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1. 10.05.2024 1
Mechanical oscillations
1. According to oscillating object: mechanical and
electromagnetic oscillations;
2. According to the type of motion: translational and rotational
oscillations;
3. According to the external action: free and forced oscillations;
4. Non-damped and damped oscillations (according to time).
Period of the oscillations (T, s);
frequency (
T
1

 ; Hz = s-1).

2. 10.05.2024 4
Simple harmonic motion (SHM)
)
cos( 0
0 
 
 t
A
x
-A A
0
X
t
T
0
A
-A
A 0
0 

 
 t - phase of oscillation;
0
0
2
T

 
0
0 2
 
- angular frequency.

3. 10.05.2024 5
Displacement, velocity and acceleration
x
t
T0/2
-A
A
3T0/2
 
0
0
cos 
 
 t
A
x
t = 0: x = A    1
cos 0
0 

 t
0
0
0 

 t 0
0 

 
0
0
0 sin 

 


 t
A
dt
dx
v
v
t
vm
-vm
dt
dv
dt
x
d
a 
 2
2
)
cos( 0
0
2
0 

 

 t
A
x
a 2
0



a
t
-am
am

4. 10.05.2024 6
x
a 2
0



ma
F 
x
m
F 2
0


 ;
2
0

m
k  kx
F 

;
0
m
k


k
m
T 
2
0 

5. Quick Check
• Identical blocks oscillate on the end of a
vertical spring, one on Earth and one on the
Moon. Where is the period of the
oscillations greater?
A. on Earth
B. on the Moon
C. same on both Earth and Moon
D. cannot be determined from the information
given
May 10, 2024 7

6. Quick Check
A block of mass m oscillates on a horizontal
spring with period T = 2.0 s. If a second identical
block is glued to the top of the first block, the
new period will be
A. 1.0 s
B. 1.4 s
C. 2.0 s
D. 2.8 s
E. 4.0 s

7. A block of mass m oscillates on a horizontal
spring with period T = 2.0 s. If a second identical
block is glued to the top of the first block, the
new period will be
A. 1.0 s
B. 1.4 s
C. 2.0 s
D. 2.8 s
E. 4.0 s
Quick Check

8. 10.05.2024 12
Mathematical pendulum
P

T

N
F

F

x
mg
F



;
0 
g


g
T 

2
0 

9. 10.05.2024 16
Energy of oscillation
;
2
2
mv
WK   
0
0
2
2
0
2
sin
2


 
 t
A
m
WK
 
)
(
2
cos
1
4
0
0
2
0
2


 

 t
A
m
WK
,
kx
F 
  


0
2
*
2
x
x
k
kxdx
A
;
*
A
WP  )
(
cos
2
0
0
2
2

 
 t
A
k
WP
))
(
2
cos
1
(
4
0
0
2

 

 t
A
k
WP

10. Energy of Mass on a Spring
May 10, 2024 17

11. 10.05.2024 18
W=WK+WP
WP(t)
WK(t)
kA2/2
kA2/4
0
0 T/2 T
W
W = WK + WP = const

12. Quick Check
A block oscillates on a very long
horizontal spring. The graph
shows the block’s kinetic energy
as a function of position. What is
the spring constant?
A. 1 N/m
B. 2 N/m
C. 4 N/m
D. 8 N/m
E. I have no idea.

13. A block oscillates on a very long
horizontal spring. The graph
shows the block’s kinetic energy
as a function of position. What is
the spring constant?
A. 1 N/m
B. 2 N/m
C. 4 N/m
D. 8 N/m
E. I have no idea.
Quick Check

14. • A 2.00-kg mass attached to a spring is
displaced 8.00 cm from the equilibrium
position. It is released and then oscillates
with a frequency of 4.00 Hz.
• What is the energy of the motion when the
mass passes through the equilibrium
position?
• What is the speed of the mass it is 2.00 cm
from the equilibrium position?
10.05.2024 21

15. 10.05.2024 22
Combination of oscillations
x

)
0
( 
t
A

0
)
0
( 
t
A

)
cos( 01
01
1
1 
 
 t
A
x
)
cos( 02
02
2
2 
 
 t
A
x
0
02
01 

 

)
cos( 0
0 
 
 t
A
x
)
cos(
2 01
02
2
1
2
2
2
1
2

 


 A
A
A
A
A
02
2
01
1
02
2
01
1
0
cos
cos
sin
sin





A
A
A
A
tg




16. 10.05.2024 23
Combination of two SHM with equal
frequencies
x
1
A

2
A


A

)
cos( 01
01
1
1 
 
 t
A
x
)
cos( 02
02
2
2 
 
 t
A
x
x1
x2

17. 10.05.2024 24
I. A1 = A2.
 
0
02
01
2
1
1
2
1
0
0
2
2
2










 










t
t
:
A

;
2
02
01
0





   
2
cos
2
cos
2 01
02
1
1
2
2
1
2
2
2
1
t
A
A
A
A
A
A










- not a SHM anymore !









 0
02
01
01
02
1
2
cos
2
cos
2 




t
t
A
x

18. 10.05.2024 25









 0
02
01
01
02
1
2
cos
2
cos
2 




t
t
A
x
A: ;
2 1
max A
A  0
min 
A
01
02 

 

p
p
p
T


2

2A1
-2A1
Tp/2 3Tp/2
t

19. 10.05.2024 26
2
1 A
A 
II.
2
1
max A
A
A 

2
1
min A
A
A 

A1+A2
A1-A2
-(A1-A2)
-(A1+A2)
- amplitude modulation

20. 10.05.2024 27
Combination of perpendicular
oscillations
)
cos( 0
1 
 
 t
A
x
)
cos( 0
2 
 
 t
A
y A2
A1
y
x
I. 0
02
01 

 

0
01
02 


,
2
1
A
A
y
x
 vai x
A
A
y
1
2


21. 10.05.2024 28
2
01
02


 

II.
t
A
x 
cos
1









2
cos
2

t
A
y
t
t 

 sin
2
cos 








1
sin
cos 2
2
2
2
1
2
2
2






A
y
A
x
t
t 

y
x
2
1
. 
 
III - Lissajous figures.

22. • https://mathcurve.com/courbes2d.gb/lissajo
us/lissajous.shtml
10.05.2024 29

23. May 10, 2024 30
Quick Check
• What is the expression relating the net
force Fnet and the acceleration a for a mass
m being acted upon by a linear restoring
force Fs plus a damping force Fγ that
depends on velocity?
A. Fnet = Fs + Fγ = ma
B. Fnet = Fs – Fγ = ma
C. Fnet = Fs = ma
D. Fnet = Fγ= ma
E. Fnet = 0
Newton’s Second Law!
F = ma
Fnet = ma = Fs + Fγ

24. 10.05.2024 31
Damped oscillations
dt
dx
r
rv
FPR 


 and ,
kx
F 

then m
a
F
F




 2
1
dt
dx
r
kx
dt
x
d
m 


2
2
- equation of damped
oscillations  
0
cos
)
( 
 
 t
t
A
x t
e
A
t
A 

 
0
)
(
x
t
t
e
A 

0


m
r
2
 damping coefficient
;
)
2
(
2
2
0
2
2
2


 



m
r
m
k
2
2
0
2





T

25. 10.05.2024 32
Energy of damped oscillations:
t
t
e
W
e
A
W 


 
 2
0
2
2
0 ~
~

26. A vertical spring with a spring constant of
2.00 N/m has a 0.3-kg mass attached to it, and
the mass moves in a medium with a damping
constant of 0.025 kg/s. The mass is released
from rest at a position 5.00 cm from the
equilibrium position.How long does it take for
the amplitude to decrease twice? How long it
take for the total energy to decrease twice?
10.05.2024 33

27. 10.05.2024 34
Forced oscillations
3
2
1 F
F
F
m
a







t
F
dt
dx
r
kx
dt
x
d
m 
cos
0
2
2



 - equation for forced
oscillations.
 
;
4
1
2
2
2
2
2
0
0



 


m
F
A ;
2
0
m
k

 .
2m
r


2
2
0
0
2







tg
)
cos( 0

 

 t
A
x )
(
f
A 

28. 10.05.2024 35
0

A
R
AR
1=0
2>1
3>2
2
0
4

 
A0

29. • Tacoma bridge collapse:
• https://www.youtube.com/watch?v=Xggxeu
FDaDU
10.05.2024 36

30. Waves
• If you throw a stone in a still pond,
it creates circular ripples that travel
along the surface of the water,
radially outward.
• If an object is floating on the
surface of the pond, it moves up and
down as the ripples move outward
underneath it, but it does not move
outward with the wave.
• If you tie a rope to a wall,
hold one end, and rapidly
move your arm up and down,
you will create a wave crest
that travels down the rope.
May 10, 2024 37

31. Waves
• A wave is an excitation that propagates through space or
some medium as a function of time but does not generally
transport matter with it.
• Wave motion is different from anything we have studied.
• Electromagnetic waves and gravitational waves do not need a
medium in which to propagate.
• Other waves need a medium in which to propagate.
• Sound travels through air but not through a vacuum.
• Many example of waves surround us in everyday life.
• In this chapter, we will study the motion of waves through
media.
May 10, 2024 38

32. 10.05.2024 39
Mechanical waves
Transverse wave - motion of the matter
particles conveying the wave are perpendicular to
the direction of propagation of the waves itself:
V
Longitudinal wave - motion of the
particles is back and forth along the
direction of propagation:
V

33. 10.05.2024 40

34. May 10, 2024 41
Quick Check
• Fans at a football game are so excited that their team is
winning that they start “the wave” in celebration.
• Which of the following 4 statements is true:
I. This wave is a traveling wave.
II. This wave is a transverse wave.
III. This wave is a longitudinal wave.
IV. This wave is a combination of a longitudinal and
transverse wave.
A. I and II
B. II only
C. III only
D. I and IV
E. I and III

35. 10.05.2024 42
 - the wavelength of the wavetrain; distance
between two adjacent points in the wave having
the same phase.
vT




v


 

v

36. 10.05.2024 45
Wave equation
0

 
t
r
v

For spherical wave:
At time instant t
at r = 0: 0
)
,
0
( 

 
 t
t
at r = r: 0
)
(
)
,
( 

 


v
r
t
t
r
;
2
T

  :


vT   0
2
)
,
( 


 

 r
T
t
t
r
A = A(r): ;
1
~ 2
r
W A2  :
1
~
r
A .
r
B
A 
 
 
0
2
cos 

 







 r
T
t
r
B
y
Wave kinematical equation:

37. 10.05.2024 46
For plane wave:
0
x
0

 
t
n

v

 
 
0
2
cos 

 

 x
T
t
A
y

38. • A sinusoidal wave travels along a string. If
the time for a particular point to move from
maximum displacement to zero
displacement is 0.17 s, what are (a) the
period, and (b) frequency? (c) If the
wavelength is 1.4 m, what is the speed of
the wave?
10.05.2024 47

39. 10.05.2024 48
Wave energy
P
K W
W
W 

;
2
2
V
dt
dy
WK 








 V
dx
dy
v
WP 








2
2
2

V
dx
dy
v
dt
dy
W
W
W P
K 



























2
2
2
2


























2
2
2
2 dx
dy
v
dt
dy
V
W 
 - energy density.

40. May 10, 2024 49
Wave Power and Intensity
• Power = Energy/ time:
• The power radiated by a wave is constant in time (all terms
on right hand side are constants).
• The definition of intensity is the power radiated per cross
sectional area is:
I =
P
A^

41. Intensity and Decibels
 Human hearing spans an extremely wide range of
intensities, from the threshold of hearing at
≈ 1 × 10–12 W/m2 (at midrange frequencies) to the
threshold of pain at ≈ 10 W/m2.
 If we want to make a scale of loudness, it’s convenient
and logical to place the zero of our scale at the threshold
of hearing.
 To do so, we define the sound intensity level,
expressed in decibels (dB), as
where I0 = 1 × 10–12 W/m2.

42. Intensity and Decibels

43. 10.05.2024 52
Standing waves
in the string
;
2
n
l

 

v
;
2 S
T
l
n
n

  n = 1, 2, 3...
T – tension;
n = 1: 1 – basic frequency
n = 2, 3 …. – overtones.
Wave speed in a stretched string:
𝑣 =
𝑇
𝑚
𝑙

44. May 10, 2024 53
Superposition Principle
• When two or more waves are simultaneously present at a
point, the resultant displacement is the sum of the
displacements of the waves.
• This is called the superposition principle.
• Wave equations are linear so that if you find two different
solutions, y1(x,t) and y2(x,t), then any linear combination of
those solutions is also a solution:
• This linear property means that wave solutions can be
added, subtracted, or combined in any linear combination
and the result is again a wave solution.

45. Superposition of Wave Pulses
May 10, 2024 54

46. Doppler effect
• https://www.youtube.com/watch?v=a3RfU
Lw7aAY
10.05.2024 55

47.  The frequencies heard by an observer moving at
speed v0 relative to a stationary sound source
emitting frequency f0 are
 The frequencies heard by a stationary observer
when the sound source is moving at speed v0 are
The Doppler Effect

48. The Doppler Effect
Doppler weather radar uses the Doppler shift of reflected radar
signals to measure wind speeds and thus better gauge the
severity of a storm.

49. A siren emits a sound wave with
frequency f0. The graph shows the
frequency you hear as you stand
at rest at x = 0 on the x-axis.
Which is the correct description of
the siren’s motion?
Quick Check
A. It moves from left to right and passes you at t = 2 s.
B. It moves from right to left and passes you at t = 2 s.
C. It moves toward you for 2 s but doesn’t reach you, then
reverses direction at t = 2 s and moves away.
D. It moves away from you for 2 s, then reverses direction
at t = 2 s and moves toward you but doesn’t reach you.

50. A siren emits a sound wave with
frequency f0. The graph shows the
frequency you hear as you stand
at rest at x = 0 on the x-axis.
Which is the correct description of
the siren’s motion?
A. It moves from left to right and passes you at t = 2 s.
B. It moves from right to left and passes you at t = 2 s.
C. It moves toward you for 2 s but doesn’t reach you, then
reverses direction at t = 2 s and moves away.
D. It moves away from you for 2 s, then reverses direction
at t = 2 s and moves toward you but doesn’t reach you.
Doppler shift to lower
frequency means it’s
moving away.
Quick Check

51. The Doppler Effect for Light Waves
 Shown is a Hubble Space
Telescope picture of a quasar.
 Quasars are extraordinarily
powerful and distant sources of
light and radio waves.
 This quasar is receding away
from us at more than 90% of the
speed of light.
 Any receding source of light is red shifted.
 Any approaching source of light is blue shifted.

52. Research on Waves
May 10, 2024 61
Electron matter waves Gravity waves