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Periodic Motion - 1
Oscillatory/Periodic Motion
Repetitive Motion
Simple Harmonic Motion SHM
special case of oscillatory.
• Object w equilibrium position has
restoring force pushing it back to
equilibrium.
Simple Harmonic Motion
A spring exerts a restoring force directily
proportional to the displacement fr. eq:
Remember Hooke’s Law
Negative sign Fnet opposite to s.
Pendulum is not SHM
• Fnet not directly opposite s
• for small displacement angles it approximates
SHM.
Vocabulary
• Period T: time required for one cycle of
periodic motion (sec).
• Frequency: number of oscillations per
unit time.
unit is Hertz:
• Equilibrium (0): the spot the mass would come
to rest when not disturbed – Fnet = 0.
• Displacement: (x) distance from equilibrium.
• Amplitude (xo) – max displacement from eq.
• Angular frequency w, multiply f by 2p.
Reported as cy/sec or rad/sec. 2p = 1 full cycle.
• Angular frequency w is a
measure that comes from
circular motion.
• It is measured in radians
per second.
• f, Hz is cycles per second.
An object makes 1 revolution per
second. How many radians does it
complete in 1 second?
• 2p rad s-1 = 1 revolution per second.
• 2p rad s-1 = also describes 1
non-circular cycle or oscillation.
To find w, from frequency f,
or period T, use:
w = 2pf
or
w=
2p
T
Ex 1. A pendulum completes 4 swings
in 0.5 seconds. What is its angular
frequency?
• T = 0.125 s
w = 2p/T = 50 rad/sec.
Free Body Diagram Mass on
Spring
• Complete sheet.
•
Simple Harmonic Motion
2 Conditions.
• 1. Acceleration/Fnet proportional to
displacement.
• 2. Acceleration/Fnet directed toward
equilibrium.
• Defining Equation for SHM
• a = -w2x
Sketch Graphs.
Simple Harmonic Motion On a
bobbing mass:
•
•
•
•
•
No friction
Weight stays constant.
Tension increases with stretch.
Fnet / accl toward equilibrium position.
Fnet increases with is proportional to
displacement.
Graphical Treatment
Equations of SHM
Displacement, x, against time
x = xo cos wt
start point at max ampl.
** Set Calculator in Radians.
Displacement against time
x = xosinwt
•
•
•
•
Speed = d/t = 2pR/T but
2p /T = w so
v = wr but r the displacement x so.
v = wx.
Velocity against time v = vocoswt
Starting where?
Midpoint = max
velocity.
Equations of Graphs
• x = xo cos wt
x = xo sin wt
• v = -vo sin wt
v = vo cos wt
• a = -aocos wt
-ao sin wt
• Released from
top.
Released equilibrium.
Ex 2. A mass on a spring is oscillating
with f = 0.2 Hz and xo = 3 cm. What is
the displacement of the mass 10.66 s
after its release from the top?
• x = xo cos wt
xo = 3 cm
 w = 2pf. = 0.4 p Hz =1.26 rad/s.
• t = 10.66 s
• x = 0.03 cos (1.26 x 10.66) = 0.019 m
• You must use radians on calculator.
SHM and Circular Motion
• Use the relationship to derive
equations.
If an object moving with constant speed in a
circular path is observed from a distant point (in
the plane of the motion), it will appear to be
oscillating with SHM.
The shadow of a pendulum bob moves with
s.h.m. when the pendulum itself is either
oscillating (through a small angle) or moving in a
circle with constant speed, as shown in the
diagram.
For any s.h.m. we can find a corresponding circular
motion.
When a circular motion "corresponds to" a given s.h.m.,
i) the radius of the circle is equal to the amplitude
of the s.h.m.
ii) the time period of the circular motion is equal
to the time period of the s.h.m.
Derive Relationship between accl & w for SHM.
From circular motion ac = v2/r and vc = 2pr/T
Oscillating systems have acceleration too.
• But w = 2p/T
•
•
•
•
•
vc = 2pr/T
vc = wr
But ac = v2/r
So ac = (wr)2/r
but r is related to displacement x.
For any displacement:
a = -w²x
ao = -w²xo
The negative sign shows Fnet & accl
direction opposite displacement.
Derivation of accl in Hamper pg 76.
Or use 2nd derivative of displacement.
Ex 3. A pendulum swings with f = 0.5 Hz.
What is the size & direction of the
acceleration when the bob has displacement
of 2 cm right?
• a = -w²x
• w = 2pf
= p
• a = -(p)2 (0.02 m) = -0.197 m/s2. left.
Ex 4: A mass is bobbing on a spring with a
period of 0.20 seconds. What is its angular
acceleration at a point where its
displacement is 1.5 cm?
•
•
•
•
•
w = 2p/T
.w = 31 rad/s
a = -w²x
a = (31rad/s)(1.5 cm) = 1480 cm/s2.
15 m/s2.
SHM, Hooke’s Law & k.
For a mass undergoing SHM on a spring,
what is the relation between angular
frequency w, and k the spring constant?
• Use Hooke’s law and make
substitutions to derive a relation in
terms of angular frequency, k, and
mass.
F= - kx.
ma = - k x.
So a = -k x
m
a = -w² x
So w2 = -k/m
To find the velocity of an oscillating mass or
pendulum at any displacement:
When the mass is at equilibrium, x = 0, and
velocity is maximum:
vo = ± wxo.
Derivation on H pg 77.
Ex 4. A pendulum swings with f = 1 Hz and
amplitude 3 cm. At what position will be its
maximum velocity &what is the velocity?
At max velocity vo = wxo.
w = 2pf = 2p(1) = 2p rad/s
vo = (2p rad/s)(0.03)
vo = 0.188 m/s
vo = 0.2 m/s
Hwk Hamper pg 75- 77 Show equations
and work, hand in virtual solar system lab.
Mechanical Universe w/questions
http://www.learner.org/resources/series42.
html?pop=yes&pid=565
Units of Chapter 4
• The Pendulum
• Damped Oscillations
• Driven Oscillations and Resonance
The Period of a Mass on a Spring
Since the force on a mass on a spring is
proportional to the displacement, and also to
the acceleration, we find that
.
Make substitutions to find the relationship
between T and k.
The Period of a Mass on a Spring
Therefore, the period is
How does T change as mass increases? Sketch it!
Period of pendulum
Energy Conservation in Oscillatory Motion
In an ideal system the total mechanical energy
is conserved. A mass on a spring:
Horizontal mass no PEg.
Determining the max KE & PE:
•
•
•
•
•
PE = ½ k x2 for a stretched spring.
So PEmax = PE = ½ k xo
KE = ½ mv2
vmax = wxo,
KEmax = ½ m(w2xo 2),
At any point:
• KE = ½ mw2 (xo2 - x2 )
• How could you determine PE from Etot?
• Subtract KE from Etot.
Since the total E will always equal the
max KE (or PE), we can calculate the
number of Joules of total E from the
KE equation:
ET = ½ mw2xo2
Ex 5: A 200-g pendulum bob is oscillating with
Amplitude = 3 cm, and f = 0.5 Hz. How much
KE will it have as it passes through the origin?
• KEmax = ½ w2xo 2,
• xo = 0.03 m
 w = p.
• KE = 8.9 x 10-4 J.
Energy Conservation in Oscillatory Motion
The total energy is constant; as the kinetic
energy increases, the potential energy
decreases, and vice versa.
Energy Conservation in Oscillatory Motion
The E transforms from potential to kinetic &
back, the total energy remains the same.
The Pendulum
A simple pendulum consists of a mass m (of
negligible size) suspended by a string or rod of
length L (and negligible mass).
The angle it makes with the vertical varies with
time as a sine or cosine.
The Pendulum
The restoring force is
actually proportional
to sin θ, whereas the
restoring force for a
spring is proportional
to the displacement x.
The Pendulum
However, for small angles, sin θ and θ are
approximately equal.
The Pendulum
Substituting θ for sin θ allows us to treat the
pendulum in a mathematically identical way to
the mass on a spring.
Therefore, we find that the period of a pendulum
depends only on the length of the string:
Damped Oscillations
In most physical situations, there is a
nonconservative force of some sort, which will
tend to decrease the amplitude of the
oscillation, and which is typically proportional
to the speed:
This causes the amplitude to decrease
exponentially with time:
Damped Oscillations
This exponential decrease is shown in the figure:
The image shows a system that is
underdamped – it goes through multiple
oscillations before coming to rest.
Damped Oscillations
A critically damped system is one that
relaxes back to the equilibrium position
without oscillating and in minimum
time;
an overdamped system will also not
oscillate but is damped so heavily that it
takes longer to reach equilibrium.
Driven Oscillations and Resonance
An oscillation can be driven by an oscillating
driving force; the f of the driving force may or may
not be the same as the natural f of the system.
Driven Oscillations and Resonance
If the driving f is close
to the natural
frequency, the
amplitude can become
quite large, especially
if the damping is small.
This is called
resonance.
Summary
• Period: time required for a motion to go
through a complete cycle
• Frequency: number of oscillations per unit time
• Angular frequency:
• Simple harmonic motion occurs when the
restoring force is proportional to the
displacement from equilibrium.
Summary
• The amplitude is the maximum displacement
from equilibrium.
• Position as a function of time:
• Velocity as a function of time:
Summary
• Acceleration as a function of time:
• Period of a mass on a spring:
• Total energy in simple harmonic motion:
Summary
• Potential energy as a function of time:
• Kinetic energy as a function of time:
• A simple pendulum with small amplitude
exhibits simple harmonic motion
Summary
• Period of a simple pendulum:
• Period of a physical pendulum:
Summary
• Oscillations where there is a nonconservative
force are called damped.
• Underdamped: the amplitude decreases
exponentially with time:
• Critically damped: no oscillations; system
relaxes back to equilibrium in minimum time
• Overdamped: also no oscillations, but
slower than critical damping
Summary
• An oscillating system may be driven by an
external force.
• This force may replace energy lost to friction,
or may cause the amplitude to increase greatly
at resonance
• Resonance occurs when the driving frequency
is equal to the natural frequency of the system