Chapter 14 Oscillations

1. Chapter 14 Oscillations To understand the physics and mathematics of oscillation. To draw and interpret oscillatory graphs. To learn the concepts of phase and phase constant To understand and use energy conservation in oscillatory system To understand the basic ideas of damping and resonance.

2. Stop to think 14.1 page 414Stop to think 14.2 page 417Stop to think 14.3 page 419Stop to think 14.4 page 423Stop to think 14.5 page 428 • Example 14.2 page 413 • Example 14.4 page 417 • Example 14.6 page 422 • Example 14.7 page 424 • Example 14.9 page 426 • Example 14.10 page 428

3. Simple Harmonic Motion • Object or systems of objects that undergo oscillatory motion are called oscillators. All these oscillators have two things in common: • 1. The oscillation takes place about an equilibrium position, and • 2. The motion is periodic. Sinusoidal oscillation Is called simple harmonic motion.

4. Period and frequency • Period T: time per cycle, units: second • Frequency f: the number of cycles per second. units: 1/s = Hz (Hertz)

5. Graph of simple harmonic motion • The amplitude A: the maximum displacement from equilibrium. • Measured A = 0.17 cm • Measured T = 1.60s • How to describe the displacement x using A, T, and t.

6. Position –vs time graph and velocity vs time graph • Position-vs time graph Velocity vs time graph

7. Angular frequency ω We define ω=2π/T= 2πf, is called angular frequency V(max) = ωA

8. Simple Harmonic Motion and Circular Motion. • Uniform circular motion projected onto one dimension is simple harmonic motion • The figure shows the x-component, when the particle does uniform circular motion • With • So ;

9. The Phase constant • In more general case, particle start phase Φo is not zero. then, • The harmonic motion function is • Φo is called the phase constant or initial phase. is called phase. When t = 0, initial condition

10. Show phase constant • The following show the oscillations by different phase constant Notice: Φo=π/3 and Φo=-π/3 have the same starting x, but different Vo

11. P14.2 • From the Figure, how we get • Amplitude • Frequency • Phase constant. • First, you write general Harmonic Wave function: Then you compare this trigonometric Function and the figure, you can get • A = 10 cm • T = 2 s, frequency f = ½ =0.5 Hz. • When t = 0 x(0) = 5cm = 10cos(Φo) cos(Φo)=0.5, Φo=±π/3. But at t = 0, the slope of curve is negative So V0 is negative, from Sin(Φo) is positive, that makes Φo=π/3.

12. Energy in simple Harmonic Motion • The mechanical energy of an object oscillating on a spring is • When x = ±A, E = ½ kA2 +0 • When x = 0 E = 0 + 1/2mV2max • From conservation of energy

13. The Dynamics of Simple Harmonic Motion • The spring force is • From Newton’s second Law • The dynamics equation: • This is second derivative equation, the solution is

14. Vertical oscillations • The equilibrium position, ΔL. • The harmonic oscillation equation should be the same on a horizontal spring. • In right figure: K = 10N/m, The spring stretch at equilibrium is given by ΔL=mg / K = 19.6 cm That is the amplitude of oscillation A = 30cm-19.6cm = 10.4 cm The initial condition y0=-A =AcosΦo Φo=π. So the oscillator function is

15. The Pendulum • Let’s look another oscillator: a pendulum

16. Small-angle Approximation • The Dynamical equation is • Using • We can write • If θ is very small sin (θ) ~ θ (θ in radians) • Then • Solution is • Or or

17. The Physical pendulum

18. Period vs Meff in log plot

19. V=-10π sin(πt+π/3)