Transcript Physics 201: Lecture 1
Physics 201: Chapter 14 – Oscillations (cont’d)
General Physical Pendulum & Other Applications Damped Oscillations Resonances 4/30/2020 Physics 201, UW-Madison 1
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Simple Harmonic Motion: Summary
k m
Force:
d
2
s
2
s dt
2
k k m m s 0 0 s
Solution:
θ =
A
cos(
t +
)
g L
L
Physics 201, UW-Madison 2
Question 1
The amplitude of a system moving with simple harmonic motion is doubled. The total energy will then be 4 times larger 2 times larger the same as it was half as much quarter as much U
at
x
1 2
kx
2
A
,
v
1 2
mv
2 0
U
1 2
kA
2
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Question 2
A glider of mass m is attached to springs on both ends, which are attached to the ends of a frictionless track. The glider moved by 0.2 m to the right, and let go to oscillate. If m = 2 kg, and spring constants are k 1 = 800 N/m and k 2 N/m, the frequency of oscillation (in Hz) is = 500 approximately 6 Hz 2 Hz 4 Hz 8 Hz 10 Hz f
2
k
/ 2
m
800 500 2 2 4 Hz
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The harmonic oscillator is often a very good approximation for (non harmonic) oscillations with small amplitude:
U
1 2
kx
2
for small x
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General Physical Pendulum
4/30/2020 Suppose we have some arbitrarily shaped solid of mass
M
hung on a fixed axis, and that we know where the CM is located
and
what the moment of inertia
I
= Mg d Mg
R about the axis is.
The torque about the rotation (
z
) axis for small is (using sin )
MgR
I
d
2
dt
2
z-axis
R xCM d d
2 2
dt
2 0 cos
t where
MgR
I
Mg
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Torsional Oscillator
Consider an object suspended by a wire attached at its CM. The wire defines the rotation axis, and the moment of inertia
I
about this axis is known. The wire acts like a “rotational spring.” When the object is rotated, the wire is twisted. This produces a torque that opposes the rotation.
In analogy with a spring, the torque produced is proportional to the displacement:
= -
k
I wire
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Torsional Oscillator…
Since
= -
k
= I
becomes
k
I d
2
dt
2
wire d
2
dt
2 2
where
k
I
I This is similar to the “mass on spring” Except I has taken the place of m (no surprise).
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Damped Oscillations
In many real systems, nonconservative forces are present The system is no longer ideal Friction/drag force are common nonconservative forces In this case, the mechanical energy of the system diminishes in time, the motion is said to be
damped
The amplitude decreases with time The blue dashed lines on the graph represent the
envelope
motion of the
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Damped Oscillation, Example
One example of damped motion occurs when an object is attached to a spring and submerged in a viscous liquid The retarding force can be expressed as
F
= b v where
b
is a constant
b
is called the
damping coefficient
The restoring force is –
kx
From Newton’s Second Law
F x
= -
k x
–
bv x
=
ma x or, 4/30/2020 Physics 201, UW-Madison 10
Damped Oscillation, Example
When b is small enough, the solution to this equation is: with (This solution applies when b<2mω 0 .)
Time constant: 4/30/2020 Physics 201, UW-Madison 11
Energy of weakly damped harmonic oscillator:
The energy of a weakly damped harmonic oscillator decays exponentially in time.
The potential energy is ½kx 2 : 4/30/2020 Time constant is the time for E to drop to 1/e .
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Types of Damping
0 is also called the
natural frequency
of the system If
F
max =
bv
max <
kA
, the system is said to be
underdamped
When
b
oscillate reaches a critical value
b c
such that
b c
/ 2 m = 0 , the system will not The system is said to be
critically damped
If
F
max =
bv
max >
kA
and
b
/2
m
> 0 , the system is said to be
overdamped
For critically damped and overdamped there is no angular frequency
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Forced Oscillations
It is possible to compensate for the loss of energy in a damped system by applying an external sinusoidal force (with angular frequency ) The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces After a driving force on an initially stationary object begins to act, the amplitude of the oscillation will increase After a sufficiently long period of time,
E
driving Then a steady-state condition is reached =
E
lost to internal The oscillations will proceed with constant amplitude
The amplitude of a driven oscillation is 4/30/2020
0 is the natural frequency of the undamped oscillator Physics 201, UW-Madison 14
Resonance
When 0 an increase in amplitude occurs This dramatic increase in the amplitude is called
resonance
The natural frequency resonance frequency 0 is also called the At resonance, the applied force is in phase with the velocity and the power transferred to the oscillator is a maximum The applied force and
v
proportional to sin (
t
+ are both ) The power delivered is
F .
v
» This is a maximum when
F
in phase and
v
are Resonance (maximum peak) occurs when driving frequency equals the natural frequency The amplitude increases with decreased damping The curve broadens as the damping increases
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Resonance Applications:
4/30/2020 Extended objects have more than one resonance frequency. When plucked, a guitar string transmits its energy to the body of the guitar. The body’s oscillations, coupled to those of the air mass it encloses, produce the resonance patterns shown.
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Question 3
A 810-g block oscillates on the end of a spring whose force constant is k=60 N/m. The mass moves in a fluid which offers a resistive force proportional to its speed – the proportionality constant is b=0.162 N.s/m.
F s
F R
ma m d
2
x
kx dt
2 Write the equation of motion and solution.
Solution:
x
(
t
)
b dx dt Ae
b
2
m t
cos(
t
) What is the period of motion?
0.730 s
k m
b
2
m
2 Write the displacement as a function of time if at
t = 0, x = 0
and at
t = 1
s,
x = 0.120
m.
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Solution:
x
(
t
) 0.182
e
0.100
t
sin(8.61
t
)
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