Chapter 11 Vibrations and Waves PDF

Chapter 11 (1) Vibrations and Waves  Units of Chapter 11 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 L...

Chapter 11 (1) Vibrations and Waves  Units of Chapter 11 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  Units of Chapter 11 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  11-1 Simple Harmonic Motion 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: (11-1)  11-1 Simple Harmonic Motion 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 11-1 Simple Harmonic Motion 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  11-1 Simple Harmonic Motion 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.  11-1 Simple Harmonic Motion Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator.  11-1 Simple Harmonic Motion Example 1: When a family of four with a total mass of 200 kg step into their 1200 kg car, the car’s springs compress 3 cm. (a) What is the spring constant? (b) How far will the car lower if loaded with 300 kg rather than 200 kg?  11-1 Simple Harmonic Motion Solution: 11-2 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: (11-3) The total mechanical energy will be conserved, as we are assuming the system is frictionless. 11-2 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: (11-4a) 11-2 Energy in the Simple Harmonic Oscillator The total energy is, therefore And we can write: (11-4c) This can be solved for the velocity as a function of position: (11-5) where 11-2 Energy in the Simple Harmonic Oscillator Example 2: A spring stretches 0.15 m when a 0.3 kg mass is gently lowered on it. The spring is then set up horizontally with the 0.3 kg mass resting on a frictionless table. The mass is pulled so that the spring is stretched 0.1 m from the equilibrium point, and released from rest. Determine (a) the spring constant, k (b) The amplitude of the horizontal oscillation (c) vmax (d) the magnitude of the velocity v when the mass is 0.05 m from equilibrium (e) the magnitude of the maximum acceleration amax of the mass 11-2 Energy in the Simple Harmonic Oscillator Solution: 11-2 Energy in the Simple Harmonic Oscillator Example 3: For the simple harmonic oscillator of example 2, determine (a) the total energy (b) The kinetic and potential energies at half amplitude (x =  A/2) 11-2 Energy in the Simple Harmonic Oscillator Solution:  11-3 The Period and Sinusoidal Nature of SHM If we look at the projection onto the x axis of an object moving in a circle of radius A at a constant speed vmax, we find that the x component of its velocity varies as: This is identical to SHM. Animation - link 11-3 The Period and Sinusoidal Nature of SHM 11-3 The Period and Sinusoidal Nature of SHM 11-3 The Period and Sinusoidal Nature of SHM Therefore, we can use the period and frequency of a particle moving in a circle to find the period and frequency: (11-7a) (11-7b) 11-3 The Period and Sinusoidal Nature of SHM Example 4: A spider of mass 0.3 g waits in its web of negligible mass. A slight movement causes the web to vibrate with a frequency of about 15 Hz. (a) Estimate the value of the spring stiffness constant, k for the web. (b) At what frequency would you expect the web to vibrate if an insect of mass 0.1 g were trapped in addition to the spider? 11-3 The Period and Sinusoidal Nature of SHM Solution: 11-3 The Period and Sinusoidal Nature of SHM We can similarly find the position as a function of time: (11-8a) (11-8b) (11-8c) 11-3 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. 11-3 The Period and Sinusoidal Nature of SHM Example 5: The displacement of an object is described by the following equation, where x is in meters and t is in seconds x = (0.3 m) cos(8.0t) Determine the oscillating object’s (a) amplitude (b) frequency (c) period (d) maximum speed (e) maximum acceleration 11-3 The Period and Sinusoidal Nature of SHM Solution:  11-3 The Period and Sinusoidal Nature of SHM The velocity and acceleration can be calculated as functions of time; the results are below, and are plotted at left. (11-9) (11-10) 11-3 The Period and Sinusoidal Nature of SHM Example 6: The cone of loudspeaker vibrates in SHM at a frequency of 262 Hz. The amplitude at the centre of the cone is A = 1.5 x 10-4 m, and at t = 0, x = A. (a) What equation describes the motion of the centre of the cone? (b) What are the velocity and acceleration as a function of time? (c) What is the position of the cone at t = 1 ms? 11-3 The Period and Sinusoidal Nature of SHM Solution: 11-4 The Simple Pendulum A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible. 11-4 The Simple Pendulum 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.  11-4 The Simple Pendulum Therefore, for small angles, we have: where The period and frequency are: (11-11a) (11-11b)  11-4 The Simple Pendulum Example 7: A geologist uses a simple pendulum that has a length of 37.1 cm and a frequency of 0.819 Hz at a particular location on the earth. What is the acceleration of gravity at this location?  11-4 The Simple Pendulum Solution: 11-4 The Simple Pendulum So, as long as the cord can be considered massless and the amplitude is small, the period does not depend on the mass.  11-5 Damped Harmonic Motion Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is small, we can treat it as an “envelope” that modifies the undamped oscillation.  11-5 Damped Harmonic Motion However, if the damping is large, it no longer resembles SHM at all. A: underdamping: there are a few small oscillations before the oscillator comes to rest. B: critical damping: this is the fastest way to get to equilibrium. C: overdamping: the system is slowed so much that it takes a long time to get to equilibrium. Animation - link  11-5 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.  11-6 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.  11-6 Forced Vibrations; Resonance The sharpness of the resonant peak depends on the damping. If the damping is small (A), it can be quite sharp; if the damping is larger (B), it is less sharp. Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it. 11-7 Wave Motion A wave travels along its medium, but the individual particles just move up and down.  11-7 Wave Motion 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.  11-7 Wave Motion Wave characteristics: Amplitude, A Wavelength, λ Frequency f and period T Wave velocity (11-12)  11-8 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).  11-8 Types of Waves: Transverse and Longitudinal Sound waves are longitudinal waves:  11-8 Types of Waves: Transverse and Longitudinal 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.  11-9 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: (11-15)  11-9 Energy Transported by Waves If a wave is able to spread out three- dimensionally from its source, and the medium is uniform, the wave is spherical. Just from geometrical considerations, as long as the power output is constant, we see: (11-16b)  11-9 Energy Transported by Waves Example: The intensity of an earthquake P wave traveling through the Earth and detected 100 km from the source is 1.0 x 106 W/m2. What is the intensity of that wave if detected 400 km from the source?  11-9 Energy Transported by Waves Solution:  11-11 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.  11-11 Reflection and Transmission of Waves 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. 11-11 Reflection and Transmission of Waves 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. 11-11 Reflection and Transmission of Waves The law of reflection: the angle of incidence equals the angle of reflection. 11-12 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. 11-12 Interference; Principle of Superposition 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. 11-13 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. 11-13 Standing Waves; Resonance Standing wave - animation 11-13 Standing Waves; Resonance The frequencies of the standing waves on a particular string are called resonant frequencies. They are also referred to as the fundamental and harmonics.  11-13 Standing Waves; Resonance The wavelengths and frequencies of standing waves are: (11-19a) (11-19b)  11-13 Standing Waves; Resonance Example: A piano string is 1.10 m long and has a mass of 9 g. a) How much tension must the string be under if it is to vibrate at a fundamental frequency of 131 Hz? b) What are the frequencies of the first four harmonics?  11-13 Standing Waves; Resonance Solution:  11-14 Refraction If the wave enters a medium where the wave speed is different, it will be refracted – its wave fronts and rays will change direction. We can calculate the angle of refraction, which depends on both wave speeds: (11-20)  11-14 Refraction The law of refraction works both ways – a wave going from a slower medium to a faster one would follow the red line in the other direction.  11-14 Refraction Example: An earthquake P wave passes across a boundary in rock where its velocity increases from 6.5 km/s to 8.0 km/s. If it strikes this boundary at 30, what is the angle of refraction?  11-14 Refraction Solution: 11-15 Diffraction When waves encounter an obstacle, they bend around it, leaving a “shadow region.” This is called diffraction.  11-15 Diffraction The amount of diffraction depends on the size of the obstacle compared to the wavelength. If the obstacle is much smaller than the wavelength, the wave is barely affected (a). If the object is comparable to, or larger than, the wavelength, diffraction is much more significant (b, c, d). 11-16 Mathematical Representation of a Traveling Wave 2 y  A sin x  To the left, we have a snapshot of a traveling wave at a single point in time. Below left, the same wave is shown traveling. 2 y  A sin x  vt    11-16 Mathematical Representation of a Traveling Wave A full mathematical description of the wave describes the displacement of any point as a function of both distance and time: (11-22)  11-16 Mathematical Representation of a Traveling Wave Example: A progressive wave is represented by the expression y = 0.25 sin(36t – 7x) where y and x are in m, and t is in s. Determine a) amplitude b) the frequency c) the wavelength d) the speed e) the direction of wave propagation f) the displacement y at a distance of x = 0.2 m when t=0  Summary of Chapter 11 For SHM, the restoring force is proportional to the displacement. The period is the time required for one cycle, and the frequency is the number of cycles per second. Period for a mass on a spring: SHM is sinusoidal. During SHM, the total energy is continually changing from kinetic to potential and back.  Summary of Chapter 11 A simple pendulum approximates SHM if its amplitude is not large. Its period in that case is: When friction is present, the motion is damped. If an oscillating force is applied to a SHO, its amplitude depends on how close to the natural frequency the driving frequency is. If it is close, the amplitude becomes quite large. This is called resonance.  Summary of Chapter 11 Vibrating objects are sources of waves, which may be either a pulse or continuous. Wavelength: distance between successive crests. Frequency: number of crests that pass a given point per unit time. Amplitude: maximum height of crest. Wave velocity:  Summary of Chapter 11 Vibrating objects are sources of waves, which may be either a pulse or continuous. Wavelength: distance between successive crests Frequency: number of crests that pass a given point per unit time Amplitude: maximum height of crest Wave velocity:  Summary of Chapter 11 Transverse wave: oscillations perpendicular to direction of wave motion. Longitudinal wave: oscillations parallel to direction of wave motion. Intensity: energy per unit time crossing unit area (W/m2): Angle of reflection is equal to angle of incidence.  Summary of Chapter 11 When two waves pass through the same region of space, they interfere. Interference may be either constructive or destructive. Standing waves can be produced on a string with both ends fixed. The waves that persist are at the resonant frequencies. Nodes occur where there is no motion; antinodes where the amplitude is maximum. Waves refract when entering a medium of different wave speed, and diffract around obstacles.