14oscillatory 2

X
A and the
0
x
O
A
on x axis at the time t
◆ the direction of the velocity:
The vector A is on the upside of x axis —— v<0 The vector A is on the underside of x axis —— v>0
x A1 cos(t 1 ) y A2 cos(t 2 )
The coordinate of the particle is (x, y) at any time t. (1) =2-1 = 0 : straight-line equation
A2
y
x y A1 A2
x 2 y 2 2 xy 2 cos(2 1 ) sin 2 (2 1 ) A12 A2 A1 A2
=2-1
= 0 y = /4
P
· Q ·
= /2 = 3/4
x
= 5/4 = 3/2 = 7/4
=
T
2


t
1 f T 2

: the phase constant (phase angle); is determined by the initial conditions.
3. The phasor diagram for simple harmonic motion:
x o
x1
x2 t
3T/8
Solution:
2 3T 3 t T 8 4
2. The superposition of two simple harmonic motions along a straight line with the different frequency. (P486-488) The amplitudes of two simple harmonic motions are the same.
Attention:
1> The net motion is a simple harmonic motion.
2> The x direction and frequency of the net motion do not change.
3> in phase and out of phase (同相和反相)
(2) = : straight-line equation
A2
y
x y A1 A2
(3) = /2 : elliptic equation
A1
A2
A1
x
x2 y 2 2 1 2 A1 A2
(4) general condition: elliptic equation
s x2 y2
2 A12 A2 cos( t )
A1
A2
A1
x
Net oscillation: simple harmonic oscillation, same frequency, net amplitude A12 A22
x A1 cos(t 1 ) y A2 cos(t 2 )

f
: the angular frequency (rad /s)
: the frequency (Hz) is determined by the system. is called natural frequency. : the phase indicates the state of motion.
Lissajous figure: If the ratio of the periods of two simple harmonic motions is the ratio of simple integer, the path of the particle is stable and closed, called Lissajous figure.
x1
2
x2 4 t
O
A
2
A
1
X A1-A2
(0= -/2)
Example 3: the x-t curves of two simple harmonic motions. The phase of the oscillation x1 is leading in comparison with the oscillation x2.
1. The simple harmonic motion:
Fs kx 2 d x 2x 0 2 dt
x(t ) A cos t
A particle attached to a spring
k m
The simple pendulum
The physical pendulum
2 2 A A1 A2 2 A1 A2 cos( 2 1 )
(1) in phase
2 1= 2k
(k=0, 1, 2,…)
so A A 1 A 2 , the net amplitude is maximum. (2) out of phase 2 1= (2k+1) (k=0, 1, 2,…)
A A A 2 A1 A2 cos( 20 10 )
A1 sin 10 A2 sin 20 Phase constant: tg A1 cos 10 A2 cos 20
Angular frequency:
x x1 x2 A cos(t )
（-3/4） （-/2） （-/4）
4. The superposition of two simple harmonic motions with the different frequency, and the directions of the motion are mutually perpendicular. The general path is complicated and unstable.
Chapter 12 Oscபைடு நூலகம்llatory Motion
§1. Simple harmonic motion (P404-413, 417-420) §2. Energy consideration in Simple harmonic motion (P413-416)
§3. The superposition of Simple harmonic motion (P486-488)
1
(A1=0.05m) (A2=0.06m)
( 0 ) 2
=10 (s-1)
Example 2: the x-t curves of two simple harmonic motions. Find the phase constant of the net motion.
x
A
0 -A/2
y A 1
-A2
0
A2
x
Ny Nx
fx 3 Tx fy 2
Ty
- A1
x A cos( m t )
0 0

4m
y A cos(n t 0 )
3 4m

2m

m
n:m
1:1
1:2 1:3
5 4m
3 2m
7 4m
2 m

4m
2:3
3:4
Decomposition of motion: A circular and elliptic motion can be decomposed into two simple harmonic motion whose directions of the oscillation are perpendicular. This can be used in the polarization of light.
2
t cos
1 2
2
t
1 2
, and both are very big.
x A(t ) cos t
A(t ) 2 A0 cos
1 2
2
t
x1
f1
t
beat frequency (拍频): the number of beats per second.
x2
f2
t
1 f 2 2
§4. Damped oscillations One common type of resistive force is proportional to the velocity and acts in the direction opposite the velocity. b is a constant. R bv

g L
m gd I
2. Mathematical representation of simple harmonic motion.
x(t ) A cos t
A : the amplitude 1 2 indicates the energy of the system. E K U kA 2 is determined by the initial conditions. T : the period
x A cos(t 0 )
◆ phase: = the angle between positive direction of x axis at the time t ◆ displacement: x(t) = the projection of