Expressions for Velocity of Sound in Different Media

Expressions for Velocity of Sound in Different Media VELOCITY OF SOUND WAVE IN STRINGS: The velocity, V of a sound wave in strings is given by the expression. V= , = mass per unit length or linear density = Where r =radius of the wire, = density of material of the string or wire and T = tension VELOCITY OF SOUND WAVE IN SOLIDS The velocity, V of a sound wave in a solid is given by the expression: Where E = Young’s modulus of the material, = density of the solid or material. VELOCITY OF SOUND WAVE IN LIQUID The velocity, V of a sound wave in a liquid is given by the expression: Where B = Bulk Modulus of the liquid, = density of the liquid. VELOCITY OF SOUND WAVE IN A GAS The velocity, V of a sound wave in a gas is given by the expression Where M = molecular mass, R = molar gas constant, = ratio of the two specific heat capacities of a gas, P = pressure and = density VELOCITY OF WATER WAVE For deep water waves, V = For shallow water waves, V= For surface ripples, V = Where = wavelength, d = depth of water, = surface tension, =density of water, g = acceleration due to gravity. The Harmonic Oscillator Consider a simple pendulum consisting of a mass-less string of length ‘l’ and a point like object of mass ‘m’ attached to one end called the bob. Suppose the string is fixed at the other end and is initially pulled out at an angle from the vertical and released from rest from the figure below. Neglect any dissipation due to air resistance or frictional forces acting at the pivot. Diagram Note Is defined with respect to the equilibrium position. When, the bob has moved to the right. When, the bob has moved to the left. Coordinate system free-body force diagram Tangential component of the gravitational force is (1) Note The 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 tangential component of acceleration is (2) Newton’s second law, , yields (3) T= (4) Simple Harmonic Motion Diagram The object is attached to one end of a spring. The other end of the spring 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). Note x>0 corresponds to an extended spring. x Therefore (5) Newton’s second law in the x-direction becomes (6) Equation 6 is called the simple harmonic oscillator equation. Because the spring force depends on the distance x, the acceleration is not constant. is constant of proportionality Energy in Simple Harmonic Motion Diagram (7) (8) It is easy to calculate the velocity for a given t value (9) And the energy associated with (10) A stretched or compressed spring has certain potential energy. Diagrams ( Hooke’s law) in order to stretch the spring from O to X one need to do work; the force changes, so we have to integrate: W= (11) Note This work is stored in the spring as its potential energy U. So, for the oscillator considered, the energy U is: U= (12) Therefore, the total energy is: (13) (14) (15) (16) Equation (16) is a famous expression for the energy of a harmonic oscillator. Note Where A is the maximum displacement. The total energy is constant in time(t), but there is continuous process of converting to kinetic energy to potential energy, and then K back to U. K reaches maximum twice every cycle (when passing through x=0)’ and U reaches maximum twice, at the turning point. Diagram0 In this graph time(t) was set to zero when the mass passed the x=0 point. Finally, we can use the principle of conservation of energy to obtain velocity for an arbitrary position by expressing the total energy position as (17) (18) (19) Example 1 A 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. The block is displaced 5.00cm from equilibrium and released from rest. Find the period of its motion Determine the maximum speed of the block What is the maximum acceleration of the block? Express the position, speed and acceleration as function of time. Example 2 A 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 amplitude of the motion is 3.0cm What is the velocity of the cart when the position is 2.00cm? Compute the kinetic energy and the potential energy of the system when the position is 2.00cm. Energy in waves Note Waves transport energy when they propagate through a medium. Consider a sinusoidal wave travelling on a string. The source of the energy is some external agent 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 element 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 same angular frequency and the same amplitude A. The kinetic energy K associated with a moving particle is: K= (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 differential relationship: (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) Note In 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 exactly 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) Note This expression shows that the rate of energy transfer by a sinusoidal wave on a string is proportional to The square of the frequency The square of the amplitude And the wave speed. Put differently, Is the rate of energy transfer in any sinusoidal wave that is proportional to the square of its amplitude. Example A taut 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 WAVES Stationary Waves Stationary 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. Note You can produce stationary wave on a rope if you tie one end of it to a wall and move the free end up and 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 amplitude Node is an area of zero amplitude. COMPARISON BETWEEN PROGRESSIVE (TRAVELLING) WAVE AND STATIONARY (STANDING) WAVE. Example3 A wave is given by the equation y= 10sin2. Find the loop length frequency, velocity and maximum amplitude of the stationary wave produced. solution

Americas Racial Freedom :: Free Essays Online

America's Racial Freedom THE ANTI-SLAVERY FAIR & THE LIBERTY BELL: The Beginnings of America’s Racial Freedom 1848 was not a year commonly associated with monumental advancements or changes in the anti-slavery cause in the United States of America. Most individuals today see slavery and the abolitionist movement as an issue that had its beginnings and endings in the 1860s. That particular decade was a complete turning point in the history of our country; however, the long hard road that finally led to the emancipation of slaves tends to be overlooked. From the moment that the first slave stepped foot upon American soil, there were individuals who believed in a greater cause – humanity. Considered radicals at the time because of their comparatively reprehensible ideals, abolitionists traveled across the United States spreading a message of freedom and equality for all mankind. Unfortunately, most of the following generations did not have the opportunity to see these movers and shakers in action. With initial appearances around 1820, one of the only strong remaining links we have to tea ch us about this cause are the manuscripts, stories, poems, and other literary works written by those that were unafraid to bear their souls in hopes of forming a new way of life in American society (â€Å"Mosaic†). The United States of America was undoubtedly a late bloomer when it came to accepting new ideas regarding the validity of slavery. British radicals had formed a group called the Anti-Slavery Society in the late eighteenth century to put an end to slavery within British colonies, a task the organization accomplished. Nearly fifty years later, the Americans adapted their own version of this group, ingeniously calling themselves the American Anti-Slavery Society (AASS) (â€Å"American Anti-Slavery Society†). The lack of ingenuity in naming this organization did not at all reflect upon the enthusiasm and hard work these individuals put into their cause. This newly established abolitionist group came into the public’s eye without any hesitation and they projected a harsh reality to sympathizers and slave owners alike. Their goal was to let their message be heard by as many Americans as possible, which led to the publication of numerous anti-slavery periodicals. These periodi cals were sold at the abolitionist fairs being held regularly all over the country. With these anti-slavery crusades in full swing, controversy also developed regarding the idea of women in leadership positions. In an era where women had just about as many rights as the black folk, many stubborn white men called their expertise into question.

To Kill A Mocking Bird :: To Kill a Mockingbird Essays

To Kill a Mockingbird is set in Maycomb County, an imaginary district in southern Alabama. The time is the years of the Great Depression in the United States. The mood of the novel is mostly light and humorous, especially when talking about the children’s antics. However, another mood throughout the novel is somber and calm, because come important issues are being valued and dealt with. Atticus’ dealings with the blacks, the negative attitudes of some other members of the community, the trial of Tom Robinson and his gruesome end, depicts a seriousness and a grave reconsideration of accepted beliefs, which is expected of the readers by the author. Atticus Finch, the father of Scout and Jem, is a highly respected and responsible citizen of Maycomb County. An attorney by profession, he has always tried to instill good values and a sense of moral in his children. Jem is a true brother to Scout, helping her out of scrapes, escorting her to school and back, guiding her at times and comforting her in general. When he is given money to buy something for himself, he buys a gift for Scout too. When he finds out that Scout has eaten the gum found in the knothole of the oak tree, he insists that she gargle her throat. When she muddles up her role in the pageant and is mortified, Jem is the one to console her. He displays much genuine concern and consideration in dealing with his unruly sister. Scout, because of her age, and being the youngest in the family, is impulsive by nature and extremely emotional too. She unthinkingly rushes into fights and scrapes, cries when her ego is hurt and is generally is rash in her actions. Conflict- The protagonist of the novel is Atticus Finch, who is the prime initiator and coordinator of various events in the novel. In his involvement with the poor whites of the community, like Walter Cunningham, as well as the deprived blacks, like Tom Robinson, he is portrayed as a just, sincere and a greatly considerate human being. He has clear-cut values and beliefs, and it is his sincere wish that his children too grow up with a broad outlook and an unprejudiced way of thinking. He is indifferent to what others have to say or think about his actions, and he is steadfast in his beliefs of equality and liberty. Bob Ewell serves as the antagonist villain in the novel, with his laid-back way of living and the utter disregard he has for other human beings.

aids awareness :: Free AIDS Essays

When AIDS first emerged, no-one could have predicted how the epidemic would spread across the world and how many millions of lives it would change. There was no real idea what caused it, and consequently, no real idea how to protect against it. Now, in 2004, we know from bitter experience that AIDS is caused by the virus HIV, and that it can devastate families, communities and whole continents. We have seen the epidemic knock decades off countries’ national development, widen the gulf between rich and poor nations and push already-stigmatized groups closer to the margins of society. We are living in an ‘international’ society, and HIV has become the first truly ‘international’ epidemic, easily crossing oceans and international borders. Just as clearly, experience shows that the right approaches, applied quickly enough with courage and resolve, can and do result in lower national HIV infection rates and less suffering for those affected by the epidemic. Globally, we have learned that if a country acts early enough, a national HIV crisis can be averted. It has also been noted that a country with a very high HIV prevalence rate will often see this rate eventually stabilise, and even decline. This indicates, among other things, that people are beginning to change risky behaviour patterns, because they have seen and known people who have been killed by AIDS. Fear is the worst, and last way of changing people’s behaviour and by the time that this happens it is usually too late to save a huge number of that country’s population. Already, more than twenty million people around the world have died of AIDS-related diseases. In 2004, 3.1 million men, women and children have died. Around twice the amount who have died until now - almost 40 million - are now living with HIV, and most of these are likely to die over the next decade or so. The most recent UNAIDS/WHO estimates show that, in 2004 alone, 4.9 million people were newly infected with HIV. It is disappointing that the global numbers of people infected with HIV continue to rise, despite the fact that effective prevention strategies already exist. Africa It is in Africa, in some of the poorest countries in the world, that the impact of the virus has been most severe. Altogether, there are now 16 countries in Africa in which more than one-tenth of the adult population aged 15-49 is infected with HIV.

Religious Freedom Restoration Act :: essays research papers

Religious Freedom Restoration Act In this paper I will describe the Religious Freedom Restoration Act. This Act was used to contradict the decision of the court case of Employment Division v. Smith, which allowed the government to forbid any religious act without giving a reason. The RFRA brought back the requirement that the government provide an adequate reason to forbid any religious act. The government once again had to show that the act was of compelling interest against the state. In 1993 one of the most important acts that has gone thorough Congress was passed (Religious Freedom, Map of the RFRA). This was the Religious Freedom Restoration Act (RFRA) of 1993 (Religious Freedom, Map of the RFRA). This act was passed to answer the 1990 court case Employment Division v. Smith (Questions and Answers, Map of the RFRA). Employment Division v. Smith was a court case in which the issue was whether â€Å"Sacramental use of peyote by members of the Native American Church was protected under the free exercise clause of the First Amendment, which provides that ‘Congress shall make no law...prohibiting the free exercise of religion'.†(Questions and Answers, Map of the RFRA). According to Justice Scalia, â€Å"if prohibiting the exercise of religion was merely the incidental effect of a generally applicable and otherwise valid provision, the First Amendment was not offended.† (Questions and Answers, Map of the RFRA). Thus, "...the government no longer had to justify most burdens on religious exercise. The free exercise clause offered protection only if a particular religious practice was singled out for discriminatory treatment. In short, free exercise was a sub category of equal protection. This placed religious rights in an inferior position to other First Amendment rights such as freedom of speech and press." (Questions and Answers, Map of the RFRA). This court case caused a series of court cases about religious freedoms (Religious Freedom, Map of the RFRA). Congress enacted the RFRA to contradict the negative affect that court cases had recently had on religious freedoms(Religious Freedom, Map of the RFRA). The RFRA is what it states it is in the title, a restoration act(Religious Freedom, Map of the RFRA). Congress decided that in Employment Division v. Smith, "the supreme court virtually eliminated the requirement that the government justify burdens on religious exercise imposed by laws neutral toward religion and the compelling interest test as set forth in prior Federal court rulings is a workable test for striking sensible balances between religious liberty and competing prior governmental interests."(Religious Freedom, Map of the RFRA) In other words, the government did not have to have a reason to impose laws

Division/Classification Essay: Three types of children Essay

Anyone who has spent time with or around children will notice that each one has a special personality all of their own. Children, like adults, have different traits that make up their personalities. Experts have researched this phenomenon in detail and classified children into different categories. The three categories that most experts agree with have been named â€Å"flexible,† â€Å"fearful,† and â€Å"feisty.† Children generally may have similar interests, but the way they interact and deal with these interests displays their personality type. The first personality type is called flexible. This is the most common of the three types. About 40 percent of all children fall into the flexible or easy group. These children usually handle feelings of anger and disappointment by reacting mildly upset. This does not mean that they do not feel mad or disappointed, they just choose to react mildly. These actions mean the flexible child is easy to take care of and be around. They usually adapt to new situations and activities quickly, are toilet-trained easily, and are generally cheerful. Flexible children are subtle in their need for attention. Rather than yelling and demanding it, they will slowly and politely let their caregiver know about the need. If they do not get the attention right away, they seldom make a fuss. They patiently wait, but they still make it known that they need the attention. These children also are easygoing, so routines like feeding and napping are regular. The next temperament is the fearful type. These are the more quiet and shy children. This makes up about 15 percent of children. They adapt slowly to new environments and take longer than flexible children when warming up to things. When presented with a new environment, fearful children often cling to something or someone familiar. Whether it is the main caregiver or a material object such as a blanket, the fearful child will cling to it until they feel comfortable with the new situation. This can result in a deep attachment of the child to a particular caregiver or object. Fearful children may also withdraw when pushed into a new situation too quickly. They may also withdraw when other children are jumping into a new project or situation they are not comfortable with. These children may tend to play alone rather than with a group. The third temperament type is called feisty. About 10 percent of children fit into this category. A feisty child expresses their opinions in a very intense way. Whether they are happy or mad, everyone around them will know how they feel. These children remain active most of the time, and this causes them to be very aggressive. Feisty children often have the tendency to have a negative persistence and will go on and on nagging, whining and negotiating if there is something they particularly want. Unlike flexible children, feisty children are irregular in their napping and feeding times, but they do not adapt well to changes in their routines. They get used to things and won’t give them up. Anything out of the ordinary could send them into some type of fit. If these children are not warned of a change, they may react very negatively. Feisty children also tend to be very sensitive to their surrounding environment. As a result, they may have strong reactions to their surroundings. Generally speaking, children can be divided into three groups, but caregivers must not forget that each child is an individual. Children may have the traits of all three of the personality groups, but they are categorized into the one they are most like. Whatever their temperament, children need to be treated according to their individual needs. When these needs are met appropriately the child will be happier, and those around the child will feel better also. Knowing the general personality types and how to react to them will help to make the caregiver’s job much easier and aid in the relief of unnecessary stress.

Howard Zinn Chapter One

Columbus has always been portrayed as an enlightened, peaceful explorer who â€Å"discovered† a new world, and became friends with the native people. Howard Zinn’s view on Columbus’s encounter with the natives is an entirely different perspective. Zinn describes Columbus as a man who is willing to torture and kill others to be able to accomplish what he wants; in this case he wanted to obtain gold and other resources to take back with him to Spain.When Columbus and his men arrived to the islands, he noticed that the natives were generous, and accommodating because they willingly traded everything they owned and brought them such things like: food, water, and gifts. Since the beginning the natives offered all of their hospitality to Columbus and his men. Columbus believed that the natives were ignorant because they had no weapons for self defense; when he showed them a sword, they had no idea how to use it and ended up cutting themselves.Columbus was more than cer tain, that he could take control over the natives, and captivate them as slaves. In his writing he wrote, â€Å"They would make fine servants†¦. With fifty men we could subjugate them all and make them do whatever we want. † (Zinn, 3) This attitude leaded to enslavement, feeling superior, and genocide by Columbus and his men towards the natives. Columbus had persuaded the king and queen of Spain to finance an expedition to the lands, and the wealth; he expected to be at the Indies and Asia.Columbus would receive ten percent of all the goods collected, governorship over new-found lands, and the fame that would go with a new title: Admiral of the Ocean Sea. Columbus believed the natives could lead him and his men to where the gold was, to be able to take it back to the King and Queen so he took many of them as prisoners on his ship. He also wanted to take them back to Spain as slaves and be able to make them do whatever they wanted them to do. Columbus took advantage of th e ignorance of the natives, and made them do hard labor for him and his men.Columbus believed that there were huge fields of gold, in the province of Cicao on Haiti. He became desperate to pay the dues back to the king and queen, so he order every native older than fourteen to collect a certain quantity of gold by three months. Once they had collected the amount, they would receive a copper to put around their neck, those without it had their hands cut off and bled to death. Many felt it was impossible to do this, and tried escaping but were always found and killed.When it was finally clear that there was no gold left, they took them all as slave labor to huge estates which are called encomiendas. Columbus’s big plan for Hispaniola since the beginning was to take advantage of the natives and take their land, and the gold he believed was located there. He built the first fort in the Western Hemisphere, and left some of his men to find and store gold there. Columbus had to ask for a little more help from their majesties, he convinced them by saying he would take them â€Å"as much gold as they need †¦ nd as many slaves as they ask† (Zinn,6 ) Columbus’s plans affected the natives, in many ways; first of all they were going to lose their land, and also they were going to be taken captive for slave labor. Howard Zinn not only introduced a new perspective on Christopher Columbus, but he changed the way I viewed things. I never knew how much killings, and torture was put upon the natives in the searching of the â€Å"New World†. I learned how the hero I thought Columbus was, in reality was more like a villain. He didn’t care what he had to do as long as he got what he wanted, even if it dealt with murders and torturing the natives.