13 Chapter Vibrations and Waves Hooke's Law Fs = - k x Fs is the spring force k is the spring constant It is a measure of the stiffness of the spring A large k indicates a stiff spring and a small k indicates a soft spring x is the displacement of the object from its equilibrium position x = 0 at the equilibrium position The negative sign indicates that the force is always directed opposite to the displacement Hooke's Law Force The force always acts toward the equilibrium position It is called the restoring force The direction of the restoring force is such that the object is being either pushed or pulled toward the equilibrium position Hooke's Law Applied to a Spring Mass System Motion of the Spring-Mass System Assume the object is initially pulled to a distance A and released from rest As the object moves toward the equilibrium position, F and a decrease, but v increases At x = 0, F and a are zero, but v is a maximum The object's momentum causes it to overshoot the equilibrium position Motion of the Spring-Mass System, cont The force and acceleration start to increase in the opposite direction and velocity decreases The motion momentarily comes to a stop at x = - A It then accelerates back toward the equilibrium position The motion continues indefinitely Simple Harmonic Motion Motion that occurs when the net force along the direction of motion obeys Hooke's Law The force is proportional to the displacement and always directed toward the equilibrium position The motion of a spring mass system is an example of Simple Harmonic Motion Simple Harmonic Motion, cont. Not all periodic motion over the same path can be considered Simple Harmonic motion To be Simple Harmonic motion, the force needs to obey Hooke's Law Amplitude Amplitude, A The amplitude is the maximum position of the object relative to the equilibrium position In the absence of friction, an object in simple harmonic motion will oscillate between the positions x = A Period and Frequency
The
period, T, is the time that it takes for the object to complete one complete cycle of motion From x = A to x = - A and back to x = A The frequency, , is the number of complete cycles or vibrations per unit time = 1 / T Frequency is the reciprocal of the period Acceleration of an Object in Simple Harmonic Motion Newton's second law will relate force and acceleration The force is given by Hooke's Law F = - k x = m a a = -kx / m The acceleration is a function of position Acceleration is not constant and therefore the uniformly accelerated motion equation cannot be applied Elastic Potential Energy A compressed spring has potential energy The compressed spring, when allowed to expand, can apply a force to an object The potential energy of the spring can be transformed into kinetic energy of the object Elastic Potential Energy, cont The energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy 2 PEs = kx The energy is stored only when the spring is stretched or compressed Elastic potential energy can be added to the statements of Conservation of Energy and Work-Energy Energy in a Spring Mass System Energy Transformations Energy Transformations, 2 Energy Transformations, 3 Energy Transformations, 4 Velocity as a Function of Position Conservation of Energy allows a calculation of the velocity of the object at any position in its motion
Speed is a maximum at x = 0 Speed is zero at x = A The indicates the object can be traveling in either direction Simple Harmonic Motion and Uniform Circular Motion Period and Frequency from Circular Motion Period
This gives the time required for an object of mass m attached to a spring of constant k to complete one cycle of its motion
Frequency
Units are cycles/second or Hertz, Hz
Angular Frequency The angular frequency is related to the frequency
The
frequency gives the number of cycles per second The angular frequency gives the number of radians per second Effective Spring 2 Mass A graph of T versus m does not pass through the origin The spring has mass and oscillates For a cylindrical spring, the effective additional mass of a light spring is 1/3 the mass of the spring Motion as a Function of Time Graphical Representation of Motion Motion Equations Remember, the uniformly accelerated motion equations cannot be used x = A cos (2t) = A cos t v = -2A sin (2t) = sin t -A 2 2 a = -4 A cos (2t) = 2 -A cos t Verification of Sinusoidal Nature Simple Pendulum Simple Pendulum, cont In general, the motion of a pendulum is not simple harmonic However, for small angles, it becomes simple harmonic In general, angles < 15 are small enough sin = Ft = - m g This force obeys Hooke's Law Period of Simple Pendulum
This shows that the period is independent of the amplitude The period depends on the length of the pendulum and the acceleration of gravity at
the
location of the pendulum Simple Pendulum Compared to a Spring-Mass System Physical Pendulum A physical pendulum can be made from an object of any shape The center of mass oscillates along a circular arc Period of a Physical Pendulum The period of a physical pendulum is given by
I is the object's moment of inertia m is the object's mass 2 For a simple pendulum, I = mL and the equation becomes that of the simple pendulum as seen before Damped Oscillations Only ideal systems oscillate indefinitely In real systems, friction retards the motion Friction reduces the total energy of the system and the oscillation is said to be damped
Damped Oscillations, cont. More Types of Damping With a higher viscosity, the object returns rapidly to equilibrium after it is released and does not oscillate The system is said to be critically damped With an even higher viscosity, the piston returns to equilibrium without passing through the equilibrium position, but the time required is longer This is said to be over damped Graphs of Damped Oscillators Wave Motion A wave is the motion of a disturbance Mechanical waves require Some source of disturbance A medium that can be disturbed Some physical connection between or mechanism though which adjacent portions of the medium influence each other All waves carry energy and momentum Types of Waves Traveling Waves Flip one end of a long rope that is under tension and fixed at one end The pulse travels to the right with a definite speed A disturbance of this type is called a traveling wave Types of Waves Transverse Types of Waves Longitudinal Other Types of Waves Waves may be a combination of transverse and longitudinal A soliton consists of a solitary wave front that propagates in isolation First studied by John Scott Russell in 1849 Now used widely to model physical phenomena Waveform A Picture of a Wave Longitudinal Wave Represented as a Sine Curve Description of a Wave A steady stream of pulses on a very long string produces a continuous wave The blade oscillates in simple harmonic motion Each small segment of the string, such as P, oscillates with simple harmonic motion
Amplitude and Wavelength Speed of a Wave v = Is derived from the basic speed equation of distance/time This is a general equation that can be applied to many types of waves Speed of a Wave on a String The speed on a wave stretched under some tension, F is called the linear density speed depends only upon the properties of the medium through which the disturbance travels
The
Interference of Waves Two traveling waves can meet and pass through each other without being destroyed or even altered Waves obey the Superposition Principle If two or more traveling waves are moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point Actually only true for waves with small amplitudes Constructive Interference Constructive Interference in a String Destructive Interference Destructive Interference in a String Reflection of Waves Fixed End Reflected Wave Free End