Mechanical Optical Laser
Electrical
AC 50 Hz
Astronomical
Biological
Bunjee jumping
Metronom
ECG
Pulsar
PERIODIC MOTIONS
PERIODIC MOTIONS
Oscillation:
A physical quantity changes periodically in space and/or time. Movement, state or change that has a periodic component.
Harmonic oscillations:
Oscillation with a single frequency. It can be described with sine (or cosine) function. Constant amplitude and period time. (Any motion that repeats itself in equal intervals of time is called periodic or
harmonic motion.)
If a particle in periodic motion moves back and forth over the same path, we call the motion oscillatory or vibratory.
HOOKE’S LAW
When an object is bent, streched or compressed by a displacement s, the restoring force F is directly proportional to the displacement (provided the elastic limit is not exceeded). F~-s 𝐹 = −𝑘𝑠 (restoring force ~ displacement) 𝑘: elastic constant
SIMPLE HARMONIC MOTION (SHM)
A body is moving with SHM if: 1. its acceleration is directly proportional to its distance from a fixed point on its path
and 2. its acceleration is always directed towards that point.
Force + acceleration
maximum
Velocity zero
𝑎 = −𝜔2𝑠(= −𝜔2r) 𝑎: acceleration of a particle 𝑠 (𝑟): displacement of the particle from the fixed point O 𝜔2: is a constant
Any system that obeys Hooke’s law will execute SHM.
KINETICS OF PERIODIC MOTIONS I. Oscillation: vertical projection of circular motion
https://www.youtube.com/watch?v=P-Umre5Np_0
https://www.youtube.com/watch?v=ipSWPnBlsD4
KINETICS OF PERIODIC MOTIONS II. Oscillation: vertical projection of circular motion
KINETICS OF PERIODIC MOTIONS IV.
CIRCULAR MOTION OSCILLATION
𝑣𝑐 =2𝜋𝑟
𝑇
𝑣𝑐 = rω
𝑣 𝑡 = 𝑣0 cos(𝜔𝑡)
𝑣0 = 𝐴𝜔
Acceleration
𝑎𝑐 = 0 𝑎𝑐𝑝 = 𝑟𝜔
2
𝑎 𝑡 = −𝑎0sin (𝜔𝑡) 𝑎0 = −𝐴𝜔
2
𝑎 = −𝐴𝜔2 sin 𝜔𝑡 = −𝑦𝜔2
Middle-position: End-position: 𝛼 = 𝜔𝑡 = 0°, 𝑣 = 𝑣0 cos 𝜔𝑡 = 𝑣0
Middle-position: 𝛼 = 𝜔𝑡 = 0°, 𝑎 = −𝑎0 sin 𝜔𝑡 = 0
𝛼 = 𝜔𝑡 = 90°, 𝑣 = 𝑣0 cos 𝜔𝑡 = 0 End-position:
𝛼 = 𝜔𝑡 = 90°, 𝑎 = −𝑎0 sin 𝜔𝑡 = −𝑎0
Wh
at
is t
he
acc
ela
rati
on
in t
he
mid
dle
- a
nd
en
d-p
osi
tio
n?
Wh
at
is t
he
velo
city
in t
he
mid
dle
- a
nd
en
d-p
osi
tio
n?
Vel
oci
ty
Acc
eler
atio
n
a
2Dx mr
DYNAMICS OF PERIODIC MOTIONS I. Force
𝐹 = 0
𝐹𝑔 = 𝐹𝑝
𝐹 = 𝑚𝑎
𝐹 = 𝐹𝑠 − 𝐷𝑥
𝐷 =𝐹
𝑥 (𝑁
𝑚)
Spring constant (D): force necessary for spring elongation.
Harmonic oscillations: displacement is proportional to the force acting on the oscillating body but its direction is opposite.
2
D
T m
2 Dx
mr
2m
TD
Dx ma
F
ω
Oscillation frequency depends only on the mass (m) and the spring constant (D), but not from the amplitude.
DYNAMICS OF PERIODIC MOTIONS II. Energy
Potential and kinetic energy changes as a body moves with SHM.
SUPERPOSITION
All kind of periodic and non-periodic oscillation could be described as the sum (or integral) of individual sinusoidal oscillations (with different frequence, amplitude or phase)
Combinations of Harmonic Motions-Interference: The phenomenon in which two (or more) waves superpose each other to form a resultant wave Two linear simple harmonic motions combined (same and perpendicular directions). The resulting motion is the sum of two independent oscillations.
DAMPING (DAMPED OSCILLATIONS)
In damped harmonic motion the mechanical energy approaches zero as time increases, being transformed into internal thermal energy associated with the damping mechanism.
DRIVEN OSCILLATION, RESONANCE
Free (self-) oscillation: oscillating system without external force oscillate with its self-frequency
Driven oscillation: the oscillation is driven by external, periodic force oscillation frequency = external excitation frequency
amplitude, phase could be different
f f0
A
Driving frequency
Resonance: driving frequency is in the nearby range of the self-frequency.
SUMMARY
• Oscillation is the vertical projection of a circular motion.
• The displacement, speed and acceleration changes periodically in time.
• Periodic movements can be described with sine or cosine functions.
• The self-oscillating frequency depends only from the spring constant and the mass.
• The maximal elastic energy equals the maximal kinetic energy in an oscillating system.
• Oscillations can be added: interference, superposition.
• Without external force the oscillation get damped.