. . . . .Chapter 9 OscillationsWe are surrounded by oscillationsmotions that repeat themselves. (1). There are swinging chandeliers, boats bobbing at anchor, and the surging pistons in the engines of cars. (2). There are oscillating guitar strings, drums, bells, diaphragms in telephones and speaker systems, and quartz crystals in wristwatches. (3). Less evident are the oscillations of the air molecules that transmit the sensation of sound, the oscillations of the atoms in a solid that convey the sensation of temperature, and the oscillations of the electrons in the antennas of radio and TV transmitters.Oscillations are not confined to material objects such as violin strings and electrons. Light, radio waves, x-rays, and gamma rays are also oscillatory phenomena. You will study such oscillations in later chapters and will be helped greatly there by analogy with the mechanical oscillations that are about to study here.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 force. Although we cannot totally eliminate such loss of mechanical energy, we can replenish the energy from some source.9.1 Simple Harmonic Motion1. The figure shows a sequence of “snapshots” of a simple oscillating system, a particle moving repeatedly back and forth about the origin of the x axis.2. Frequency: (1). One important property of oscillatory motion is its frequency, or number of oscillations that are completed each second. (2). The symbol for frequency is f, and (3) its SI unit is hertz (abbreviated Hz), where 1 hertz = 1 Hz = 1 oscillation per second = 1 s-1.3. Period: Related to the frequency is the period T of the motion, which is the time for one complete oscillation (or cycle). That is .4. Any motion that repeats itself at regular intervals is called period motion or harmonic motion. We are interested here in motion that repeats itself in a particular way. It turns out that for such motion the displacement x of the particle from the origin is given as a function of time by , in which are constant. The motion is called simple harmonic motion (SHM), the term that means that the periodic motion is a sinusoidal of time.5. The quantity , a positive constant whose value depends on how the motion was started, is called the amplitude of the motion; the subscript m stands for maximum displacement of the particle in either direction. 6. The time-varying quantity is called the phase of the motion, and the constant is called the phase constant (or phase angle). The value of depends on the displacement and velocity of the particle at t=0.7. It remains to interpret the constant . The displacement must return to its initial value after one period T of the motion. That is, must equal for all t. To simplify our analysis, we put . So we then have . The cosine function first repeats itself when its argument (the phase) has increased by rad, so that we must have . It means . The quantity is called the angular frequency of the motion; its SI unit is the radian per second. 8. The velocity of SHM: (1). Take derivative of the displacement with time, we can find an expression for the velocity of the particle moving with simple harmonic motion. That is, . (2). The positive quantity in above equation is called the velocity amplitude. 9. The acceleration of SHM: Knowing the velocity for simple harmonic motion, we can find an expression for the acceleration of the oscillation particle by differentiating once more. Thus we have The positive quantity is called the acceleration amplitude. We can also to get , which is the hallmark of simple harmonic motion: the acceleration is proportional to the displacement but opposite in sign, and the two quantities are related by the square of the angular frequency. 9.2 The Force Law For SHM1. Once we know how the acceleration of a particle varies with time, we can use Newtons second law to learn what force must act on the particle to give it that acceleration. For simple harmonic motion, we have . This result-a force proportional to the displacement but opposite in sign-is something like Hooks law for a spring, the spring constant here being .2. We can in fact take above equation as an alternative definition of simple harmonic motion. It says: Simple harmonic motion is the motion executed by a particle of mass m subject to a force that is proportional to the displacement of the particle but opposite in sign.3. The block-spring system forms a linear simple harmonic oscillator (linear oscillator for short), where linear indicates that F is proportional to x rather than to some other power of x. (1). The angular frequency of the simple harmonic motion of the block is . (2). The period of the linear oscillator is .9.3 Energy in Simple Harmonic Motion1. The potential energ