1. Chapter 16 Oscillations Guitar string; drum; diaphragms in telephones and speaker systems; quartz crystal in wristwatches …. Oscillations in the real world are usually damped; that is, the motion dies out gradually, transferring mechanical energy to thermal energy by the action of frictional forces. Although we cannot totally eliminate such loss of mechanical energy, we can replenish the energy from some source. For example, by swinging your legs or torso you can “pump” a swing to maintain or increase the oscillations. In doing this, you transfer biochemical energy to mechanical energy of the oscillating system. Oscillations motions that repeat themselves. Fundamental physical phenomena

2. §1 Simple Harmonic Motion A “snapshots” of a simple oscillating system, a particle moving repeatedly back and forth about the origin of an x axis. (a)A sequence of “snapshots” (taken at equal time intervals) showing the position of a particle as it oscillates back and forth about the origin along an x axis, between the limits +x m and -x m. The vector arrows are scaled to indicate the speed of the particle. (b) A graph of x as a function of time for the motion of (a).

3. One important property of oscillatory motion is its frequency, or number of oscillations that are completed each second. The symbol for frequency is f, and its SI unit is the hertz (abbreviated Hz), where 1 hertz=1 Hz=1 oscillation per second=1 s -1. Related to the frequency is the period T of the motion, which is the time for one complete oscillation (or cycle); that is, Any motion that repeats itself at regular intervals is called periodic motion or harmonic motion.

4. A simple harmonic motion is described by following function where x m, , and  are constants. They are basic quantities of a simple harmonic motion. x(t) is a period function, that means x(t) = x(t +T). For simplicity, we consider  =0. Then we have

5. The cosine function first repeats itself when its argument (the phase) has increased by 2  rad (radian), then we have that is

6. T=T’ x m ’>x m  =  ’=0 T=2T’ x m ’=x m  =  ’=0 T=T’ x m ’=x m

7. The Velocity of SHM From the definition The Acceleration of SHM In SHM, the acceleration is proportional to the displacement but opposite in Sign, and the two quantities are related by the square of the angular frequency. a(t)= -   x(t).

8. Mathematically, the spring force According to Newton’s second law SHM is the motion executed by a particle of mass m subject to a force that is proportional to displacement of the particle but opposite in sign. Sample Problem 16-2 :

9. §2 Energy in Simple Harmonic Motion Potential Energy Kinetic Energy or Obviously, the total energy is given by

10. = constant Sample Problem:A block of mass M, at rest on a horizontal frictionless table, is attached to a rigid support by a spring of constant k. A bullet of mass m and velocity strikes the block as shown in the figure. The bullet is embedded in the block. Determine (a) the speed of the block immediately after the collision and (b) the amplitude of the resulting simple harmonic motion.

11. Solution: The problem consists of two distinct parts: the completely inelastic collision (which is assumed to occur instantaneously, the bullet embedding itself in the block before the block moves through significant distance) followed by simple harmonic motion (of mass m +M attached to a spring of spring constant k). (a) Momentum conservation readily yields (b) Since v’ occurs at the equilibrium position, then v’ = v m for the simple harmonic motion. The relation v m =  x m can be used to solve for x m, or we can pursue the alternate (though related) approach of energy conservation. Here we choose the latter:

12. which simplifies to §3 Pendulums A massless string of length L is fixed at a ceiling. An apple is hanged at the other end, and swings back and forth a small distance. The simple Pendulum

13. Forces acting on the apple are: The apple swings with a very small angle .  =0 is the so called equilibrium position. It is easy to see that The torque of the system is where the minus sign indicates that the torque reduces .

14. According to  = I  mL 2  is the rotational inertia. We have Considering that  is very small, and sin  ≈  we have with The equation is the same as the simple harmonic oscillation.

15. The Physical Pendulum A real pendulum, usually called a physical pendulum, can have a complicated distribution of mass, much different from that of a simple pendulum. Does a physical pendulum also undergo SHM? If so, what is its period?

16. According to the same method as discussed in simple pendulum, one can have A physical pendulum can be used to measure the gravitational acceleration. How to do that? For a uniform rod, I=(1/3)mL 2, and h=L/2. So we have It is equivalent to a simple pendulum of L 0 =2L/3.

17. §4 Simple Harmonic Motion and Uniform Circular Motion Considering various relations of a simple harmonic motion Position: Velocity: Acceleration: You may find that the equations look like to have some relations with a circular motion. Actually, Galileo first discovered a simple harmonic motion from his observation of four moons of Jupiter, each of them moves forth and back relative to the planet. (see book.)

18. Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs. In figure (a), the radium of the circle is x m, the projection in the x axis of the position vector of point P’ at time t can be written as follows

19. Similarly, from the figures (b) and (c), we can have the corresponding projections on x axis are and respectively. §5 Damped Simple Harmonic Motion A block with mass m oscillates vertically on a spring with spring constant k. From the block, a rod extends to a vane (both assumed massless) that is submerged in a liquid. As the vane moves up and down, the liquid exerts an inhibiting drag force on it and thus on the entire oscillating system.

20. The liquid exerts a damping force d that is proportional in magnitude to the velocity of the vane and block (an assumption that is accurate if the vane moves slowly). Then, for components along the x axis, we have where b is a damping constant that depends on the characteristics of both the vane and the liquid and has the SI unit of kilogram per second. The minus sign indicates that opposes the motion. Then we have a damped harmonic motion, This equation can be solved analytically and the result is

21. We can regard the equation as a cosine function whose amplitude, whish is, gradually decreases with time. where

22. §6 Forced Oscillation and Resonance For a swing, if someone pushes the swing periodically, the swing has forced, or driven, oscillations. Two angular frequencies are associated with a system undergoing driven oscillations: (1) the natural angular frequency  of the system, which is the angular frequency at which it would oscillate if it were suddenly disturbed and then left to oscillate freely, and (2) the angular frequency  d of the external driving force causing the driven oscillations. The general equation is

23. One special solution is where the constant is to be determined. In other words, we might suppose that if we kept pushing back and forth, the mass would follow back and forth in step with the force. For the case b=0, it is easy to have That is, m oscillates at the same frequency as the force, but with amplitude which depends on the frequency of the force, and also upon the frequency of the natural motion of the oscillator.

24. §7 Superposition of Oscillations Superposition of Two Simple Harmonic Oscillation in the Same Direction I. With the same frequencies where

25. 11   The resultant oscillation is related to the phase difference  2 -  1. We have Then we have maximum oscillation amplitude. If

26. we have II. With two different frequencies Suppose x m1 = x m2 = A, we have

27. Superposition of Two Perpendicular Simple Harmonic Oscillation By eliminating the parameter t, This is an ellipse equation. The path of the particle depends on the phase difference  2 -  1 : (1)  2 -  1 =0: y x

28. (2)  2 -  1 =  (3)  2 -  1 =  /2 If x m =y m =A,

29. Assignments: 16 — 9E 16 — 19p 16 — 37E