Chapter 17 Oscillations. 17-1 Oscillating Systems. Each day we encounter many kinds of oscillatory motion, such as swinging pendulum of a clock , a person bouncing on a trampoline, a vibrating guitar string , and a mass on a spring . They have common properties:
Each day we encounter many kinds of oscillatory motion,such as swinging pendulum of a clock, a person bouncing on a trampoline,a vibrating guitar string, and a mass on a spring.
They have common properties:
the force always acts in a direction to restore the
system to its equilibrium position. Such a force is
called a “restoring force(恢复力)”.
3. The number of cycles per unit time is called the
Unit: period (s)
frequency(Hz, SI unit), 1 Hz = 1 cycle/s
4. The magnitude of the maximum displacement
from equilibrium is called the amplitude of the motion.
1. Simple harmonic motion
An oscillating system which can be described in
terms of sine and cosine functions is called a
“simple harmonic oscillator” and its motion is called
“simple harmonic motion”.
2. Equation of motion of the simple harmonic oscillator
Fig 17-5 shows a simple harmonic oscillator,
consisting of a spring of force constant K acting on
origin is chosen at here
a body of mass m that slides on a frictionless horizontal surface. The body moves in x direction.
Eq(17-4) is called the “equation of motion of the simple harmonic oscillator”. It is the basis of many complex oscillator problems.
Rewrite Eq(17-4) as
We write a tentative solution to Eq(17-5) as
3. Find the solution of Eq. (17-4)
Putting this into Eq(17-5) we obtain
Therefore, if we choose the constant such that
Eq(17-6) is in fact a solution of the equation of
motion of a simple harmonic oscillator.
If we increase the time by in Eq(17-6), then
Therefore is the period of the motion T.
The quantity is called the angular frequency.
is the maximum value of displacement. We call it
the amplitude of the motion.
The quantity is called phase of the motion.
is called “phase constant （常相位)”.
and are determined by the initial position and
velocity of the particle. is determined by the system.
How to understand ?
How to compare the phases of two SHOs with same ?
(a) same: ,
(b) same: ,
(c) same: ,
When the displacement is a maximum in either
direction, the speed is zero, because the velocity
must now change its direction.
1.The potential energy
2.The kinetic energy
3. The total mechanical energy E is
Fig 17-8 (b)
At the equilibrium position , .
Eq(17-14) can be written quite generally as
In Fig 17-5, m=2.43kg, k=221N/m, the block is
stretched in the positive x direction a distance of
11.6 cm from equilibrium and released. Take time
t=0 when the block is released, the horizontal
surface is frictionless.
(a) What is the total energy?
(b) What is the maximum speed of the block?
(c) What is the maximum acceleration?
(d) What is the position, velocity, and acceleration
(c) The maximum acceleration occurs just at the
instant of release, when the force is greatest
So at t=0.215s
In Fig17-5, m=2.43kg, k=221N/m, when the block
m is pushed from equilibrium to x=0.0624m, and
its velocity , the external force is
removed and the block begins to oscillate on the
horizontal frictionless surface. Write an equation
for x (t) during the oscillation.
Setting this equal to , we have
To find the phase constant , we still need to use the
information give for t=0:
So only will give the correct initial velocity.
1. The torsional oscillator(扭转振子)
Fig 17-9 shows a torsional oscillator.
If the disk is rotated in the horizontal (xy)
plane, the reference line op will move
to the OQ, and the wire oo’ will be
twisted. The twisted wire will exert a
restoring torque on the disk, tending to
return the system to its equilibrium.
Here is constant ( the Greek letter Kappa ), and
is called torsional constant.
The equation of motion for such a system is
where is the rotational inertia of the disk about z
axis. Using Eq(17-17) we have
Eq(17-19) and (17-5) are mathematically identical.
The solution should be a simple harmonic oscillation in the angle coordinate ,
A torsional oscillator is also called torsional pendulum(扭摆). The Cavendish balance, used to measure the gravitational force constant G, is a torsional pendulum.
Fig(17-10) shows a simple pendulum of length L
and particle mass m.
The restoring force is:
If the is small,
In Fig. 17-11 a body of irregular shape is pivoted about a horizontal frictionless axis through P
For small angular displacement .
(a)The rotational inertia can be found from Eq(17-28).
Center of oscillation (振动中心 )”
Suppose the mass of the physical pendulum were concentrated at one point with distance L from the pivot, it will form a simple pendulum.
The resulting simple pendulum would have same period as the original physical pendulum.
The point O is called the “center of oscillation” of the physical pendulum.
If an impulsive force in the plane of oscillation acts at the center of oscillation, no effect of this force is felt at the pivot point P.
A thin uniform rod of mass M=0.112kg and
length L=0.096m is suspended by a wire that
passes through its center and is perpendicular
to its length. The wire is twisted and the rod
set oscillating. The period is found to be 2.14s.
When a flat body in the shape of an equilateral
triangle is suspended similarly through its
center of mass, the period is found to be 5.83m.
Find the rotational inertia of the triangle about this axis.
The rotational inertia about its Cm is
The simple pendulum having the same period has
a length (Eq(17-30))
The center of oscillation of the disk pivoted at P is
therefore at o, a distance below the point of support.
You may check that the period of the pendulum pivoted at O is the same as that pivoted at P.
Torsional pendulum clock
Simple harmonic motion can be described as a
projection of uniform circular motion along a
diameter of the circle.
1. Fig17-14 shows a particle
P in uniform circular motion.
with x axis, and the x component of is
This is of course identical to Eq(17-6) for the
displacement of the simple harmonic oscillator.
point P is , the x component of is
3. The centripetal acceleration is , and its x
component is (17-33)
Eqs(17-32) and (17-33) are identical with Eqs(17-
11) for simple harmonic motion, again with
replace by r.
So the projection of uniform circular motion along the y direction also gives simple harmonic motion.
On the contrary, the combination of two simple harmonic motions at right angles, with identical amplitude and
frequencies, can form a uniform circular motion.
Up to this point we have assumed that no frictional
force act on the system.
For real oscillator, there may be friction, air resistance act on the system, the amplitude will decrease.
1. This loss in amplitude is called “damping” and
the motion is called “damped harmonic motion”.
where is called the “damping time constant” or
the “mean lifetime” of the oscillator.
is the time necessary for the amplitude to drop to 1/e of its initial value.
Eq(17-37) shows that the mechanical energy of
the oscillator decreases exponentially with time.
The energy decreases twice as rapidly as the
Assume the damping force is , where b is a
positive constant called the “damping constant”.
With , Newton’s second law gives
If , that is , damping slows down
the motion. This case is called underdamping (欠阻尼)
Comparing Eqs(17-39) and (17-36) we have
(b) When , , the motion decays
exponentially to zero with no oscillation at all.
This condition is called “critical damping (临界阻尼)”. The lifetime has its smallest possible value, .
(c) When , the motion also decays
exponentially to zero with no oscillation, called overdamping （过阻尼）.
Oscillations of a system carried out under the action of an external periodical force, such as
or a successive action of an external non-periodical force.
,Which frequency will the forced oscillation system take?
Forced oscillation system takes the frequency
of the external force, namely .
The amplitude of the forced oscillation can increase much as approaches .
This condition is known as “resonance” and is called “resonant angular frequency”.
In the case with damping, the rate at which energy is provided by the driving force exactly matches the rate at which energy is dissipated by the damping force.
Theory of anything (任意之理)
Where string theory stands today
Peter Woit (Princeton)
Not even wrong!