CHAPTER 11 VIBRATIONS AND WAVES http://www.physicsclassroom.com/class/waves/u10l1a.html UNITS Simple Harmonic Motion Energy in the Simple Harmonic Oscillator The Period and Sinusoidal Nature of SHM The Simple Pendulum Damped Harmonic Motion Forced Vibrations; Resonance Wave Motion Types of Waves: Transverse and Longitudinal Energy Transported by Waves Intensity Related to Amplitude and Frequency Reflection and Transmission of Waves Interference; Principle of Superposition Standing Waves; Resonance Refraction Diffraction Mathematical Representation of a Traveling Wave Simple Harmonic Motion If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system. HOOK S LAW F = -kx If a spring is stretched or compressed a small distance, x, from its unstretched position and then released, this force law for springs is known as Hook s Law. 1
SIMPLE HARMONIC MOTION The direction of the restoring force is such that the mass is being either pulled or pushed toward the equilibrium position. As the object moves toward x = 0, the magnitude of the force decreases (because x decreases) and reaches zero at x = 0. Velocity reaches its maximum at x = 0. Acceleration is maximum at the greatest distance from x = 0. We assume that the surface is frictionless. There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 on the previous figure). The force exerted by the spring depends on the displacement. The minus sign on the force indicates that it is a restoring force it is directed to restore the mass to its equilibrium position. k is the spring constant. The force is not constant, so the acceleration is not constant either. Displacement is measured from the equilibrium point Amplitude is the maximum displacement A cycle is a full to-and-fro motion; this figure shows half a cycle Period is the time required to complete one cycle Frequency is the number of cycles completed per second If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force. 2
Example 1: Spring Constant A spring is suspended vertically with a 0.5kg mass attached to it. If the spring is displaced 2.00 cm what is the spring constant? Solution Strategy GIVEN Mass = 0.5 kg Distance stretched = 2.0 cm mg FS kx k d FIND Spring constant = k k 2 (0.5 kg)(9.80 m / s ) 2.00x10 2 m 245N AP Equations of Motion The equation that gives the object s position as a function of time is referred to as the equation of motion. The equation of motion with a constant linear acceleration is 1 2 x xo vot at 2 The acceleration is not constant in SHM, so kinematic equations do not apply. From the reference circle the y-coordinate of the object is given by y Asin But the object moves with a constant angular velocity of magnitude. In terms of the angular distance, assuming that 0 o at t = 0, we have t, so y Asin t 3
The angular speed (in rad/s) is called the angular frequency of the oscillating object, since 2 f, where f is the frequency of revolution or rotation of the object. We can write this as 2 t y Asin(2 ft) Asin T The Period and Sinusoidal Nature of SHM The top curve is a graph of the previous equation. The bottom curve is the same, but shifted ¼ period so that it is a sine function rather than a cosine. The equations of motion T 2 1 f 2 m k k m period of object oscillating on a spring frequency of mass oscillating on a spring k m angular frequency of mass oscillating on a spring T 2 L g period of a simple pendulum Energy in the Simple Harmonic Oscillator We already know that the potential energy of a spring is given by: The total mechanical energy is then: 4
The total mechanical energy will be conserved, as we are assuming the system is frictionless. Energy in the Simple Harmonic Oscillator If the mass is at the limits of its motion, the energy is all potential. If the mass is at the equilibrium point, the energy is all kinetic. We know what the potential energy is at the turning points: Energy in the Simple Harmonic Oscillator The total energy is, therefore 1 2 ka, and we can write: 1 2 1 2 1 2 mv kx ka 2 2 2 This can be solved for the velocity as a function of position: Where Example 2: A spring stretches 0.150 m when a 0.300-kg mass is hung from it. The spring is then stretched an additional 0.100 m from this equilibrium point, and released. 5
a) Determine the spring constant k. Since the spring stretches 0.150 m when 0.300 kg is hung from it, k = 2 F mg (0.300 kg)(9.80 m / s ) k 19.6 N / m x x 0.150m b) Determine the amplitude of the oscillation. Since the spring is stretched 0.100 m from equilibrium and is given no initial speed, A = 0.100 m. c) Determine the maximum velocity vo. The maximum velocity is attained as the mass passes through the equilibrium point where all the energy is kinetic. 1 1 mv 0 0 ka 2 2 2 2 o k 19.6 N / m vo A (0.100 m) 0.808 m / s m 0.300kg d) Determine the magnitude of the velocity, v, when the mass is 0.050m from the equilibrium. 2 2 2 (0.050 m) v vo 1 x / A (0.808 m / s) 1 0.700 m / s 2 (0.100 m) e) Calculate the magnitude of the maximum acceleration of the mass. f) Determine the total energy. ka (19.6 N / m)(0.100 m) a 6.53 m / s m 0.300kg 2 1 1 (19.6 / )(0.100 ) 9.80 10 2 2 2 2 2 E ka N m m x J g) Determine the kinetic and potential energies at half amplitude. 1 2 2 2 PE kx 2.5x10 J 2 KE E PE 7.3x10 J 6
Example 3: What is the period and frequency of a 1400-kg car that has springs at a spring 4 constant of 6.5x10 N / m? T m 1400kg 2 6.28 0.92s f 1/ T 1.09Hz 4 k 6.5x10 N / m So the mass is at y = -0.10m at t =3.1s. But what is its direction of motion? In t = 3.1s, the mass has gone through 3.1s/5.0 s = 0.62, or 62%, of a period or cycle, so it is moving downward. [The motion is up (¼ cycle) and back (1/4 cycle) to yo = 0 in ½, or 50%, of the cycle, therefore downward during the next ¼ cycle]. The Simple Pendulum 1 1 T 5.0s f 0.20Hz In order to be in SHM, the restoring force must be proportional to the negative of the displacement. Here we have: which is proportional to sin θ and not to θ itself. However, if the angle is small, sin θ θ. Example 6: Estimate the length of the pendulum in a grandfather clock that ticks once per second. 2 2 T (1.0 s) 2 L g = (9.80 m / s ) 0.25m 2 2 4 4 7
Damped Harmonic Motion There are systems where damping is unwanted, such as clocks and watches. Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings. Forced Vibrations; Resonance Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system. If the frequency is the same as the natural frequency, the amplitude becomes quite large. This is called resonance. Example would be pushing a child on a swing. If you push with a frequency equal to the natural frequency of the swing, the amplitude increases greatly. This illustrates that at resonance, relatively little effort is required to obtain a large amplitude. The tenor Enrico Caruso was able to shatter a crystal because the resonance created a large enough amplitude that the glass exceeds its elastic limit and breaks. Tacoma Narrows Oakland Freeway Bridge 1940 1989 Wave Motion 8
A wave travels along its medium, but the individual particles just move up and down. All types of traveling waves transport energy. Study of a single wave pulse shows that it is begun with a vibration and transmitted through internal forces in the medium. Continuous waves start with vibrations too. If the vibration is SHM, then the wave will be sinusoidal. Wave characteristics: Amplitude, A Wavelength, λ Frequency f and period T Wave velocity A wave crest travels a distance of one wavelength,, in one period, T. v f The velocity of a wave depends on the properties of the medium in which it travels. For waves of small amplitude, the relationship is v F T m/ L 9
Example 7: Wave on a wire. A wave whose wavelength is 0.30m is traveling down a 300-m-long wire whose total mass is 15 kg. If the wire is under a tension of 1000N, what is the velocity and frequency of this wave? FT 1000N v 140 m / s m / L (15 kg) /(300 m) v 140 m / s f 470Hz 0.30m Types of Waves: Transverse and Longitudinal The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal). Sound waves are longitudinal waves: Earthquakes produce both longitudinal and transverse waves. Both types can travel through solid material, but only longitudinal waves can propagate through a fluid in the transverse direction, a fluid has no restoring force. Surface waves are waves that travel along the boundary between two media. 10
Energy Transported by Waves Just as with the oscillation that starts it, the energy transported by a wave is proportional to the square of the amplitude. Definition of intensity: The intensity is also proportional to the square of the amplitude: Intensity Related to Amplitude and Frequency By looking at the energy of a particle of matter in the medium of the wave, we find: Then, assuming the entire medium has the same density, we find: Therefore, the intensity is proportional to the square of the frequency and to the square of the amplitude. Reflection and Transmission of Waves A wave reaching the end of its medium, but where the medium is still free to move, will be reflected (b), and its reflection will be upright. A wave hitting an obstacle will be reflected (a), and its reflection will be inverted. A wave encountering a denser medium will be partly reflected and partly transmitted; if the wave speed is less in the denser medium, the wavelength will be shorter. 11
Two- or three-dimensional waves can be represented by wave fronts, which are curves of surfaces where all the waves have the same phase. Lines perpendicular to the wave fronts are called rays; they point in the direction of propagation of the wave. The law of reflection: the angle of incidence equals the angle of reflection. Interference; Principle of Superposition The superposition principle says that when two waves pass through the same point, the displacement is the arithmetic sum of the individual displacements. In the figure below, (a) exhibits destructive interference and (b) exhibits constructive interference. These figures show the sum of two waves. In (a) they add constructively; in (b) they add destructively; and in (c) they add partially destructively. 12
Standing Waves; Resonance Standing waves occur when both ends of a string are fixed. In that case, only waves which are motionless at the ends of the string can persist. There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value. The frequencies of the standing waves on a particular string are called resonant frequencies. They are also referred to as the fundamental and harmonics. The wavelengths and frequencies of standing waves are: 13
v f WAVE CHARACTERISTICS A change in the frequency of a wave does not affect the speed of a wave. This means that the wavelength must change in order to keep the equation balanced. An increase in the wave frequency will cause a decrease in the wavelength. period 1 frequency If the frequency > the period < If the frequency < the period > If the period > the frequency < If the period < the frequency > The amplitude of the reflected pulse is less than the amplitude of the incident pulse. The frequency of a wave is not altered by crossing a boundary. If a wave is refracted the speed of a wave and the wavelength is changed. If the frequency of a wave increases, the distance between crests decreases. If the frequency of a wave increases, the distance between successive crests decreases. 14
CHAPTER 11 12 VIBRATION & WAVES CONCEPTS 1. A characteristic common to sound waves and light waves is that they transfer energy. 2. As the energy imparted to a mechanical wave increases, the maximum displacement of the particles in the medium increases. 3. A single vibratory disturbance which moves from point to point in a material is known as a pulse. 4. As a pulse travels through a stretched heavy rope attached to a light rope its speed will increase as it travels through the light rope. 5. In a longitudinal wave is the disturbance parallel to the direction of wave travel. 6. In a nondispersive medium, the speed of a light wave depends on the nature of the medium. 7. The diagram to the right shows a transverse water wave moving in the direction shown by the velocity vector v. At the instant shown, a cork at point P on the water s surface is moving toward B. 15
8. Whether or not a wave is longitudinal or transverse may be determined by its ability to be polarized. 9. A sound wave can not be polarized. 10. Light is to brightness as sound is to loudness. 11. As a wave is refracted, the frequency of the wave will remain unchanged. 12. The graph that best represents the relationship between the frequency and period of a wave is C. 13. As the period of a wave decreases, the wave s frequency increases. 1 T f 14. Two waves have the same frequency. It must also have identical periods for both waves. 15. If the frequency of a sound wave in air at STP remains constant, its energy can be varied by changing its amplitude. 16. Only coherent wave sources produce waves that have a constant phase relation. 17. In the diagram to the right, the distance between points A and B on a wave is 0.10 m. This wave must have a wavelength of 0.20 m. 18. As a wave enters a medium, there may be a change in the wave s speed. 19. An observer detects an apparent change in the frequency of sound waves produced by an airplane passing overhead. This phenomenon illustrates the Doppler Effect. 20. As observed from the Earth, the light from a star is shifted toward lower frequencies. This is an indication that the distance between the Earth and the star is increasing. 16
21. The vibrating tuning fork shown in the diagram below produces a constant frequency. The tuning fork is moving to the right at a constant speed, and observers are located at points A, B, C, and D. The observer at D hears the lowest frequency. 22. Interference occurs when two or more waves pass simultaneously through the same region in a medium. 23. An opera singer s voice is able to break a thin crystal glass if the singer s voice and the glass have the same natural frequency. 24. Standing waves are produced by the interference of two waves with the same frequency and amplitude, but opposite directions. 25. When the stretched string of the apparatus represented to the right is made to vibrate, point P does not move. Point P is most probably at the location of a node. 26. The distance between successive antinodes in the standing wave pattern shown to the right is equal to ½ wavelength. 27. Maximum destructive interference between two waves occurs when the waves are out of phase by 180 degrees. 28. The diagram to the right represents a group of light waves emitted simultaneously from a single source. The light waves would be classified as both monochromatic and coherent. 29. To the nearest order of magnitude the speed of light is 10 6 times greater than the speed of sound. 30. Refraction of a wave is caused by a change in the wave s speed. 17
31. The change in the direction of a wave when it passes obliquely from one medium to another is called refraction. 32. As a periodic wave travels from one medium to another, the period and frequency of that wave cannot change. 18