Kinematics and Dynamics - Real-life applications
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K INEMATICS : H OW O BJECTS M OVE
By the sixteenth century, the Aristotelian world-view had become so deeply ingrained that few European thinkers would have considered the possibility that it could be challenged. Professors all over Europe taught Aristotle's precepts to their students, and in this regard the University of Pisa in Italy was no different. Yet from its classrooms would emerge a young man who not only questioned, but ultimately overturned the Aristotelian model: Galileo Galilei (1564-1642.)Challenges to Aristotle had been slowly growing within the scientific communities of the Arab and later the European worlds during the preceding millennium. Yet the ideas that most influenced Galileo in his break with Aristotle came not from a physicist but from an astronomer, Nicolaus Copernicus (1473-1543.) It was Copernicus who made a case, based purely on astronomical observation, that the Sun and not Earth was at the center of the universe.
Galileo embraced this model of the cosmos, but was later forced to renounce it on orders from the pope in Rome. At that time, of course, the Catholic Church remained the single most powerful political entity in Europe, and its endorsement of Aristotelian views—which philosophers had long since reconciled with Christian ideas—is a measure of Aristotle's impact on thinking.
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GALILEO'S REVOLUTION IN PHYSICS.
After his censure by the Church, Galileo was placed under house arrest and was forbidden to study astronomy. Instead he turned to physics—where, ironically, he struck the blow that would destroy the bankrupt scientific system endorsed by Rome. In 1638, he published Discourses and Mathematical Demonstrations Concerning Two New Sciences Pertaining to Mathematics and Local Motion, a work usually referred to as Two New Sciences. In it, he laid the groundwork for physics by emphasizing a new method that included experimentation, demonstration, and quantification of results.In this book—highly readable for a work of physics written in the seventeenth century—Galileo used a dialogue, an established format among philosophers and scientists of the past.
The character of Salviati argued for Galileo's ideas and Simplicio for those of Aristotle, while the genial Sagredo sat by and made occasional comments. Through Salviati, Galileo chose to challenge Aristotle on an issue that to most people at the time seemed relatively settled: the claim that objects fall at differing speeds according to their weight.
In order to proceed with his aim, Galileo had to introduce a number of innovations, and indeed, he established the subdiscipline of kinematics, or how objects move. Aristotle had indicated that when objects fall, they fall at the same rate from the moment they begin to fall until they reach their "natural" position. Galileo, on the other hand, suggested an aspect of motion, unknown at the time, that became an integral part of studies in physics: acceleration.
S CALARS AND V ECTORS
Even today, many people remain confused as to what acceleration is. Most assume that acceleration means only an increase in speed, but in fact this represents only one of several examples of acceleration. Acceleration is directly related to velocity, often mistakenly identified with speed.In fact, speed is what scientists today would call a scalar quantity, or one that possesses magnitude but no specific direction. Speed is the rate at which the position of an object changes over a given period of time; thus people say "miles (or kilometers) per hour." A story problem concerning speed might state that "A train leaves New York City at a rate of 60 miles (96.6 km/h). How far will it have traveled in 73 minutes?"
Note that there is no reference to direction, whereas if the story problem concerned velocity—a vector, that is, a quantity involving both magnitude and direction—it would include some crucial qualifying phrase after "New York City": "for Boston," perhaps, or "northward." In practice, the difference between speed and velocity is nearly as large as that between a math problem and real life: few people think in terms of driving 60 miles, for instance, without also considering the direction they are traveling.
RESULTANTS.
One can apply the same formula with velocity, though the process is more complicated. To obtain change in distance, one must add vectors, and this is best done by means of a diagram. You can draw each vector as an arrow on a graph, with the tail of each vector at the head of the previous one. Then it is possible to draw a vector from the tail of the first to the head of the last. This is the sum of the vectors, known as a resultant, which measures the net change.Suppose, for instance, that a car travels east 4 mi (6.44 km), then due north 3 mi (4.83 km). This may be drawn on a graph with four units along the x axis, then 3 units along the y axis, making two sides of a triangle. The number of sides to the resulting shape is always one more than the number of vectors being added; the final side is the resultant. From the tail of the first segment, a diagonal line drawn to the head of the last will yield a measurement of 5 units—the resultant, which in this case would be equal to 5 mi (8 km) in a northeasterly direction.
VELOCITY AND ACCELERATION.
The directional component of velocity makes it possible to consider forms of motion other than linear, or straight-line, movement. Principal among these is circular, or rotational motion, in which an object continually changes direction and thus, velocity. Also significant is projectile motion, in which an object is thrown, shot, or hurled, describing a path that is a combination of horizontal and vertical components.Furthermore, velocity is a key component in acceleration, which is defined as a change in velocity. Hence, acceleration can mean one of five things: an increase in speed with no change in direction (the popular, but incorrect, definition of the overall concept); a decrease in speed with no change in direction; a decrease or increase of speed with a change in direction; or a change in direction with no change in speed. If a car speeds up or slows down while traveling in a straight line, it experiences acceleration. So too does an object moving in rotational motion, even if its speed does not change, because its direction will change continuously.
D YNAMICS : W HY O BJECTS M OVE
GALILEO'S TEST.
To return to Galileo, he was concerned primarily with a specific form of acceleration, that which occurs due to the force of gravity. Aristotle had provided an explanation of gravity—if a highly flawed one—with his claim that objects fall to their "natural" position; Galileo set out to develop the first truly scientific explanation concerning how objects fall to the ground.According to Galileo's predictions, two metal balls of differing sizes would fall with the same rate of acceleration. To test his hypotheses, however, he could not simply drop two balls from a rooftop—or have someone else do so while he stood on the ground—and measure their rate of fall. Objects fall too fast, and lacking sophisticated equipment available to scientists today, he had to find another means of showing the rate at which they fell.
This he did by resorting to a method Aristotle had shunned: the use of mathematics as a means of modeling the behavior of objects. This is such a deeply ingrained aspect of science today that it is hard to imagine a time when anyone would have questioned it, and that very fact is a tribute to Galileo's achievement. Since he could not measure speed, he set out to find an equation relating total distance to total time. Through a detailed series of steps, Galileo discovered that in uniform or constant acceleration from rest—that is, the acceleration he believed an object experiences due to gravity—there is a proportional relationship between distance and time.
With this mathematical model, Galileo could demonstrate uniform acceleration. He did this by using an experimental model for which observation was easier than in the case of two falling bodies: an inclined plane, down which he rolled a perfectly round ball. This allowed him to extrapolate that in free fall, though velocity was greater, the same proportions still applied and therefore, acceleration was constant.
POINTING THE WAY TOWARD NEWTON.
The effects of Galileo's system were enormous: he demonstrated mathematically that acceleration is constant, and established a method of hypothesis and experiment that became the basis of subsequent scientific investigation. He did not, however, attempt to calculate a figure for the acceleration of bodies in free fall; nor did he attempt to explain the overall principle of gravity, or indeed why objects move as they do—the focus of a subdiscipline known as dynamics.At the end of Two New Sciences, Sagredo offered a hopeful prediction: "I really believe that… the principles which are set forth in this little treatise will, when taken up by speculative minds, lead to another more remarkable result…." This prediction would come true with the work of a man who, because he lived in a somewhat more enlightened time—and because he lived in England, where the pope had no power—was free to explore the implications of his physical studies without fear of Rome's intervention. Born in the very year Galileo died, his name was Sir Isaac Newton (1642-1727.)
NEWTON'S THREE LAWS OF MOTION.
In discussing the movement of the planets, Galileo had coined the term inertia to describe the tendency of an object in motion to remain in motion, and an object at rest to remain at rest. This idea would be the starting point of Newton's three laws of motion, and Newton would greatly expand on the concept of inertia.The three laws themselves are so significant to the understanding of physics that they are treated separately elsewhere in this volume; here they are considered primarily in terms of their implications regarding the larger topic of matter and motion.
Introduced by Newton in his Principia (1687), the three laws are:
- First law of motion: An object at rest will remain at rest, and an object in motion will remain in motion, at a constant velocity unless or until outside forces act upon it.
- Second law of motion: The net force acting upon an object is a product of its mass multiplied by its acceleration.
- Third law of motion: When one object exerts a force on another, the second object exerts on the first a force equal in magnitude but opposite in direction.
MASS AND GRAVITATIONAL ACCELERATION.
The first law establishes the principle of inertia, and the second law makes reference to the means by which inertia is measured: mass, or the resistance of an object to a change in its motion—including a change in velocity. Mass is one of the most fundamental notions in the world of physics, and it too is the subject of a popular misconception—one which confuses it with weight. In fact, weight is a force, equal to mass multiplied by the acceleration due to gravity.It was Newton, through a complicated series of steps he explained in his Principia, who made possible the calculation of that acceleration—an act of quantification that had eluded Galileo. The figure most often used for gravitational acceleration at sea level is 32 ft (9.8 m) per second squared. This means that in the first second, an object falls at a velocity of 32 ft per second, but its velocity is also increasing at a rate of 32 ft per second per second. Hence, after 2 seconds, its velocity will be 64 ft (per second; after 3 seconds 96 ft per second, and so on.
Mass does not vary anywhere in the universe, whereas weight changes with any change in the gravitational field. When United States astronaut Neil Armstrong planted the American flag on the Moon in 1969, the flagpole (and indeed Armstrong himself) weighed much less than on Earth. Yet it would have required exactly the same amount of force to move the pole (or, again, Armstrong) from side to side as it would have on Earth, because their mass and therefore their inertia had not changed.
B EYOND M ECHANICS
The implications of Newton's three laws go far beyond what has been described here; but again, these laws, as well as gravity itself, receive a much more thorough treatment elsewhere in this volume. What is important in this context is the gradually unfolding understanding of matter and motion that formed the basis for the study of physics today.After Newton came the Swiss mathematician and physicist Daniel Bernoulli (1700-1782), who pioneered another subdiscipline, fluid dynamics, which encompasses the behavior of liquids and gases in contact with solid objects. Air itself is an example of a fluid, in the scientific sense of the term. Through studies in fluid dynamics, it became possible to explain the principles of air resistance that cause a leaf to fall more slowly than a stone—even though the two are subject to exactly the same gravitational acceleration, and would fall at the same speed in a vacuum.
EXTENDING THE REALM OF PHYSICAL STUDY.
The work of Galileo, Newton, and Bernoulli fit within one of five major divisions of classical physics: mechanics, or the study of matter, motion, and forces. The other principal divisions are acoustics, or studies in sound; optics, the study of light; thermodynamics, or investigations regarding the relationships between heat and other varieties of energy; and electricity and magnetism. (These subjects, and subdivisions within them, also receive extensive treatment elsewhere in this book.)Newton identified one type of force, gravitation, but in the period leading up to the time of Scottish physicist James Clerk Maxwell (1831-1879), scientists gradually became aware of a new fundamental interaction in the universe. Building on studies of numerous scientists, Maxwell hypothesized that electricity and magnetism are in fact differing manifestations of a second variety of force, electromagnetism.
MODERN PHYSICS.
The term classical physics, used above, refers to the subjects of study from Galileo's time through the end of the nineteenth century. Classical physics deals primarily with subjects that can be discerned by the senses, and addressed processes that could be observed on a large scale. By contrast, modern physics, which had its beginnings with the work of Max Planck (1858-1947), Albert Einstein (1879-1955), Niels Bohr (1885-1962), and others at the beginning of the twentieth century, addresses quite a different set of topics.Modern physics is concerned primarily with the behavior of matter at the molecular, atomic, or subatomic level, and thus its truths cannot be grasped with the aid of the senses. Nor is classical physics much help in understanding modern physics. The latter, in fact, recognizes two forces unknown to classical physicists: weak nuclear force, which causes the decay of some subatomic particles, and strong nuclear force, which binds the nuclei of atoms with a force 1 trillion (10 12 ) times as great as that of the weak nuclear force.
Things happen in the realm of modern physics that would have been inconceivable to classical physicists. For instance, according to quantum mechanics—first developed by Planck—it is not possible to make a measurement without affecting the object (e.g., an electron) being measured. Yet even atomic, nuclear, and particle physics can be understood in terms of their effects on the world of experience: challenging as these subjects are, they still concern—though within a much more complex framework—the physical fundamentals of matter and motion.
WHERE TO LEARN MORE
Ballard, Carol. How Do We Move? Austin, TX: Raintree Steck-Vaughn, 1998.Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.
Fleisher, Paul. Objects in Motion: Principles of Classical Mechanics. Minneapolis, MN: Lerner Publications, 2002.
Hewitt, Sally. Forces Around Us. New York: Children's Press, 1998.
Measure for Measure: Sites That Do the Work for You (Web site). <http://www.wolinskyweb.com/measure.html> (March 7, 2001).
Motion, Energy, and Simple Machines (Web site). <http://www.necc.mass.edu/MRVIS/MR3_13/start.ht ml> (March 7, 2001).
Physlink.com (Web site). <http://www.physlink.com> (March 7, 2001).
Rutherford, F. James; Gerald Holton; and Fletcher G. Watson. Project Physics. New York: Holt, Rinehart, and Winston, 1981.
Wilson, Jerry D. Physics: Concepts and Applications, second edition. Lexington, MA: D. C. Heath, 1981.
Read more: http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Kinematics-and-Dynamics-Real-life-applications.html#ixzz1SFEp6neJ
In this tutorial we will examine some of the elementary ideas concerning vectors. The reason for this introduction to vectors is that many concepts in science, for example, displacement, velocity, force, acceleration, have a size or magnitude, but also they have associated with them the idea of a direction. And it is obviously more convenient to represent both quantities by just one symbol. That is the vector.
Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude. This is shown in Panel 1. . If we denote one end of the arrow by the origin O and the tip of the arrow by Q. Then the vector may be represented algebraically by OQ. 
. The line and arrow above the Q are there to indicate that the symbol represents a vector. Another notation is boldface type as: Q.
. Even though their lengths are identical, their directions are exactly opposite, in fact OQ = -QO.
#1 Two vectors, A and B are equal if they have the same magnitude and direction, regardless of whether they have the same initial points, as shown in 
We can now define vector addition. The sum of two vectors, A and B, is a vector C, which is obtained by placing the initial point of B on the final point of A, and then drawing a line from the initial point of A to the final point of B , as illustrated in Panel 4. This is sometines referred to as the "Tip-to-Tail" method.
Vector subtraction is defined in the following way. The difference of two vectors, A - B , is a vector C that is, C = A - B
Any quantity which has a magnitude but no direction associated with it is called a "scalar". For example, speed, mass and temperature.
.The quantity 
is read as "a hat" or "a unit".
or it is sometimes denoted by 
. Similarly in the y-direction we use 
or sometimes 
. Any two-dimensional vector can now be represented by employing multiples of the unit vectors, 
+ Fy'
. Which representation to use will depend on the particular problem that you are faced with.
.For example
.
and 
. It is clear that there must be a relation between these unit vectors and those of the Cartesian system.
.
And in particular we have 
, since the angle between a vector and itself is 0 and the cosine of 0 is 1.
Alternatively, we have 
, since the angle between 
,
then 
.
Because the other terms involved,
and 
. Now what is the angle between these two vectors?
.But 
.