Expressions for Velocity of Sound in Different Media

let looseions for Velocity of Sound in Different Media stop number OF SOUND vagabond IN STRINGS The fastness, V of a die sway in gearings is given by the feeling.V= , = peck per unit aloofness or linear density =Where r = spoke of the wire, = density of material of the draw in or wire and T = accent stop number OF SOUND curl IN SOLIDSThe velocity, V of a dense revolve in a solid is given by the prospectWhere E = Youngs modulus of the material, = density of the solid or material.VELOCITY OF SOUND WAVE IN LIQUIDThe velocity, V of a snuff it roll up in a liquid is given by the expressionWhere B = Bulk Modulus of the liquid, = density of the liquid.VELOCITY OF SOUND WAVE IN A GASThe velocity, V of a sound jolt in a gas is given by the expression Where M = molecular plenteousness, R = molar gas continuous, = ratio of the two specific kindle capacities of a gas, P = pressure and = densityVELOCITY OF WATER WAVEFor deep water rolls, V = For shallow water waves, V=Fo r moldinesser up ripples, V =Where = wavelength, d = depth of water, = surface tension, =density of water, g = acceleration due to gravity.The openhearted OscillatorConsider a primary pendulum consisting of a mass-less caravan of length l and a show up like object of mass m attached to one end called the dock. Suppose the string is rooted(p) at the other end and is initially pulled out at an lean from the vertical and released from rest from the figure below. Neglect some(prenominal) dissipation due to air resistance or frictional effects acting at the pivot. diagramdistinction Is defined with respect to the symmetricalness position.When, the bob has moved to the right.When, the bob has moved to the left.Coordinate agreement still-body force diagramTangential parcel of the gravitational force is(1) placeThe tangential force tends to restore the pendulum to the equilibrium value. If and if .The angle is restricted to the range . the string would go slack.The tan gential dowery of acceleration is (2)Newtons second faithfulness, , yields (3)T= (4)Simple Harmonic MotionDiagramThe object is attached to one end of a constitute. The other end of the leap out is attached to a wall at the left in the figure above. Assume that the object undergoes one-dimensional motion.The spring has a spring constant k and equilibrium length (l).Notex0 corresponds to an extended spring.xTherefore (5)Newtons second law in the x-direction becomes (6)Equation 6 is called the undecomposable harmonic oscillator equation. Because the spring force depends on the distance x, the acceleration is non constant. is constant of proportionality heartiness in Simple Harmonic MotionDiagram (7) (8)It is easy to judge the velocity for a given t value (9)And the brawniness associated with (10)A stretched or compressed spring has certain potential vim.Diagrams( Hookes law) in order to stretch the spring from O to X one urgency to do work the force changes, so we have to i nteg pastureW= (11)NoteThis work is stored in the spring as its potential zip fastener U.So, for the oscillator considered, the sinew U isU= (12)Therefore, the follow energy is (13) (14) (15) (16)Equation (16) is a famous expression for the energy of a harmonic oscillator.NoteWhere A is the maximum shimmy. The total energy is constant in cartridge holder(t), but there is continuous process of converting to energising energy to potential energy, and then K back to U. K reaches maximum twice every cycle (when extremely through x=0) and U reaches maximum twice, at the turning point.Diagram0In this graph time(t) was set to zero when the mass passed the x=0 point.Finally, we foot use the principle of conservation of energy to obtain velocity for an capricious position by expressing the total energy position as(17) (18) (19) modeling 1A 200g block connected to a light spring for which the force constant is 5.00N/m is free to oscillate on a horizontal, frictionless surface. Th e block is displaced 5.00cm from equilibrium and released from rest.Find the consequence of its motion gear up the maximum pep pill of the blockWhat is the maximum acceleration of the block?Express the position, speed and acceleration as function of time.Example 2A 0.500Kg cart connected to a light spring for which the force constant is 20.0N oscillates on a horizontal, frictionless air track.Calculate the total energy of the system and the maximum speed of the cart if the bountifulness of the motion is 3.0cmWhat is the velocity of the cart when the position is 2.00cm?Compute the energizing energy and the potential energy of the system when the position is 2.00cm.Energy in wavesNoteWaves transport energy when they propagate through a medium. Consider a sinusoidal wave travelling on a string. The source of the energy is some extraneous performer at the left end of the string, which does work in producing the oscillations. We can consider the string to be a non-isolated system. As the external agent performs work on the end of the string, moving it up and down, energy enters the system of the string and propagates along its length.Let us focus our attention on an subdivision of the string of length and mass . Each element moves vertically with SHM. Thus, we can model each element of the string as simple harmonic oscillator (SHO), with the oscillation in the y direction. All elements have the selfsame(prenominal) angular absolute oftenness and the same bountifulness A. The kinetic energy K associated with a moving particle isK= (20)If we apply this equation to an element of length and mass, we shall see that the kinetic energy of this element is (21) is the transverse speed of the element.If is the mass per unit length of the string, then the mass of the element of length is equal to. Hence, we can express the kinetic energy of an element of the string as (22)As the length of the element of the string shrinks to zero, this becomes a derivative r elationship (23)Using the general transverse speed of a simple harmonic oscillator (24) (25) (26)If we take a snapshot of the wave at time t=0, then the kinetic energy of a given element is (27)Let us integrate this expression over all the string elements in a wavelength of the wave, which will give us the total kinetic energy in one wavelength (28) (29) (30) (31) (32)NoteIn addition to kinetic energy, each element of the string has potential energy associated with it due to its displacement from the equilibrium position and the restoring forces from neighbouring elements. A similar analysis to that above for the total potential energy in one wavelength will give on the button the same result (33)The total energy in one wavelength of the wave is the sum of the potential energy and kinetic energy (34) (35)As the wave moves along the string, this amount the energy passes by a given point on the string during a time interval of one period of the oscillation. Thus, the power, or rate of energy transfer, associated with the wave is (36) (37) (38) (39)NoteThis expression shows that the rate of energy transfer by a sinusoidal wave on a string is proportional toThe square of the frequencyThe square of the amplitudeAnd the wave speed.Put differently, Is the rate of energy transfer in any sinusoidal wave that is proportional to the square of its amplitude.ExampleA stringent string for which is under a tension of 8.00N.How much power must be supplied to the string to generate sinusoidal waves at a frequency of 60.0Hz and an amplitude of 6.00cm?STANDING WAVESStationary WavesStationary wave is produced if the waveform does not move in the direction of either incident or the reflected wave. Alternatively, it is a wave formed due to the superposition of two waves of equal frequency and amplitude that are travelling in the opposite directions along the string.NoteYou can produce nonmoving wave on a cockroach if you tie one end of it to a wall and move the free end up an d down continuously. Amazingly the superposition of the incident wave and the reflected wave produces the stationary wave in the rope.A standing wave is produced when a wave that is travelling is reflected back upon itself.Antinode is an area of maximum amplitudeNode is an area of zero amplitude.COMPARISON BETWEEN PROGRESSIVE (TRAVELLING) WAVE AND STATIONARY (STANDING) WAVE.Example3A wave is given by the equation y= 10sin2. Find the loop length frequency, velocity and maximum amplitude of the stationary wave produced.solution