Transcript 幻灯片 1 - SJTU

Chapter 9 Oscillatory Motion
Main Points of chapter 9
• The kinematics of simple harmonic motion
• Connection to circular motion
• The dynamics of simple harmonic motion s
• Energy
• Simple and physical pendulums
• Damped and driven harmonic motion
9-1 The Kinematics of Simple Harmonic Motion
Any motion that repeats itself at regular intervals
is called periodic motion
Examples: circular motion, oscillatory motion
We know that if we
stretch a spring with a
mass on the end and let
it go, the mass will
oscillate back and forth
(if there is no friction).
k
k
k
m
m
m
This oscillation is called Simple Harmonic Motion
The position of the object is
• angular frequency ω: determined by the
inertia of the moving objects and the restoring
force acting on it .
SI: rad/s
• amplitude A: The maximum distance of
displacement to the equilibrium point
• phase wt+f,
phase angle (constant) f
The value of A and f depend on the displacement
and velocity of the particle at time t = 0 (the initial
conditions)
Period T: the time for one complete oscillation
(or cycle);
Frequency f: number of oscillations that are
completed each second.
The red curve differs from the blue curve
(a) only in that its
amplitude is greater
(b) only in that its
period is T´ = T/2
(c) only in that f = -p/4
rad rather than zero
a phase difference
T = 2p/w
A
p
p
A
p
p

p
p

f
p
p
f
x
A1
A2
o
- A2
-A1
x
x2
x1
They are in phase
T
A1
A2
t o
- A2
-A1
x1
T
x2
They have a phase
difference of p
t
Relations Among Position, Velocity, and Acceleration
in Simple Harmonic Motion
We can take derivatives to
find velocity and acceleration:
v(t) leads x(t) by p/2
v(t) is phase –shifted to the left from x(t) by p/2
x(t) lags behind v(t) by p/2
x(t) is phase –shifted to the right from v(t) by -p/2
a(t) is phase –shifted to the left from x(t) by p
In SHM, the acceleration is proportional to the
displacement but opposite in sign, and the two quantities
are related by the square of the angular frequency.
9-2 A Connection to Circular Motion
A reference particle P´ moving in
a reference circle of radius A with
steady angular velocity w . Its
projection P on the x axis
executes simple harmonic motion.
Simple harmonic motion is the projection of
uniform circular motion on a diameter of the circle
in which the latter motion occurs.
demo

ACT A mass oscillates up & down on a spring.
Its position as a function of time is shown below.
Write down the displacement as the function of
time
y(t)(cm)
4
2
1
y
t (s)
9-3 Springs and Simple Harmonic Motion
The block–spring system forms a linear simple
F = -kx
harmonic oscillator
Hooke’s law
k
a
Combining with Newton’s second law
m
x
a differential equation for x(t)
Simple harmonic motion is the motion executed by a
particle of mass m subject to a force that is
proportional to the displacement of the particle but
opposite in sign ( a restoring force).
Solution
The period of the motion is independent of the amplitude
The initial conditions
t=0；x=x0，v=v0
X=0
X=A; v=0; a=-amax
X=0; v=-vmax; a=0
X=-A; v=0; a=amax
X=0; v=vmax; a=0
X=A; v=0; a=-amax
X=-A
X=A
•Another solution is
ok
•
is equivalent to
where
 Example A block whose mass m is 680 g is
fastened to a spring whose spring constant k is 65
N/m. The block is pulled a distance x = 11 cm from
its equilibrium position at x = 0 on a frictionless
surface and released from rest at t = 0. (a) What are
the angular frequency, the frequency, and the period
of the resulting motion? (b) What is the amplitude
of the oscillation? (c) What is the maximum speed
vm of the oscillating block, and where is the block
when it occurs? (d) What is the magnitude am of the
maximum acceleration of the block? (e) What is the
phase constant f for the motion? (f) What is the
displacement function x(t) for the spring–block
system?
Solution
At equilibrium point
 Example At t = 0, the displacement x(0) of the block
in a linear oscillator is -8.50 cm. The block's velocity
v(0) then is -0.920 m/s, and its acceleration a(0) is
+47.0 m/s2. (a) What is the angular frequency w of
this system? (b) What are the phase constant f and
amplitude A?
Solution
155
-25
Correct phase constant is1550
Additional Constant Forces
Solution
Simple harmonic motion with the same frequency,
but equilibrium point is shifted from x=0 to x=x1
Vertical Springs
Choose the origin at
equilibrium position
Fs
mg
Solution
Simple harmonic motion with equilibrium point at y=0

ACTA mass hanging from a vertical spring is
lifted a distance d above equilibrium and released
at t = 0. Which of the following describes its
velocity and acceleration as a function of time?
(a) v(t) = -vmax sin(wt)
a(t) = -amax cos(wt)
(b) v(t) = vmax sin(wt)
a(t) = amax cos(wt)
k
(c) v(t) = vmax cos(wt)
y
a(t) = -amax cos(wt)
t=0
(both vmax and amax are positive numbers)
m
d
0
9-4 Energy and Simple Harmonic Motion
This is not surprising since there are only
conservative forces present, hence the total
energy is conserved.
(a)Potential energy U(t), kinetic
energy K(t), and mechanical energy
E as functions of time t for a linear
harmonic oscillator. They are all
positive. U(t) and K(t) peak twice
during every period
(b)Potential energy U(x), kinetic
energy K(x), and mechanical energy
E as functions of position x for a
linear harmonic oscillator with
amplitude xm. For x = 0 the energy is
all kinetic, and for x = ±xm it is all
potential. The mechanical energy is
conserved
Note
The potential energy and the kinetic energy
peak twice during every period
The mechanical energy is conserved for a linear
harmonic oscillator
The dependence of energy on the square of the
amplitude is typical of Simple Harmonic Motion
ACT In Case 1 a mass on a spring oscillates back
and forth. In Case 2, the mass is doubled but the
spring and the amplitude of the oscillation is the
same as in Case 1.
In which case is the maximum potential energy of
the mass and spring the biggest?
A. Case 1
B. Case 2
C. Same
Look at time of maximum displacement x = A
Energy = ½ k A2 + 0 Same for both!
It’s Not Just About Springs
Besides springs, there are many other systems that
exhibit simple harmonic motion. Here are some
examples:
Almost all systems that are in stable equilibrium
exhibit simple harmonic motion when they
depart slightly from their equilibrium position
For example, the potential between H atoms in
an H2 molecule looks something like this:
U
x
If we do a Taylor expansion of this function about
the minimum, we find that for small displacements,
the potential is quadratic:
since x0 is minimum of potential
U
then
U
x0
x
Restoring force
x
Identifying SHM
c, c’ positive constant
Transport Tunnel
A straight tunnel with a frictionless interior is dug
through the Earth. A student jumps into the hole
at noon. What time does he get back?
x
g = 9.81 m/s2
RE = 6.38 x 106 m
He gets back 84 minutes later, at 1:24 p.m.
• Strange but true: The period of oscillation does not
depend on the length of the tunnel. Any straight
tunnel gives the same answer, as long as it is
frictionless and the density of the Earth is constant.
• Another strange but true fact: An object orbiting
the earth near the surface will have a period of the
same length as that of the transport tunnel.
g = w2R
9.81 = w2 6.38(10)6
m
w = .00124 s-1
so T =
= 5067 s
84 min
Example The potential energy of a diatomic molecule
whose two atoms have the same mass, m, and are
separated by a distance r is given by the formula
where A and e are positive constants. (a)
Find the equilibrium separation of the two atoms. (b)
Show that if the atoms are slightly displaced, then they
will undergo simple harmonic motion about the
equilibrium position. Calculate the angular frequency
of the harmonic motion.
Solution
(a) The equilibrium separation occurs where the
potential energy is a minimum, so we set
(b) We do a Taylor expansion of this U(r) function
about the equilibrium separation
This has the form of the elastic potential energy, so
the motion will be simple harmonic
The spring constant
The reduced mass
The angular frequency is
9-5 The Simple Pendulum
a simple pendulum consists of a pointlike mass m
(called the bob of the pendulum) suspended from one
end of an unstretchable, massless string of length l
that is fixed at the other end
T
mg
If  is small
Solution
with
The motion of a simple pendulum swinging through
only small angles is approximately SHM.
The period of small-amplitude pendulum is
independent of the amplitude --- the pendulum clock
The horizontal displacement
The energy of a simple pendulum:
For small 
The total energy is conserved
 ACT
You are sitting on a swing. A friend gives you a
small push and you start swinging back & forth
with period T1.
 Suppose you were standing on the swing rather
than sitting. When given a small push you start
swinging back & forth with period T2.
Which of the following is true:

(a) T1 = T2
(b) T1 > T2
(c) T1 < T2
You make a pendulum shorter, it
oscillates faster (smaller period)
ACT A pendulum is hanging vertically from the
ceiling of an elevator. Initially the elevator is at
rest and the period of the pendulum is T. Now the
pendulum accelerates upward. The period of the
pendulum will now be
1. greater than T
2. equal to T
3. less than T
“Effective g” is larger when accelerating upward
(you feel heavier)
9-6 More About Pendulums
The Physical Pendulum
Any object, if suspended and then displaced so
the gravitational force does no run through the
center of mass, can oscillate due to the torque.
If  is small
T
Mg
Solution
with
The period of a physical pendulum is independent
of it’s total mass—only how the mass is distributed
matters
For a simple pendulum
ACT A pendulum is made by hanging a thin
hoola-hoop of diameter D on a small nail.
What is the angular frequency of oscillation of the
hoop for small displacements? (ICM = mR2 for a hoop)
(a)
pivot (nail)
(b)
D
(c)
 Example In Figure below, a meter stick swings
about a pivot point at one end, at distance h from its
center of mass. (a) What is its period of oscillation T?
(b) What is the distance L0 between the pivot point O
of the stick and the center of oscillation of the stick?
Solution
 Example In Figure below , a penguin (obviously
skilled in aquatic sports) dives from a uniform board
that is hinged at the left and attached to a spring at the
right. The board has length L = 2.0 m and mass m = 12
kg; the spring constant k is 1300 N/m. When the penguin
dives, it leaves the board and spring oscillating with a
small amplitude. Assume that the board is stiff enough
not to bend, and find the period T of the oscillations.
Solution
Choose the equilibrium position as the origin
F
O
mg
y
T is independent of the board’s length
 Example A block of mass m is attached to a spring of
constant k through a disk of mass M which is free to
rotate about its fixed axis. Find the period of small
oscillations
M
Solution
Choose the equilibrium
T’
T
position as the origin
T o
T’
k
m
mg
x
9-7 Damped Harmonic Motion
A pendulum does not go on swinging forever. Energy
is gradually lost (because of air resistance) and the
oscillations die away. This effect is called damping.
Look at drag force that is proportional to velocity;
b is the damping coefficient:
Then the equation of motion is:
Damping factor
If a is small
Solution
Natural frequency
t
Life time
The larger the value of t ,the slower the exponential
As b increases, w’ decreases
When
Some systems have so much damping that no real
oscillations occur. The minimum damping needed
for this is called critical damping
x
critical damping
critical damping
heavy damping
Over (heavy) damping o
t
light damping
(light) damping
The time of the critical damping takes for the
displacement to settle to zero is a minimum
Example For the damped oscillator: m = 250 g,
k = 85 N/m, and b = 70 g/s. (a) What is the period
of the motion? (b) How long does it take for the
amplitude of the damped oscillations to drop to
half its initial value? (c) How long does it take for
the mechanical energy to drop to one-half its initial
value?
Solution
9-8 Driven Harmonic Motion
In damped harmonic motion, a mechanism such as
friction dissipates or reduces the energy of an
oscillating system, with the result that the amplitude of
the motion decreases in time.
Now, applying a driving force
Equation of motion becomes:
solution
After long times
The condition for the maximum of A
If b=0
In the absence of damping, if the frequency of the force
matches the natural frequency of the system , then the
amplitude of the oscillation reaches a maximum. This
effect is called resonance
For small b, the total width
at half maximum
peak becomes broader as b increases:
The role played by the frequency of a driving force is a
critical one. The matching of this frequency with a
natural frequency of vibration allows even a relatively
weak force to produce a large amplitude vibration
Examples
• Breaking glass
• The collapse of the Tacoma Narrows Bridge
Turbulent winds set up standing waves in the
Tacoma Narrows suspension bridge leading to its
collapse on November 7, 1940, just four months
after it had been opened for traffic
demo
Summary of chapter 9
• Simple Harmonic Motion
• Springs
Summary of chapter 9 Cont.
• Energy
• Simple and physical pendulums
Summary of chapter 9 Cont.
• Damped and driven harmonic motion


If b=0
resonance