Download Unit 4 – Chapter 7: Oscillatory Motion Requires a Set of Conditions

Unit 4 – Chapter 7: Oscillatory Motion Requires a
Set of Conditions
Examples of oscillatory motion
Pendulum, Vibrating spring, string, vibrating
ruler, strobe …..
7.1 Period and frequency
Period is time for one complete cycle,
oscillation
Frequency is number of oscillations in one
second.
f=1/T
T=1/f
Oscillatory motion – Motion with period of
each cycle is constant. It is repeated and
predictable.
A piston in the engine makes oscillatory
motion.
camshaft
Valve springs in
car engines
valve
http://www.youtube.com/watch?v=60QX5RY_ohQ
http://library.thinkquest.org/C006011/english/sites/ottomotor.php3?v=2
Ex:A pendulum takes 0.25 s to swing from right to
left. What is its period and frequency of oscillation
Solve 1-3 Workbook
7.2 Simple Harmonic motion
SHM is an oscillatory motion that has a
constant period and frequency. It repeats itself
in a predictable way.
Examples of simple harmonic motion (SHM)
1.
A swing (pendulum)
2. Vertically oscillating mass attached to a
spring
Pendulum and vertical spring http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=44
Oscillating Vertical Spring http://upload.wikimedia.org/wikipedia/commons/9/9d/Simple_harmonic_oscillator.gif
A mass attached to a horizontal
spring
3.
http://webphysics.davidson.edu/physlet_resources/gustavus_physlets/HorizontalSpring.html
Hooke’s Law for springs
 A mass bobbing up and down on a spring
executes "periodic motion" - motion that repeats
over a specific "period" of time.
Frestoring
Displacements From
Equilibrium
(Extension /
Compression)
Equilibrium Position
Frestoring = 0
Frestoring
Experiment: Springs
Purpose: To find a relationship between the extension of
a spring and the restoring force.
Procedure: A spring's extension was measured for masses
hung from the spring.
Data:
Trial
Frestoring =
Mass
mg
(kg)
(N)
Extension
(x)
(m)
1
2
0.050
0.50
0.025
0.100
1.00
0.05
3
0.150
1.50
0.075
4
0.200
2.00
0.10
5
0.250
2.50
0.125
Conclusions:
Hooke's Law:
Spring force is called a restoring force because it is trying
to restore the spring to its equilibrium position where
x=0. It causes the mass to make simple harmonic motion
 For a spring (or any elastic material), the restoring
force is directly proportional to the extension (or
compression).
F
x
restoring
or
F
 k x, where 'k"is the
restoring
"spring cons tan t"
for that spring.
- the larger the "k", the "stiffer" the spring!
When a spring is extended or compressed, work is done:
Ex: When a 450 g mass is attached to a vertical spring, the
spring stretches a distance of 0.560 m. What is the spring
constant?
Note: use free body diagram
Simple Harmonic Motion of Horizontal Mass-Spring
System
Do cart spring demo
Ex: A spring with spring constant of 50 N/m is pulled
horizontally on a frictionless surface to a distance of 2.1 m
from equilibrium. What is the restoring force?
2.1 m
Solve Q# 6-8 Workbook
For SHM:
 The restoring force acts in the opposite direction to the
displacement.
 At the extreme points of SHM, the displacement is at
max and called amplitude A where force and
acceleration are also max and v=0
xmax=A
 At equilibrium (x=0), v is max and Force (net force) and
acce are zero.
http://webphysics.davidson.edu/physlet_resources/gustavus_physlets/HorizontalSpring.html
Simple Harmonic Motion of a Pendulum.
 All pendulums we discuss are ideal means frictionless
 Motion of the pendulum and the spring are similar
Frestoring= -mg sin θ
1 FT= mg cos θ
4
𝜃
V=0
FRestoring= -max
a= - max
V= max
FRestoring=0
a=0
𝜃
Fg cos 𝜃
Fg sin 𝜃
Fg
2
5
V= - max
FRestoring=0
a=0
𝜃
V=0
FRestoring= -max
a= - max
𝜃
Fg sin 𝜃
Fg
3
V=0
FRestoring=max
a=max
𝜃
𝜃
Fg sin 𝜃
Fg cos 𝜃
Ex: A 2.90-kg pendulum bob is attached to a 3.20 mlong string. The pendulum is pulled sideways until
the string makes an angle of 20˚ with the vertical.
What is the restoring force acting on the
pendulum?
Ex: A 4.00-kg pendulum bob is suspended by a 2.50m-long string. The pendulum is pulled sideways
until the restoring force acting on the pendulum is
8.95 N. What angle does the string make with the
vertical?
Solve Q# 9-11 Workbook
Graphing Simple Harmonic Motion
How a restoring force affects the displacement, velocity,
and acceleration of the mass when a mass is pulled
horizontally on a frictionless surface and released.
F= -max
a=-max
v=0
-x
F=0
a=0
v=-max
F= +max
a=+max
v=0
F
t
+x
a
t
F=0
a=0
v=+max
v
F= -max
a=-max
v=0
Equilibrium position
t
Any object obeys Hooke’s Law makes SHM and
has a restoring force (F=-Kx).
The restoring force is opposite to x.
Solve Q#12-15 Workbook
7.3 Position, Velocity, Acceleration, and Time
Relations
Acceleration of a mass-spring system
In a horizontally oscillating mass attached to a
spring,
http://www.edumedia-sciences.com/a266_l2-shm.html
http://webphysics.davidson.edu/physletprob/ch4_tour/4.1.tour_4.html
-x
+x
FRestoring
x
-x
a
x
-x
v
-x
x
Acceleration of SHM

Since only force acting on the mass is the
restoring force, then according to Newton 2nd
Law’
Fnet=FR
m a = - Kx
kx
a
m
a is the instantaneous acceleration at any given
distance in m/s2
k is spring constant in N/m
x is position from equilibrium in m
m is mass in kg

The more the k and x the more the
acceleration

The more the mass the less the acceleration
Ex: A 1.54 kg mass is attached to a horizontal spring
with a spring constant of 95.0 N/m. What is the
acceleration of the mass when the mass is pulled
8.00 cm from the equilibrium on a frictionless
surface?
Solve Workbook 16-21
Maximum speed of a mass-spring system occurs
when the mass is at equilibrium position and zero at
maximum positions.
Since the system is frictionless and the system is
isolated, then energy is conserved:
Ek max  E p
1 2
1 2
mvmax  kxmax
2
2
1 2
1 2
mvmax  kA
2
2
vmax
k
A
m
When x is at max position,
x=A(amplitude)
In your formula sheet
vmax= maximum speed in m/s
A is amplitude (max displacement) in m
K spring constant in N/m
m is mass in kg

If the amplitude increases, max speed
increases.

If the spring constant increases, max speed
increases.

If the mass increases, max speed decreases.
Ex: A 50 g mass is attached to a spring (4.00 N/m).
The mass oscillates with amplitude of 1.12 m. What
is the maximum speed?
Solve Q#22- 31 workbook
Period of a Mass-Spring System
http://webphysics.davidson.edu/physletprob/ch4_tour/4.1.tour_4.html
http://www.edumedia-sciences.com/recherche.php?q=oscillation
SHM is like circular motion, then
2r
v
T
vmax
k
A
m
2A
k
A
T
m
m
T  2
k
T= period of SHM in s
m= mass in kg
K= spring constant in N/m
Factors affecting the period of an oscillation for a
spring:
If mass is increased the period increases
If spring constant increases (stiffer spring) period
decreases.
The period of a spring and a pendulum does not
depend on the displacement, x. If you double x to
2x, the period, T, stays the same since restoring
force increases.
Ex: A mass of 3.20 kg is attached to a horizontal
spring (k=55 N/m) that oscillates with amplitude of
60 cm. Find:
a. Acceleration of the mass at 45 cm
b.Max speed
c. The period
Ex: A vertical mass oscillates with a period of 1.25 s.
The spring constant is 15.0 N/m. What is the mass?
Solve Q#32-37 Workbook
Period of a Pendulum
l
T  2
g
T= period in s
l = length of pendulum in m
g= gravitation acceleration (9.81 m/s2 If on Earth)
Factors affecting period of a pendulum:
When on Earth, the length of the pendulum is the
only factor affecting period of pendulum. The more
the length the longer the period. On planets with
less gravitational acceleration, the period increases.
Mass has no effect on the period as also the
displacement.
Ex: What is the period of a 100 cm pendulum on
Earth?
Ex: What is the gravitational field strength (g) on
Mercury, if a 50 cm pendulum has a period of 2.28
s.
7.4 Applications of Simple Harmonic
Motion
The natural frequency of vibration of an object is
called Resonant Frequency. Resonant frequency
depends on the physical properties of the object
like m, k, l. Like after let go of a swing, your arms,
string, rattling of dishes....
All pendulums with same length have the same
Resonant Frequency.
Objects will continue to oscillate at same frequency
only if there is no friction.
Damped oscillations lose their energy to friction
causing amplitude to reduce until the pendulum
stops.
http://www.lon-capa.org/~mmp/applist/damped/d.htm
http://www.edumedia-sciences.com/recherche.php?q=oscillation
Forced Frequency – when an external force is
applied to an oscillating object. Parents giving a
push to their kid on a swing. This to compensate the
effect of friction.
Mechanical Resonance – when a periodic force is
applied to an oscillating object. Amplitude of
oscillation could increase and cause harm. A bridge
could break from the wind. Ringing a church bell.
http://www.acoustics.salford.ac.uk/feschools/waves/shm3.htm
http://www.pbs.org/wgbh/nova/bridge/tacoma3.html
Solve Q# 34-