Chapter 16: Oscillatory Motion and Waves PDF

Chapter 16: Oscillatory Motion and waves Introduction A child in a swing, the cone inside a speaker, a guitar, atoms in a crystal, the motion of chest cavities, and the beating of hearts all have in common? They all oscillate—-that is, they move back and forth between two points. Many...

Chapter 16: Oscillatory Motion and waves Introduction A child in a swing, the cone inside a speaker, a guitar, atoms in a crystal, the motion of chest cavities, and the beating of hearts all have in common? They all oscillate—-that is, they move back and forth between two points. Many systems oscillate, and they have certain characteristics in common. All oscillations involve force and energy. You push a child in a swing to get the motion started. The energy of atoms vibrating in a crystal can be increased with heat. You put energy into a guitar string when you pluck it.  Some oscillations create waves. A guitar creates sound waves. You can make water waves in a swimming pool by slapping the water with your hand. Some, such as water waves, are visible. Some, such as sound waves, are not. But every wave is a disturbance that moves from its source and carries energy. Other examples waves include earthquakes and visible light. Even subatomic particles, such as electrons, can behave like waves. Hooke’s Law: Stress and Strain Newton’s first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate. Consider, for example, plucking a plastic ruler to the left as shown in Figure. The deformation of the ruler creates a force in the opposite direction, known as a restoring force.  Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.  The simplest oscillations occur when the restoring force is directly proportional to displacement. 𝐹 = −𝑘𝑥 Here, F is the restoring force, x is the displacement from equilibrium or deformation, and k is a constant related to the difficulty in deforming the system. The minus sign indicates the restoring force is in the direction opposite to the displacement. The force constant k is related to the rigidity (or stiffness) of a system— the larger the force constant, the greater the restoring force, and the stiffer the system. The units of k are Newtons per meter (N/m). Energy in Hooke’s Law of Deformation Here, we generalize the idea to elastic potential energy for a deformation of any system that can be described by Hooke’s law. Hence, 1 2 𝑃𝐸𝑒𝑙 = 𝑘𝑥 2 where 𝑃𝐸𝑒𝑙 is the elastic potential energy stored in any deformed system that obeys Hooke’s law and has a displacement x from equilibrium and a force constant k. Period and Frequency in Oscillations Periodic motion A motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by an object on a spring moving up and down. Period The time to complete one oscillation or the time for some event whether repetitive or not; SI unit is seconds  Frequency is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is 1 𝑓= 𝑇 The SI unit for frequency is the cycle per second, which is defined to be a hertz (Hz): 𝑐𝑦𝑐𝑙𝑒 1 1 𝐻𝑧 = 1 𝑜𝑟 1 𝐻𝑧 = 𝑠𝑒𝑐 𝑠 1. Fish are hung on a spring scale to determine their mass, (a) What is the force constant of the spring in such a scale if it the spring stretches 8.00 cm for a 10.0 kg load? (b) What is the mass of a fish that stretches the spring 5.50 cm? (c) How far apart are the half-kilogram marks on the scale? 2. Find the frequency of a tuning fork that takes 2.50 x 10-3 s to complete one oscillation. Simple Harmonic Motion: A Special Periodic Motion Simple Harmonic Motion (SHM) Is an oscillatory motion for a system where the net force can be described by Hooke’s law, and such a system is called a simple harmonic oscillator. Amplitude: The maximum displacement from equilibrium is called the amplitude X. The units for amplitude and displacement are the same but depend on the type of oscillation. What is so significant about simple harmonic motion? The period and frequency of a simple harmonic oscillator are independent of amplitude. Factors do affect the period of a simple harmonic oscillator: 1. The period is related to how stiff the system is!! A very stiff object has a large force constant k, which causes the system to have a smaller period. 2. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period.  The period of a simple harmonic oscillator is given by 𝑚 𝑇 = 2𝜋 𝑘 and, because f = 1/T, the frequency of a simple harmonic oscillator is 1 𝑘 𝑓= 2𝜋 𝑚 Note that neither T nor f has any dependence on amplitude. 3. A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass? The Simple Pendulum A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure.  From Figure that the net force on the bob is tangent to the arc and equals −𝑚𝑔 sin 𝜃. (The weight mg has components 𝑚𝑔 cos 𝜃 along the string and 𝑚𝑔 sin 𝜃 tangent to the arc.) Tension in the string exactly cancels the component 𝑚𝑔 cos 𝜃 parallel to the string. This leaves a net restoring force back toward the equilibrium position at 𝜃 = 0. Thus, for angles less than about 15o, the restoring force F is 𝐹 = −𝑘𝑥 Using this equation, we can find the period of a pendulum for amplitudes less than about 15o. For the simple pendulum: 𝐿 𝑇 = 2𝜋 𝑔 4. What is the length of a pendulum that has a period of 0.500 s? 5. How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot? Energy and the Simple Harmonic Oscillator The energy stored in the deformation of a simple harmonic oscillator is a form of potential energy given by 1 𝑃𝐸𝑒𝑙 = 𝑘 𝑥 2 2 Conservation of energy for these two forms is: 𝐾𝐸 + 𝑃𝐸𝑒𝑙 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Or 1 2 1 2 𝑚𝑣 + 𝑘𝑥 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2 2  Namely, for a simple pendulum we replace the velocity with v = Lω, the spring constant with k = mg/L, and the displacement term with x = Lθ. Thus, 1 1 𝑚𝐿 𝜔 + 𝑚𝑔𝐿𝜃 2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2 2 2 2 In the case of undamped simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates. So, for the simple example of an object on a frictionless surface attached to a spring,  The motion starts with all of the energy stored in the spring. As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position. It is then converted back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on. This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits.  The maximum velocity depends on three factors. 𝑘 𝑣𝑚𝑎𝑥 = 𝑋 𝑚 Maximum velocity is directly proportional to amplitude. Maximum velocity is also greater for stiffer systems, Finally, the maximum velocity is smaller for objects that have larger masses.  Uniform Circular Motion and Simple Harmonic Motion There is an easy way to produce simple harmonic motion by using uniform circular motion. Figure shows one way of using this method. A ball is attached to a uniformly rotating vertical turntable, and its shadow is projected on the floor as shown.  The shadow undergoes simple harmonic motion. Hooke’s law usually describes uniform circular motions ( constant) rather than systems that have large visible displacements. So, observing the projection of uniform circular motion, as in Figure, is often easier than observing a precise large-scale simple harmonic oscillator.  A point P moving on a circular path with a constant angular velocity is undergoing uniform circular motion. Its projection on the x-axis undergoes simple harmonic motion. Also shown is the velocity of this point around the circle, vmax , and its projection, which is v. Note that these velocities form a similar triangle to the displacement triangle. Thus, the period of the motion is the same as for a simple harmonic oscillator. We have determined the period for any simple harmonic oscillator using the relationship between uniform circular motion and simple harmonic motion. 6. (a) What is the maximum velocity of an 85.0-kg person bouncing on a bathroom scale having a force constant of 1.50 x 106 N/m, if the amplitude of the bounce is 0.200 cm? (b) What is the maximum energy stored in the spring? Waves A wave is a disturbance that propagates or moves from the place it was created. For water waves, the disturbance is in the surface of the water. For sound waves, the disturbance is a change in air pressure, perhaps created by the oscillating cone inside a speaker. For earthquakes, there are several types of disturbances, including disturbance of Earth’s surface and pressure disturbances under the surface. Even radio waves are most easily understood using an analogy with water waves.  Water waves exhibit characteristics common to all waves, such as amplitude, period, frequency and energy. All wave characteristics can be described by a small set of underlying principles. Let us start by considering the simplified water wave in Figure.  An idealized ocean wave passes under a sea gull that bobs up and down in simple harmonic motion. The wave has a wavelength λ, which is the distance between adjacent identical parts of the wave. The up and down disturbance of the surface propagates parallel to the surface at a speed vw. This movement of the wave is actually the disturbance moving to the right, not the water itself. We define wave velocity to be the speed at which the disturbance moves. Wave velocity is sometimes also called the propagation velocity or propagation speed, because the disturbance propagates from one location to another.  The speed of propagation is the distance the wave travels in a given time, which is one wavelength in the time of one period. In equation form, that is λ 𝑣𝑤 = 𝑡 𝑜𝑟 𝑣𝑤 = 𝑓𝜆 This fundamental relationship holds for all types of waves. For water waves, 𝑣𝑤 is the speed of a surface wave; for sound, 𝑣𝑤 is the speed of sound; and for visible light, 𝑣𝑤 is the speed of light, for example. 7. Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency? 8. Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake it the bridge twice per second, what is the propagation speed of the waves? 9. What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at 0.800 m/s? Transverse and Longitudinal Waves The wave in shown in figure propagates in the horizontal direction while the surface is disturbed in the vertical direction. Such a wave is called a transverse wave or shear wave; in such a wave, the disturbance is perpendicular to the direction of propagation. In contrast, in a longitudinal wave or compressional wave, the disturbance is parallel to the direction of propagation.  Waves may be transverse, longitudinal, or a combination of the two. (Water waves are actually a combination of transverse and longitudinal.) The waves on the strings of musical instruments are transverse—so are electromagnetic waves, such as visible light.  Sound waves in air and water are longitudinal. Their disturbances are periodic variations in pressure that are transmitted in fluids. Fluids do not have appreciable shear strength, and thus the sound waves in them must be longitudinal or compressional. Sound in solids can be both longitudinal and transverse. Earthquake waves under Earth’s surface also have both longitudinal and transverse components (called compressional or Pwaves and shear or S- waves, respectively).  Most waves do not look very simple, They look more like the waves in Figure below. When two or more waves arrive at the same point, they superimpose themselves on one another. More specifically, the disturbances of waves are superimposed when they come together a phenomenon called superposition.  Figure shows two identical waves that arrive at the same point exactly in phase The crests of the two waves are precisely aligned, as are the troughs This superposition produces pure constructive interference. Because the disturbances add, pure constructive interference produces a wave that has twice the amplitude of the individual waves, but has the same wavelength  Figure shows two identical waves that arrive exactly out of phase that is, precisely aligned crest to trough producing pure destructive interference Because the disturbances are in the opposite direction for this superposition, the resulting amplitude is zero for pure destructive interference the waves completely cancel.  While pure constructive and pure destructive interference do occur, they require precisely aligned identical waves. The superposition of most waves produces a combination of constructive and destructive interference and can vary from place to place and time to time. Sound from a stereo, for example, can be loud in one spot and quiet in another. Varying loudness means the sound waves add partially constructively and partially destructively at different locations. Standing Waves The waves move through each other with their disturbances adding as they go by. If the two waves have the same amplitude and wavelength, then they alternate between constructive and destructive interference. The resultant looks like a wave standing in place and, thus, is called a standing wave. 10. If a spring is stretched beyond its elastic limit, A. it will break. B. it will snap back into its original shape. C. it will remain deformed. D. its density will be forever changed. 11. The distance from the top of one wave crest to the next is called the A. speed. B. amplitude. C. period. D. wavelength. 12. (a) A novelty clock has a 0.0100-kg mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? (b) How many joules of kinetic energy does the object have at its maximum velocity? THANK YOU