indicates that it is a
restoring force—it is directed to restore the mass to
its equilibrium position.
• k is the spring constant
• The force is not constant, so the acceleration is not
constant either
Springs are like Waves and Circles
The amplitude, A, of a wave is the
same as the displacement ,x, of a
spring. Both are in meters.
CREST
Equilibrium Line
Trough
Period, T, is the time for one revolution or
in the case of springs the time for ONE
COMPLETE oscillation (One crest and
trough). Oscillations could also be called
vibrations and cycles. In the wave above
we have 1.75 cycles or waves or
vibrations or oscillations.
Simple Harmonic Motion—Spring
Oscillations
If the spring is hung
vertically, the only
change is in the
equilibrium position,
which is at the point
where the spring force
equals the
gravitational
force.
Displacement in SHM
x
m
x = -A
x=0
x = +A
• Displacement is positive when the position is
to the right of the equilibrium position (x = 0)
and negative when located to the left.
• The maximum displacement is called the
amplitude A.
Velocity in SHM
v (-)
v (+)
m
x = -A
x=0
x = +A
• Velocity is positive when moving to the right
and negative when moving to the left.
• It is zero at the end points and a maximum
at the midpoint in either direction (+ or -).
Acceleration in SHM
+a
-x
+x
-a
m
x = -A
x=0
x = +A
• Acceleration is in the direction of the
restoring force. (a is positive when x is
negative, and negative when x is positive.)
F  ma  kx
• Acceleration is a maximum at the end points
and it is zero at the center of oscillation.
Acceleration vs. Displacement
a
v
x
m
x = -A
x=0
x = +A
Given the spring constant, the displacement, and
the mass, the acceleration can be found from:
F  ma  kx or
 kx
a
m
Note: Acceleration is always opposite to displacement.
Energy in Simple Harmonic Motion
We already know that the potential energy of a spring is
given by:
U = ½ kx2
The total mechanical energy is then:
The total mechanical energy will be conserved, as we are
assuming the system is frictionless.
Energy in Simple Harmonic Motion
If the mass is at the limits of its
motion, the energy is all potential.
If the mass is at the equilibrium
point, the energy is all kinetic.
We know what the potential energy
is at the turning points:
(11-4a)
Energy in Simple Harmonic Motion
The total energy is, therefore ½ kA2
And we can write:
This can be solved for the maximum
velocity which is given by making
total energy equal to only Kinetic:
The Period and Sinusoidal Nature
of SHM
If we use calculus we can find that the period of a mass
and ideal spring to be:
11-3 The Period and Sinusoidal Nature
of SHM
We can similarly find the position as a function of time
(note the diagram is 90 degrees out of phase):
© 2014 Pearson Education, Inc.
The Period and Sinusoidal Nature
of SHM
The top curve is a graph of
the previous equation.
The bottom curve is the
same, but shifted ¼ period
so that it is a sine function
rather than a cosine.
The Period and Sinusoidal Nature
of SHM
The velocity and acceleration can
be calculated as functions of
time; the results are below, and
are plotted at left.