Oscillations and Waves PDF

Oscillations and Waves Department of Physics and Materials Science (DPMS) Thapar Institute of Engineering & Technology (Deemed to be University)  Contents in this topic Simple Harmonic Motion (SHM)—Spring Oscillations Mathematical Representation of Simple Harmonic M...

Oscillations and Waves Department of Physics and Materials Science (DPMS) Thapar Institute of Engineering & Technology (Deemed to be University)  Contents in this topic Simple Harmonic Motion (SHM)—Spring Oscillations Mathematical Representation of Simple Harmonic Motion Energy in Simple Harmonic Motion The Period and Sinusoidal Nature of SHM The Simple Pendulum Damped Oscillations  Oscillations and Waves As a sound wave travels through the air, elements of the air oscillate back and forth; as a water wave travels across a pond, elements of the water oscillate up and down and backward and forward. To explain many other phenomena in nature, we must understand the concepts of oscillations and waves. Periodic motion: Repeating motion of an object in which the object continues to return to a given position after a fixed time interval. Oscillatiory Motion: The back and forth movements of an object.  Simple Harmonic Motion We will focus our attention on a special case of periodic motion called simple harmonic motion (SHM) What is SIMPLE in Simple Harmonic motion? Simple harmonic motion (SHM) is a harmonic motion with Single frequency, Well defined amplitude. Simple harmonic motion forms a basic building block for more complicated periodic motion. All periodic motions can be modeled as combinations of simple harmonic motions. Simple harmonic motion also forms the basis for our understanding of mechanical waves. Sound waves, seismic waves, waves on stretched strings, and water waves are all produced by some source of oscillation. Simple Harmonic Motion-Spring Oscillations 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. Simple Harmonic Motion-Spring Oscillations Displacement is measured from the equilibrium point and 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 Simple Harmonic Motion-Spring Oscillations 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) 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 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. Simple Harmonic Motion-Spring Oscillations Applying Newton’s second law: Solution: So, an object moves with simple harmonic motion whenever its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium. 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.  Displacement v/s time Pure Sine-like curve Amplitude Equil. point period (=T) Displacement, velocity and acceleration v/s time  SHM (Spring Oscillator) Following Equations form the basis of the mathematical representation of simple harmonic motion.  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. s = Lθ F = -mg sin θ 𝟐 𝟐 -mg sin θ Energy Conservation in Oscillatory Motion In an ideal system with no non-conservative forces, the total mechanical energy is conserved. We already know that the PE and KE of a spring is given by: U = ½ kx2 K = ½ mv2 The total mechanical energy will be conserved, as we are assuming the system is frictionless. Energy in Simple Harmonic Motion 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. So total energy is Energy in Simple Harmonic Motion This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same. Energy in Simple Harmonic Motion  Electrical oscillator: LC circuit When a capacitor is connected The Eqn. of voltage due to the to an inductor the combination Instantaneous current I can be is an LC circuit. written as; 𝟐 𝟐 𝟐 𝟐    Simple harmonic oscillation Differential eqn for SHM 𝟐 𝟐 𝟐 𝟎 Solution 𝟎 Electrical oscillator (Tank circuit) 𝟐 𝟐 𝟐 𝟎 Solution 𝒎𝒂𝒙 𝟎  Energy in Electrical oscillator: LC circuit The potential energy ½ kx2 stored in a stretched spring is analogous to the electric potential energy Q2/2C stored in the capacitor. The kinetic energy ½ mv2 of the moving block is analogous to the magnetic energy ½ LI2 stored in the inductor, which requires the presence of moving charges. Assume the circuit resistance to be zero, so no energy is transformed to internal energy and none is transferred out of the system of the circuit. Therefore, the total energy of the system must remain constant in time.  Energy in Electrical oscillator: LC circuit An LC circuit, oscillating at its natural resonant frequency, can store electrical energy. A capacitor stores energy in the electric field (E) between its plates, depending on the voltage across it, and an inductor stores energy in its magnetic field (B), depending on the current through it. Energy in Electrical oscillator: LC circuit Energy in Electrical oscillator: LC circuit 𝟐 + 𝟐 + 𝟐  + 𝒎𝒂𝒙  Energy in oscillators Energy stored in oscillators 𝟐 𝟐 𝟐 𝟐 𝟐 𝑪 𝑳 BUT these are the idealized phenomenon. There are always some energy losses in nature. So the harmonic motions diminish with time.  Damped Harmonic Motion Damped harmonic motion is harmonic motion with a frictional or drag force. Consequently, the mechanical energy of the system diminishes in time, and the motion is said to be damped.  Damped Harmonic Motion If the damping is large, it no longer resembles SHM 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. Systems where damping is wanted/unwanted Damping is wanted: Automobile, shock absorbers Damping is unwanted Clocks and watches  Damped Harmonic Motion The retarding force is often observed when an object moves. Retarding force : R = – bv (where b is a damping coefficient) Restoring force : F = – kx (where k is a spring constant) 𝑥 2 2 2 2 2 Differential equation for 2 damped harmonic motion  Damped Harmonic Motion 2 2 As we know that x has two properties: (1) x is a function of time and (2) x decreases exponentially in damping. So we suppose the following form of it: 2  Damped Harmonic Motion C. If r2 > , No Oscillation Slower than exponential decay  Damped Harmonic Motion 2 = r2 – B. If r2 ~ , 2 1 2 2 2 2 2 1 2 2 1 2 2 1 1 2 1 2 2 1 No Oscillation Exponential decay  Damped Harmonic Motion 2 A. If r2 < , 2  Damped Harmonic Motion When the retarding force is small, the oscillatory character of the motion is preserved but the amplitude decreases in time, with the result that the motion ultimately ceases. Any system that behaves in this way is known as a damped oscillator. Amplitude decays exponentially represents the angular frequency in the absence of a retarding force (the undamped oscillator) and is called the natural frequency of the system  Electrical Oscillator: LCR circuit 2 2 C L 2 R 2 Q /  Electrical Oscillator: LCR circuit / Q Damping Critical condition There exists a critical resistance value Overdamping above which no oscillations occur. A system with R = Rc is said to be critically damped. When R exceeds Rc, the system is said to be overdamped Electrical Oscillator: Mechanical Oscillator  Characterization of damping Logarithmic decrement Relaxation time Amplitude relaxation time Energy relaxation time Quality factor  Logarithmic Decrement (d) Logarithmic decrement measures the rate at which the amplitude of the oscillatory motion decay. Let xn and xn+1 be the two successive maxima corresponding to amplitudes, separated by a time period T. If the maximum x1 is occurring at t1 = t, then x2 will occur at t2 = t + T (T=2p/). ( ) ( ) ( ) d = rT here r = b/2m  Relaxation Time (t) Measure of time (ta) during which the amplitude of an oscillatory motion decays to 1/e of its initial value is known as amplitude relaxation time. In similar words The time (te) during which the energy of an oscillatory motion decays to 1/e of its initial value is known as energy relaxation time. when t = t0 , when t = t0 , when t = t0 + ta , when t = t0 + te , ( ta) ( te) ta te ta = 1/r 2te = ta te = 1/2r d = rT = T/ta d = rT = T/2te  Quality Factor (Q) The quality factor measures the quality of the oscillator; less the losses (due to damping), more the quality. It is also called the figure of merit and is defined as: 2p the energy stored in the damped harmonic oscillator to the energy loss per cycle. energy stored in damped SHM energy loss per cycle For a full cycle dt = T  Quality Factor (Q) The quality factor measures the quality of the oscillator; less the losses (due to damping), more the quality. It is also called the figure of merit and is defined as: 2p the energy stored in the damped harmonic oscillator to the energy loss per cycle. angular velocity speed te by which angular position of mass (in simple pendulum) is varying. te energy relaxation time time in which energy of oscillator decreases to E/e. te angle turned during which energy decays to E/e.  Characterization of damping Logarithmic decrement Relaxation time Amplitude relaxation time ta = 1/r Energy relaxation time te = 1/2r 2te = ta d = rT = T/ta Quality factor te  Examples of damping: Eddy Currents An emf and a current are induced in a circuit by a changing magnetic flux. If a conductor and a magnetic field are in relative motion, the magnetic force on charged particles in the conductor causes circulating currents. These circulating currents are called eddy currents. These are induced in bulk pieces of metal moving through a magnetic field.  Application: Braking system The braking systems on many subway and rapid-transit cars make use of electro-magnetic induction and eddy currents. An electromagnet attached to the train is positioned near the steel rails. The braking action occurs when a large current is passed through the electro- magnet. The relative motion of the magnet and rails induces eddy currents in the rails, and the direction of these currents produces a drag force on the moving train. Also, as a safety measure, some power tools use eddy currents to stop rapidly spinning blades once the device is turned off.  Application: Induction stove An ac current in a coil in the stove top produces a changing magnetic field at the bottom of a metal pan. The changing magnetic field gives rise to a current in the bottom of the pan. Because the pan has resistance, the current heats the pan. If the coil in the stove has low resistance it doesn’t get hot but the pan does. An insulator won’t heat up on an induction stove. Nevertheless, some believe induction stoves are hazardous.