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
k
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 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
w
t+
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
p p
A
f
T = 2
p / w
A
p p p p p f p
A
1
A
2
o - A
2
-A
1
x x
2
x
1 They are in phase
T A
1
A
2
t o - A
2
-A
1
x x x
2 1 They have a phase difference of
p
T 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
steady angular velocity
w
with . 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
t (s)
1
y
9-3 Springs and Simple Harmonic Motion The block–spring system forms
a linear simple harmonic oscillator
F = -kx k
a
Hooke’s law
m
Combining with Newton’s second law
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=x 0
，
v=v 0
X=0 X=-A X=A X=A; v=0; a=-a max X=0; v=-v max ; a=0 X=-A; v=0; a=a max X=0; v=v max ; a=0 X=A; v=0; a=-a max
• •
Another solution is is equivalent to
ok
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
v m
of the oscillating block, and where is the block when it occurs? (d) What is the magnitude a
m
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/s 2 . (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 is155 0
Additional Constant Forces
Solution Simple harmonic motion with the same frequency, but equilibrium point is shifted from x=0 to x=x 1
Vertical Springs
Choose the origin at equilibrium position F s mg Solution Simple harmonic motion with equilibrium point at y=0
ACT
A 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) = -v max sin(
w
t) a(t) = -a max cos(
w
t)
(b)
v(t) = v max sin(
w
t) a(t) = a max cos(
w
t)
(c)
v(t) = v max cos(
w
t) a(t) = -a max cos(
w
t)
(both
v max
and
a max
are positive numbers)
k t = 0 m y 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 x
m
. For x = 0 the energy is all kinetic, and for x =
±
x m
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 A 2 + 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 H 2 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 x
0
is minimum of potential
U
then
U x 0
Restoring force
x
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/s 2 R E = 6.38 x 10 6
He gets back 84 minutes later, at 1:24 p.m.
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 =
w 2
R 9.81 =
w 2
m 6.38(10) 6
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 T
1
.
Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T
2
. Which of the following is true:
(a)
T 1 = T 2
(b)
T 1 > T 2
(c)
T 1 < T 2
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? (I
CM = mR 2
for a hoop)
(a) (b) (c) pivot (nail)
D
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 L 0 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
Solution
Choose the equilibrium position as the origin T’ M T T’ k T m o 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:
Solution If
a
is small Damping factor 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
critical damping
x
critical damping heavy damping Over (heavy) damping
o
light damping
t
(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