Chapter 14 Periodic Motion

Goals for Chapter 14 To describe oscillations in terms of amplitude, period, frequency and angular frequency To do calculations with simple harmonic motion To analyze simple harmonic motion using energy To apply the ideas of simple harmonic motion to different physical situations To analyze the motion of a simple pendulum To examine the characteristics of a physical pendulum To explore how oscillations die out To learn how a driving force can cause resonance

What causes periodic motion? If a body attached to a spring is displaced from its equilibrium position, the spring exerts a restoring force on it, which tends to restore the object to the equilibrium position. This force causes oscillation of the system, or periodic motion. Figure 14.2 at the right illustrates the restoring force Fx.

Characteristics of periodic motion The amplitude, A, is the maximum magnitude of displacement from equilibrium. The period, T, is the time for one cycle. The frequency, f, is the number of cycles per unit time. The angular frequency, , is 2π times the frequency:  = 2πf. The frequency and period are reciprocals of each other: f = 1/T and T = 1/f. Follow Example 14.1.

Simple harmonic motion (SHM) When the restoring force is directly proportional to the displacement from equilibrium, the resulting motion is called simple harmonic motion (SHM). An ideal spring obeys Hooke’s law, so the restoring force is Fx = –kx, which results in simple harmonic motion.

Simple harmonic motion viewed as a projection Simple harmonic motion is the projection of uniform circular motion onto a diameter, as illustrated in Figure 14.5 below. SHM

Characteristics of SHM For a body vibrating by an ideal spring: Follow Example 14.2 and Figure 14.8 below. (a) Find force constant k of the spring (b) Find angular frequency, frequency, and period of oscillation

Displacement as a function of time in SHM The displacement as a function of time for SHM with phase angle  is x = Acos(t + ). (See Figure at right.) Changing m, A, or k changes the graph of x versus t, as shown below. change f

Displacement, Velocity and Acceleration The displacement as a function of time for SHM with phase angle  is: As always, velocity is the time-derivative of displacement: Likewise, acceleration is the time-derivative of velocity (or the second derivative of displacement):

Graphs of displacement, velocity, and acceleration The graph below shows the effect of different phase angles. The graphs below show x, vx, and ax for  = π/3.

Behavior of vx and ax during one cycle Figure 14.13 at the right shows how vx and ax vary during one cycle. Example 14.3: A glider attached to a spring is given an initial displacement 0.015 m and a push with velocity +0.40 m/s. Find the period, amplitude and phase of the resulting motion. Write equations for the displacement, velocity and acceleration as a function of time.

E = ½ mvx2 + ½ kx2 = ½ kA2 = constant Energy in SHM The total mechanical energy E = K + U is conserved in SHM: E = ½ mvx2 + ½ kx2 = ½ kA2 = constant

Energy diagrams for SHM Figure 14.15 below shows energy diagrams for SHM.

Energy and momentum in SHM Example 14.5: A block of mass M attached to a horizontal spring with force constant k is moving in SHM with amplitude A1. As the block passes through its equilibrium position, a lump of putty of mass m is dropped from a small height and sticks to it. Find the new amplitude and period of the motion. Repeat part (a) if the putty is dropped onto the block when it is at one end of its path.

Angular SHM A coil spring (see Figure 14.19 below) exerts a restoring torque z = –, where  is called the torsion constant of the spring. The result is angular simple harmonic motion.

The simple pendulum Other systems can show SHM. Consider a simple pendulum that consists of a point mass (the bob) suspended by a massless, unstretchable string. If the pendulum swings with a small amplitude  with the vertical, its motion is simple harmonic, where the restoring force is the component of gravity along the arc of the motion.

The physical pendulum A physical pendulum is any pendulum that uses an extended body instead of a point-mass bob. For small amplitudes, its motion is simple harmonic. (See Figure 14.23 at the right.) Example 14.9: A uniform rod of length L is pivoted about one end, what is its period of oscillation?

Damped oscillations Real-world systems have some dissipative forces that decrease the amplitude. Such dissipative forces are typically proportional to the speed v, and appear in the force equation with minus sign: The decrease in amplitude is called damping and the motion is called damped oscillation. The mechanical energy of a damped oscillator decreases continuously. The general solution is:

Chapter 15 Mechanical Waves

Goals for Chapter 15 To study the properties and varieties of mechanical waves To relate the speed, frequency, and wavelength of periodic waves To interpret periodic waves mathematically To calculate the speed of a wave on a string To calculate the energy of mechanical waves To understand the interference of mechanical waves To analyze standing waves on a string To investigate the sound produced by stringed instruments

Types of mechanical waves A mechanical wave is a disturbance traveling through a medium. Figure 15.1 below illustrates transverse waves and longitudinal waves.

Periodic waves For a periodic wave, each particle of the medium undergoes periodic motion. The speed of the wave is not the same as the speed of the particles. The wavelength  of a periodic wave is the length of one complete wave pattern. The speed of any periodic wave of frequency f is v = f. Example 15.1: The speed of sound in air at 20° C is 344 m/s. What is the wavelength of a sound wave in air at 20° C if the frequency is 262 Hz?

Periodic transverse waves For the transverse waves shown here in Figures 15.3 and 15.4, the particles move up and down, but the wave moves to the right. This difference in direction of the waves and particles is why the wave is called a transverse wave. Note that the restoring force is transverse to the direction of the wave propagation.

Periodic longitudinal waves For the longitudinal waves shown here in Figures 15.6 and 15.7, the particles oscillate back and forth along the same direction that the wave moves. The restoring force (pressure) is in the same direction as the wave propagation.

Mathematical description of a wave The wave function, y(x,t), gives a mathematical description of a wave. In this function, y is the displacement of a particle at time t and position x. The wave function for a sinusoidal wave moving in the +x-direction is y(x,t) = Acos(kx – t), where k = 2π/ is called the wave number. For transverse waves, y might represent the height of the wave at location x. For longitudinal waves, y might represent the pressure at location x.

Graphing the wave function The graphs in Figure 15.9 to the right look similar, but they are not identical. Graph (a) shows the shape of the string at t = 0, but graph (b) shows the displacement y as a function of time at t = 0. Starting with y(x = 0, t) = A coswt, the wave moving at speed v to the right will cause the motion at a point x to be delayed by time t = x/v. Thus But l = v/f = 2p v/w, and T = 2p /w so We define k = 2p/l = wavenumber, so wave function

Derivatives of y: wave equation Starting with , take partial derivative with respect to time to get y component of velocity: Likewise, take another partial derivative to get y component of acceleration: We can also take partial derivatives with respect to x (instead of t) to get: If we take the ratio of these two equations, we have: Rearranging gives the wave equation:

Particle velocity and acceleration in a sinusoidal wave The graphs in Figure 15.10 below show the velocity and acceleration of particles of a string carrying a transverse wave.

The speed of a wave on a string When a portion of a string under tension is displaced, a wave will be launched along the string. The text gives two explanations of how to derive the velocity of the wave, but I find both to be relatively tedious and uninspiring. It is probably not surprising that the velocity will depend on the tension force F and on the weight of the string (actually the mass/unit length m). Think of a guitar whose strings are all at the same tension. The thicker strings will produce a lower tone due to the slower speed of the wave (we will discuss standing waves shortly). Here is the expression for the propagation speed: Example 15.3: One end of a 2.00 kg rope is tied to a support at the top of a mine shaft 80.0 m deep. The rope is stretched taut by a 20.0 kg box of rocks at the bottom. What is the speed of a transverse wave on the rope? If a point on the rope is in transverse SHM with f = 2.00 Hz, how many cycles of the wave are there in the rope’s length?

Boundary conditions When a wave reflects from a fixed end, the pulse inverts as it reflects. See Figure 15.19(a) at the right. When a wave reflects from a free end, the pulse reflects without inverting. See Figure 15.19(b) at the right.

Wave interference and superposition Interference is the result of overlapping waves. Principle of super-position: When two or more waves overlap, the total displacement is the sum of the displace-ments of the individual waves. Study Figures 15.20 and 15.21 at the right.

Standing waves on a string Waves traveling in opposite directions on a taut string interfere with each other. The result is a standing wave pattern that does not move on the string. Destructive interference occurs where the wave displacements cancel, and constructive interference occurs where the displacements add. At the nodes no motion occurs, and at the antinodes the amplitude of the motion is greatest.

Photos of standing waves on a string Some time exposures of standing waves on a stretched string.

The formation of a standing wave In Figure 15.24, a wave to the left combines with a wave to the right to form a standing wave. Both are moving waves, one in the +x direction and one in the –x direction, but their sum creates a standing wave. Using the identities for cosine of the sum and difference of two angles we easily find the standing wave equation Why is this not a traveling wave?

Normal modes of a string For a taut string fixed at both ends, the possible wavelengths are n = 2L/n and the possible frequencies are fn = n v/2L = nf1, where n = 1, 2, 3, … f1 is the fundamental frequency, f2 is the second harmonic (first overtone), f3 is the third harmonic (second overtone), etc. Figure 15.26 illustrates the first four harmonics.

Standing waves and musical instruments A stringed instrument is tuned to the correct frequency (pitch) by varying the tension. Longer strings produce bass notes and shorter strings produce treble notes. For a stringed instrument with f1 = v/2L, since , the fundamental frequency (pitch) of a string is