1. Fundamentals of Physics School of Physical Science and Technology
Mechanics
(Bilingual Teaching)
张昆实
School of Physical Science and Technology
Yangtze University

2. Chapter 14 Oscillations 16-1 Oscillations 16-2 Simple Harmonic Motion
16-3 The Force Law for Simple Harmonic Motion
16-4 Energy in Simple Harmonic Motion
16-5 An Angular Simple Harmonic Oscillator
16-6 Pendulums
16-7 Simple Harmonic Motion and Uniform
Circular Motion
16-8 Damped Simple Harmonic Motion
16-9 Forced Oscillations and Resonance

3. We are surounded by oscillations motions that repeat themselves.
There are swinging chandeliers, boats bobbing at anchor, and the surging pistons in the engines of cars.

4. 16-1 Oscillations
There are oscillating guitar strings, drums,
bells, diaphragms in telephones and
speaker systems,
and quartz crystals in wristwatches.

5. 16-1 Oscillations
Less evident are the oscillations of the air molecules that transmit the sensation of sound,
the oscillations of the atoms in a solid that convey the sensation of temperature,

6. 16-1 Oscillations
the oscillations of the electrons in the antennas of radio and TV transmitters that convey information.

7. 16-2 Simple Harmonic Motion
In a oscillating system a particle moves repeatidly back and forth about the origin of an axis.
Properties of oscillation
frequency (symbol ): the number of oscillations completed in one second.
SI unit:
1 hertz = 1 Hz = 1 oscillation per second = 1 s-1
period (symbol ): the time for one complete oscillation (sycle)
(16-2)
harmonic motion: Any motion that repeats itself at regular intervals is called periodic motiom or harmonic motion.

8. 16-2 Simple Harmonic Motion
Simple Harmonic Motio (SHM)
The displacement of the particle from the origin is given as a function of time by
(16-3)
( displacement )

9. 16-2 Simple Harmonic Motion
The amplitude of the motion : is a positive constant whose value depends on how the motion was started. The subscript stands for maximum because the amplitude is the magnitude of the maximum displacement of the particle in either direction.
The cosine function in Eq.16-3 varies between the limits , so the displacement x(t) varies between the limits

10. 16-2 Simple Harmonic Motion
The time-varying quantity ( ) in Eq.16-3 is called the phase of the motion, and the constant is called the phase constant (or phase angle).
The value of depends on the displace- ment and velocity of the particle at time i For the plots of Fig.16-3a, the phase constant is zero.

11. 16-2 Simple Harmonic Motion
The constant is called the angular frequency of the motion.
Note, the desplacement must return to its initial value after one period of the motion; that is must equal for all .
For simplicity, let then
(16-4)

12. 16-2 Simple Harmonic Motion
For simplicity, let then
(16-4)
The cosine function first repeats itself when its phase has increased by ,
so Eq gives us
(16-5)
From
(16-2)
The SI unit of angular frequency is the radian per second

13. 16-2 Simple Harmonic Motion
Compare for two simple harmonic motions
(a) Different amplitude
(b) Different period
(c) Different phase
constant

14. 16-2 Simple Harmonic Motion
The Velocity of SHM
(16-3)
( displacement )
By differentiating
(16-6)
(velocy)
positive quantity is called the velocity amplitude :
The curve of is shifted to the left from the curve of by one-quarter period.

15. 16-2 Simple Harmonic Motion
The Acceleration of SHM
(velocy)
(16-6)
By differentiating
(16-7)
(acceleration)
positive quantity is the acceleration amplitude :
Varies between the limits

16. 16-2 Simple Harmonic Motion
The Acceleration of SHM
(16-3)
(16-6)
(16-7)
Combine Eqs and 16-7
(16-8)
In SHM, the acceleration is pro- portional to the displacement but opposite in sign, and the two quantities are related by the square of angular frequen- cy.
漆安慎 《力学》P290 例题1

17. 16-3 The Force Law for Simple Harmonic Motion
Combine Newton’s second law and Eq.16-8, the force acting on a body in SHM can be found
(16-9)
is a restoring force that is proportional to the displacement but opposite in sign.
(16-10)
It is Hooke’s law
For a spring, the spring constant here being
(16-11)
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.

18. 16-3 The Force Law for Simple Harmonic Motion
The block-spring system forms a linear simple harmonic oscillator. (linear means )
(16-12)
By Eq.16-11, the angular frequency of the SHM is
( angular frequency)
Combining Eq.16-5
(16-13)
( period )
Equations 16-12, 16-5 and tell us that a large angular frequency (and thus a small period ) goes with a stiff spring (large ) and a light block (small ).

19. 16-4 Energy in Simple Harmonic Motion
Consider the mechanical energy of the oscillator
The potential energy of a linear oscillator is associated entirely with the spring. Its value depends on how much the spring is stret- ched or compressed (no ).
(16-18)
The kinetic energy of a linear oscillator is associa- ted entirely with the block. Its value depends on how fast the block is moving (no ).
(16-19)

20. 16-4 Energy in Simple Harmonic Motion
Consider the mechanical energy of the oscillator
Substitute for
(16-20)
The mechanical energy of the oscillator is
(16-21)
The mechanical energy of a linear oscillator is indeed constant and independent of time.

21. 16-4 Energy in Simple Harmonic Motion
Potential energy , kinetic energy and mechanical energy as a function of time for a linear harmonic oscillator.
All energies are positive.
Potential energy , kinetic energy and mechanical energy as a function of position for a linear harmonic oscillator with amplitude

22. 16-5 An Angular Simple Harmonic Oscillator
The device in Fig.16-7 is called torsion pendulum. The element of elasticity is associated with the twisting of a suspension wire.
If we rotate the desk by some angular displacement from its rest position (where the reference
line is at ) and release it, it will oscillate about that position in angular simple harmonic motion.
Rotating the desk through an angle in either direction introduces a restoring torque given by
(16-22)

23. 16-5 An Angular Simple Harmonic Oscillator
restoring torque
(16-22)
Constant (kappa) is called the torsion constant, that depends on the length, diameter, and material of the suspension wire.
(16-22)
comparing
(16-10)
Hooke’s law
Hooke’s law (angular form)
linear SHM
angular SHM
The rotational inertial of the oscillating desk
period
(16-13)
(16-23)
( torsion pendulum )
漆安慎 《力学》P286

24. 16-6 Pendulums
Discussing a class of simple harmonic oscillators in which the springiness is associated with the gravitational force.
the Simple Pendulum
an unstretchable, massless string of length L .
a bob of the pendulum (mass ) can swing back and forth freely in the virtical plane.
Two forces act on the bob :
from the sthing
gravitational force
tangential
radial

25. 16-6 Pendulums
The tangential component produces a restoring torque about the pendulum’s pivot point, because it always acts opposite the displacement of the bob so as to bring the bob back toward its equilibrium position
(16-24)
(16-25)
take
For small
then
(16-26)
(16-8)

26. 16-6 Pendulums but they are opposite in sign
Hallmark of SHM
Hallmark of SHM
but they are opposite in sign
Thus as the bob moves to the right, its acceleration to the left increaces until it stopes and bigins moving to the left. The same thing happens when it is on the left, and so on, as it swing back and forth in SHM.
Period of simple pendulum
(16-27)

27. 16-6 Pendulums The rotational inertia of the pendulum is (16-27)
(16-28)
(Period of simple pendulum, small amplitude)
漆安慎《力学》P285

28. 16-5 An Angular Simple Harmonic Oscillator
The Physical Pendulum: Is the real pendulum with complicated distribution of mass (Fig ).
the gravitational force acts at its center of mass C at a distance h from the pivot point O.
simple pendulum
tangetial component
moment arm
period
for small
in SHM in SHM
physical pendulum

29. 16-5 An Angular Simple Harmonic Oscillator
If a physical pendulum and a simple pendulum has the same period T,
center of oscillation
the length of the simple pendulum is L0,
the point along the physical pendulum at distance L0 from the pivot point O is called the center of oscillation of the physical pendulum for the given suspension point.

30. 16-5 An Angular Simple Harmonic Oscillator
Measuring
Take the pendulum to be a uniform rod of length L, suspended from one end. The period of the physical pendulum is
(16-29)
(16-30)
Sample Problem 16-5 P357
(16-32)
(16-31)

31. 16-7 Simple Harmonic Motion and Uniform Circular Motion
Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs.
(16-3)
Vector rotates counterclockwise with uniform angular speed , at time it makes an angle of
with axis. The projection of ’ end( )
on the axis is point P and its displacement is
( SHM )
(16-6)
The speed of is ; its projection on the axis is
( SHM )
The radial acceleration of is ; its projection on the axis is
(16-7)
( SHM )

32. 16-8 Damped Simple Harmonic Motion
When the motion of an oscillator is reduced by an external force, the oscillator and its motion are said to be damped.
Fig shows a damped oscillator, where a block (m) oscillates vertically on a spring (k). A rod and a vane (massless) is fixed to the block. The vane is submerged in a liquid. As the vane moves up and down, the liquid exerts a drag force on the osillating system.
The forces acting on the system:
(16-37)
damping force
(for small v)
b is a damping constant. SI unit: kg/s
Newton’s second law:
restoring force
The gravitational force is negligible compared to and .
(16-38)

33. 16-8 Damped Simple Harmonic Motion
Newton’s second law:
(16-38)
Substituting for and for ,
get the differential equation
(16-39)
(16-40)
The solution of this equation is If
(16-3)
damped oscillator angular frequency If
undamped oscillator angular frequency
mechanical energy
mechanical energy

34. 16-9 Forced Oscillations and Resonance
Two angular frequencies are associated with a system undergoing forced (driven) oscillations .
1. the natural angular frequency of the system
2. the angular frequency of the external driving force causing the driven oscillations.
Such a forced oscillator oscillates at the angular
frequency of the driving force, and its displacement is
(16-43)
The amplitude depends on a com- plicated function of and The velocity amplitude of the oscilla-tions is greatest when
(16-44)
(resonance)

35. 16-9 Forced Oscillations and Resonance
(16-44)
(resonance)
this condition is called resonance
During resonance the amplitude of the oscilla-tions is also approximately greatest.
Fig shows hwo the displacement amplitude of an oscillator depends on the angular frequency
of the driving force, for three values of the damping coefficient
Note that for all three the amplitude is approximately greatest when that is, when the resonance condition
is satisfied.
The curves show that less damping gives a taller and narrower rasonance peak.

36. 9.4 同方向不同频率简谐振动的合成
漆安慎《力学》P299

37. 9.4同方向不同频率简谐振动的合成 拍的形成
漆安慎《力学》P302

38. 9.4 振动合成原理的模拟 方波的傅立叶合成与分解
9.4 振动合成原理的模拟 方波的傅立叶合成与分解
漆安慎《力学》P308

39. 9.4相互垂直的简谐振动的合成 利萨如图形1
漆安慎《力学》P307
1:1
1:2
1:3

40. 9.4相互垂直的简谐振动的合成 利萨如图形2
漆安慎《力学》P307
1:2
2:3
4:5