PREMED COURSE, 14/08/2015 OSCILLATIONS

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1 PREMED COURSE, 14/08/2015 OSCILLATIONS

2 PERIODIC MOTIONS Mechanical Metronom Laser Optical Bunjee jumping Electrical Astronomical Pulsar Biological ECG AC 50 Hz

3 Another biological exampe PERIODIC MOTIONS

4 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.

5 SINE AND COSINE FUNCTIONS

6 MEASURING ANGLES IN RADIAN θ = s r

7 ANGULAR VELOCITY P; Time t X; t=0 ω = θ t [rad/s] v = rω

8 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

9 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 = ω 2 s(= ω 2 r) a: acceleration of a particle s (r): displacement of the particle from the fixed point O ω 2 : is a constant Any system that obeys Hooke s law will execute SHM.

10 EXAMPLES OF BODIES MOVING WITH SHM

11 KINETICS OF PERIODIC MOTIONS I. Oscillation: vertical projection of circular motion

12 KINETICS OF PERIODIC MOTIONS II. Oscillation: vertical projection of circular motion

13 KINETICS OF PERIODIC MOTIONS III. (hertz) (sec)

Acceleration CIRCULAR MOTION v c = 2πr T v c = rω a c = 0 a cp = rω 2 v t v 0 = Aω OSCILLATION = v 0 cos(ωt) a t = a 0 sin (ωt) a

14 What is the accelaration in the middle- and end-position? What is the velocity in the middle- and end-position? Velocity Acceleration KINETICS OF PERIODIC MOTIONS IV. Acceleration CIRCULAR MOTION v c = 2πr T v c = rω a c = 0 a cp = rω 2 v t v 0 = Aω OSCILLATION = v 0 cos(ωt) a t = a 0 sin (ωt) a 0 = Aω 2 a = Aω 2 sin ωt = yω 2 Middle-position: α = ωt = 0, v = v 0 cos ωt = v 0 End-position: α = ωt = 90, v = v 0 cos ωt = 0 Middle-position: α = ωt = 0, a = a 0 sin ωt = 0 End-position: α = ωt = 90, a = a 0 sin ωt = a 0

Force DYNAMICS OF PERIODIC MOTIONS I. Harmonic oscillations: displacement is proportional to the force acting on the oscillating body but its direction is opposite.

15 Force DYNAMICS OF PERIODIC MOTIONS I. Harmonic oscillations: displacement is proportional to the force acting on the oscillating body but its direction is opposite. D = F x (N m ) F = 0 F g = F p F = ma F = F s Dx Spring constant (D): force necessary for spring elongation. a 2 Dx mr 2 Dx ω mr 2 D T m T Dx 2 F ma m D Oscillation frequency depends only on the mass (m) and the spring constant (D), but not from the amplitude.

16 THE SIMPLE PENDULUM T = 2π l g g: acceleration due to gravity; l: length

17 Energy DYNAMICS OF PERIODIC MOTIONS II. 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)

18 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.

19 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.

periodic force oscillation frequency = external excitation frequency amplitude, phase could be

20 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 A f 0 Driving frequency f Resonance: driving frequency is in the nearby range of the self-frequency.

21 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.

11/17/10. Chapter 14. Oscillations. Chapter 14. Oscillations Topics: Simple Harmonic Motion. Simple Harmonic Motion

11/17/10. Chapter 14. Oscillations. Chapter 14. Oscillations Topics: Simple Harmonic Motion. Simple Harmonic Motion 11/17/10 Chapter 14. Oscillations This striking computergenerated image demonstrates an important type of motion: oscillatory motion. Examples of oscillatory motion include a car bouncing up and down,