Transcript periodic motion - Grade 12 Physics

Chapter 11
Vibrations and Waves
Ms. Hanan
11-1 Simple Harmonic Motion
Objectives
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Identify the conditions of simple harmonic
motion.
•
Explain how force, velocity, and acceleration
change as an object vibrates with simple
harmonic motion.
•
Calculate the spring force using Hooke’s law.
Vocabulary
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Periodic Motion
Simple Harmonic Motion
Period
Amplitude
Hooke’s Law
Pendulum
Oscillation
Vibration
Spring Constant
Displacement
Periodic Motion
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Motion where a body travels along the same path in a
repeated, back and forth manner
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Also called oscillation and vibration
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We are surrounded by oscillations – motions that repeat
themselves (periodic motion)
• Grandfather clock pendulum, boats bobbing at anchor,
oscillating guitar strings, pistons in car engines
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Understanding periodic motion is essential for the study of
waves, sound, alternating electric currents, light, etc.
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An object in periodic motion experiences restoring forces
or torques that bring it back toward an equilibrium position
Periodic Motion
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Those same forces cause the object to “overshoot”
the equilibrium position
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Think of a block oscillating on a spring or a pendulum
swinging back and forth past its equilibrium position
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Examples of periodic motion:
R
L
/5
m
r
L
/2
k
m
Example 1
Mass-Spring System
a
a
a
Equil. position
a
Example 2
Simple Pendulum
a
a
Equil. position
a
a
Example 3
Floating Cylinder
Equil. position
a
a
a
a
Hooke’s Law Force
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LEQ
k
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x
m
The force always
acts toward the
equilibrium
position
The direction of
the restoring
force is such
that the object
Fs=kx is being either
pushed or pulled
toward the
equilibrium
position
Hooke’s Law Reviewed
F  kx
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When x is positive
F is negative
;
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When at equilibrium (x=0),
F = 0 ;
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When x is negative
F is positive
;
,
,
Stretched and Equilibrium
11
Equilibrium and Compressed
12
Motion of the Spring-Mass System
• Assume the object is initially pulled to a
distance A and released from rest
• As the object moves toward the
equilibrium position, F and a decrease,
but v increases
• At x = 0, F and a are zero, but v is a
maximum
• The object’s momentum causes it to
overshoot the equilibrium position
Graphing x vs. t
A
T
A : amplitude (length, m)
T : period (time, s)
Sample Problem A – P. 370
If a mass of 0.55 kg attached to a vertical
spring stretches the spring 2.0 cm from its
equilibrium position, what is the spring
constant?
Givens:
m = 0.55 kg
x = -2.0 cm
x = -0.02 m
g = 9.81 m/s2
Unknowns:
k = ?
X = -2.0 cm
Step 1: Choose the equation/situation:
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When the mass is attached to the spring, the equilibrium
position changes.
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At the new equilibrium point, the net force acting on the mass
is zero.
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By Hooke’s Law, the Spring Force must be equal and opposite
of the weight of the mass
FSpring  kx
Fg  mg
Fnet  FSpring  Fg  0
(kx)  (mg )  0
kx  mg
 mg
k
x
 mg
k
x
Step 2: Substitute the known values into this equation.
m = 0.55 kg
x = -0.02 m
g = 9.81 m/s2
 (0.55kg)(9.81m / s )
k
 0.02m
2
k  270 N / m
Assignments
• Class-work:
Practice A , page 371, questions 1,
2, and 3.
• Homework:
Review and Assess; Page 396: # 8
and 9
Due next class
Elastic Potential Energy
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The energy stored in a stretched or
compressed spring or other elastic material
is called elastic potential energy
PEs = ½kx2
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The energy is stored only when the spring is
stretched or compressed
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Elastic potential energy can be added to the
statements of Conservation of Energy and
Work-Energy
Energy Transformations
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The block is moving on a frictionless surface
The total mechanical energy of the system is
the kinetic energy of the block
Energy Transformations, 2
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The spring is partially compressed
The energy is shared between kinetic energy
and elastic potential energy
The total mechanical energy is the sum of the
kinetic energy and the elastic potential energy
Energy Transformations, 3
• The spring is now fully compressed
• The block momentarily stops
• The total mechanical energy is stored as
elastic potential energy of the spring
Energy Transformations, 4
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When the block leaves the spring, the total
mechanical energy is in the kinetic energy of
the block
The total energy of the system remains
constant
Simple Pendulum
Restoring force of a pendulum is a
Component of the bob’s weight
x 2  L2
x
F  mg sin 
x
x
sin  

x 2  L2 L
mg
F
x
L
Looks like Hooke’s law (k  mg/L)
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When oscillations are small, the motion is called
simple harmonic motion (shm) and can be described
by a simple sine curve.
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The pendulum’s potential energy is gravitational, and
increases as the pendulum’s displacement increases.
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Gravitational potential energy is equal to zero at
the pendulum’s equilibrium position. PEg = mgh
Assignments
• Class-work:
Practice section review page 375,
questions 1, 2, 3, and 4.
• Homework:
Vibrations and Waves Problem A,
Hooke’s Law Additional Practice
Sheet, even questions. Due Sunday
20/2/11