LAW OF UNIVERSAL GRAVITATION

Newton's Law of Universal Gravitation

     Isaac Newton compared the acceleration of the moon to the acceleration of objects on earth. Believing that gravitational forces were responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center. But distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation

F_net = m • a

      Newton knew that the force that caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth. So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance that separates the centers of the earth and the object.

But Newton's law of universal gravitation extends gravity beyond earth. Newton's law of universal gravitation is about the universality of gravity. Newton's place in the Gravity Hall of Fame is not due to his discovery of gravity, but rather due to his discovery that gravitation is universal. ALL objects attract each other with a force of gravitational attraction. Gravity is universal. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance that separates their centers. Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as

     Since the gravitational force is directly proportional to the mass of both interacting objects, more massive objects will attract each other with a greater gravitational force. So as the mass of either object increases, the force of gravitational attraction between them also increases. If the mass of one of the objects is doubled, then the force of gravity between them is doubled. If the mass of one of the objects is tripled, then the force of gravity between them is tripled. If the mass of both of the objects is doubled, then the force of gravity between them is quadrupled; and so on.

Since gravitational force is inversely proportional to the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces. So as two objects are separated from each other, the force of gravitational attraction between them also decreases. If the separation distance between two objects is doubled (increased by a factor of 2), then the force of gravitational attraction is decreased by a factor of 4 (2 raised to the second power). If the separation distance between any two objects is tripled (increased by a factor of 3), then the force of gravitational attraction is decreased by a factor of 9 (3 raised to the second power).

The proportionalities expressed by Newton's universal law of gravitation are represented graphically by the following illustration. Observe how the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation.

      Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below.

      The constant of proportionality (G) in the above equation is known as the universal gravitation constant. The precise value of G was determined experimentally by Henry Cavendish in the century after Newton's death. (This experiment will be discussed later in Lesson 3.) The value of G is found to be

G = 6.673 x 10^-11 N m²/kg²

      The units on G may seem rather odd; nonetheless they are sensible. When the units on G are substituted into the equation above and multiplied by m₁• m₂ units and divided by d² units, the result will be Newtons - the unit of force.

Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance. As a first example, consider the following problem.

Sample Problem #1 Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a 70-kg physics student if the student is standing at sea level, a distance of 6.38 x 106 m from earth's center.

      The solution of the problem involves substituting known values of G (6.673 x 10^-11 N m²/kg²), m₁ (5.98 x 10²⁴ kg), m₂ (70 kg) and d (6.38 x 10⁶ m) into the universal gravitation equation and solving for F_grav. The solution is as follows:

Sample Problem #2 Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a 70-kg physics student if the student is in an airplane at 40000 feet above earth's surface. This would place the student a distance of 6.39 x 106 m from earth's center.

      The solution of the problem involves substituting known values of G (6.673 x 10^-11 N m²/kg²), m₁ (5.98 x 10²⁴ kg), m₂ (70 kg) and d (6.39 x 10⁶ m) into the universal gravitation equation and solving for F_grav. The solution is as follows:

      Two general conceptual comments can be made about the results of the two sample calculations above. First, observe that the force of gravity acting upon the student (a.k.a. the student's weight) is less on an airplane at 40 000 feet than at sea level. This illustrates the inverse relationship between separation distance and the force of gravity (or in this case, the weight of the student). The student weighs less at the higher altitude. However, a mere change of 40 000 feet further from the center of the Earth is virtually negligible. This altitude change altered the student's weight changed by 2 N that is much less than 1% of the original weight. A distance of 40 000 feet (from the earth's surface to a high altitude airplane) is not very far when compared to a distance of 6.38 x 10⁶ m (equivalent to nearly 20 000 000 feet from the center of the earth to the surface of the earth). This alteration of distance is like a drop in a bucket when compared to the large radius of the Earth. As shown in the diagram below, distance of separation becomes much more influential when a significant variation is made.

      The second conceptual comment to be made about the above sample calculations is that the use of Newton's universal gravitation equation to calculate the force of gravity (or weight) yields the same result as when calculating it using the equation presented in Unit 2:

F_grav = m•g = (70 kg)•(9.8 m/s²) = 686 N

Both equations accomplish the same result because (as we will study later in Lesson 3) the value of g is equivalent to the ratio of (G•M_earth)/(R_earth)².

Newton's Laws of Motion

Sir Isaac Newton First Law of Motion Second Law of Motion Third Law of Motion Review Newton's Laws Quiz Quiz Answers Hot Wheels Lab Balloon Racers

 

Sir Isaac Newton

Sir Isaac Newton was one of the greatest scientists and mathematicians that ever lived. He was born in England on December 25, 1643. He was born the same year that Galileo died. He lived for 85 years. Isaac Newton was raised by his grandmother. He attended Free Grammar School and then went on to Trinity College Cambridge. Newton worked his way through college. While at college he became interested in math, physics, and astronomy. Newton received both a bachelors and masters degree.
While Newton was in college he was writing his ideas in a journal. Newton had new ideas about motion, which he called his three laws of motion. He also had ideas about gravity, the diffraction of light, and forces. Newton's ideas were so good that Queen Anne knighted him in 1705. His accomplishments laid the foundations for modern science and revolutionized the world. Sir Isaac Newton died in 1727.
In this lesson you will develop an understanding of each of Newton's Three Laws of Motion. 

According to Newton's first law...

An object at rest will remain at rest unless acted on by an unbalanced force. An object in motion continues in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This law is often called "the law of inertia".

What does this mean? This means that there is a natural tendency of objects to keep on doing what they're doing. All objects resist changes in their state of motion. In the absence of an unbalanced force, an object in motion will maintain this state of motion.

Let's study the "skater" to understand this a little better.
What is the motion in this picture?

What is the unbalanced force in this picture?

What happened to the skater in this picture?



This law is the same reason why you should always wear your seatbelt.




Now that you understand
Newton's First Law of Motion,
let's go on to his Second Law of Motion. 

According to Newton's second law...

Acceleration is produced when a force acts on a mass. The greater the mass (of the object being accelerated) the greater the amount of force needed (to accelerate the object).

What does this mean? Everyone unconsiously knows the Second Law. Everyone knows that heavier objects require more force to move the same distance as lighter objects.




However, the Second Law gives us an exact relationship between force, mass, and acceleration. It can be expressed as a mathematical equation:

or
FORCE = MASS times ACCELERATION



This is an example of how Newton's Second Law works:
Mike's car, which weighs 1,000 kg, is out of gas. Mike is trying to push the car to a gas station, and he makes the car go 0.05 m/s/s. Using Newton's Second Law, you can compute how much force Mike is applying to the car.


Answer = 50 newtons


This is easy, let's go on to
Newton's Third Law of Motion 

According to Newton's third law...

For every action there is an equal and opposite re-action.

What does this mean?

This means that for every force there is a reaction force that is equal in size, but opposite in direction. That is to say that whenever an object pushes another object it gets pushed back in the opposite direction equally hard.



Let's study how a rocket works to understand
Newton's Third Law.

The rocket's action is to push down on the ground with the force of its powerful engines, and the reaction is that the ground pushes the rocket upwards with an equal force.

UP, UP, and AWAY!

 

THERMODYNAMICS

 THERMODYNAMICS

     Thermodynamics is the science of energy conversion involving heat and other forms of energy, most notably mechanical work. It studies and interrelates the macroscopic variables, such as temperature, volume and pressure, which describe physical, thermodynamic systems.

Historically, thermodynamics developed out of a desire to increase the efficiency of early steam engines, particularly through the work of French physicist Nicolas Léonard Sadi Carnot (1824) who believed that engine efficiency was the key that could help France win the Napoleonic Wars.^[1] Scottish physicist Lord Kelvin was the first to formulate a concise definition of thermodynamics in 1854:^[2]

Thermo-dynamics is the subject of the relation of heat to forces acting between contiguous parts of bodies, and the relation of heat to electrical agency.

The initial application of thermodynamics to mechanical heat engines was extended early on to the study of chemical systems. Chemical thermodynamics studies the nature of the role of entropy in the process of chemical reactions and provided the bulk of expansion and knowledge of the field.^{[3][4][5][6][7][8][9][10][11]} Other formulations of thermodynamics emerged in the following decades. Statistical thermodynamics, or statistical mechanics, concerned itself with statistical predictions of the collective motion of particles from their microscopic behavior. In 1909, Constantin Carathéodory presented a purely mathematical approach to the field in his axiomatic formulation of thermodynamics, a description often referred to as geometrical thermodynamics. Although these levels of description required increasingly difficult mathematical tools, and are therefore often taught independently, modern thermodynamics is practiced as an amalgamation of all descriptions, without intentional separation of view points.

 Introduction

     Central to thermodynamics are the concepts of system and surroundings.^[7][12] A thermodynamic system is a macroscopic physical object, explicitly specified in terms of macroscopic physical and chemical variables which describe its macroscopic properties. The macroscopic variables of thermodynamics have been recognized in the course of empirical work in physics and chemistry.^[8] They are of two kinds, extensive and intensive.^[7][13] Examples of extensive thermodynamic variables are total mass and total volume. Examples of intensive thermodynamic variables are temperature, pressure, and chemical concentration; intensive thermodynamic variables are defined at each spatial point and each instant of time in a system. Physical macroscopic variables can be mechanical or thermal.^[13] Temperature is a thermal variable; according to Guggenheim, "the most important conception in thermodynamics is temperature."^[7] The surroundings of a thermodynamic system are other thermodynamic systems that can interact with it. An example of a thermodynamic surrounding is a heat bath, which is considered to be held at a prescribed temperature, regardless of the interactions it might have with the system.

The macroscopic variables of a thermodynamic system can under some conditions be related to one another through equations of state. They can be combined to express internal energy and thermodynamic potentials, which are useful for determining conditions for equilibrium and spontaneous processes.

Thermodynamics describes how systems change when they interact with one another or with their surroundings. This can be applied to a wide variety of topics in science and engineering, such as engines, phase transitions, chemical reactions, transport phenomena, and even black holes. The results of thermodynamics are essential for other fields of physics and for chemistry, chemical engineering, aerospace engineering, mechanical engineering, cell biology, biomedical engineering, materials science, and economics, to name a few.^[14][15] Many of the empirical facts of thermodynamics are comprehended in its four laws, principles that can also be taken as an axiomatic basis for it. The first law specifies that energy can be exchanged between physical systems as heat and thermodynamic work.^[16] The second law concerns a quantity called entropy, that expresses limitations on the amount of thermodynamic work that can be delivered to an external system by a thermodynamic process.^[17]

     Thermodynamic facts can often be explained by viewing macroscopic objects as assemblies of very many microscopic or atomic objects that obey Hamiltonian dynamics.^[18][7][13] The microscopic or atomic objects exist in species, the objects of each species being all alike. Because of this likeness, statistical methods can be used to account for the macroscopic properties of the thermodynamic system in terms of the properties of the microscopic species. Such explanation is called statistical thermodynamics; also often it is also referred to by the term 'statistical mechanics', though this this term can have a wider meaning, referring to 'microscopic objects', such as economic quantities, that do not obey Hamiltonian dynamics.^[13]

This article is focused mainly on classical thermodynamics which primarily studies systems in thermodynamic equilibrium. Non-equilibrium thermodynamics is often treated as an extension of the classical treatment, but statistical mechanics has brought many advances of the field.

The history of thermodynamics as a scientific discipline generally begins with Otto von Guericke who, in 1650, built and designed the world's first vacuum pump and demonstrated a vacuum using his Magdeburg hemispheres. Guericke was driven to make a vacuum in order to disprove Aristotle's long-held supposition that 'nature abhors a vacuum'. Shortly after Guericke, the English physicist and chemist Robert Boyle had learned of Guericke's designs and, in 1656, in coordination with English scientist Robert Hooke, built an air pump.^[20] Using this pump, Boyle and Hooke noticed a correlation between pressure, temperature, and volume. In time, Boyle's Law was formulated, which states that pressure and volume are inversely proportional. Then, in 1679, based on these concepts, an associate of Boyle's named Denis Papin built a steam digester, which was a closed vessel with a tightly fitting lid that confined steam until a high pressure was generated.

     Later designs implemented a steam release valve that kept the machine from exploding. By watching the valve rhythmically move up and down, Papin conceived of the idea of a piston and a cylinder engine. He did not, however, follow through with his design. Nevertheless, in 1697, based on Papin's designs, engineer Thomas Savery built the first engine, followed by Thomas Newcomen in 1712. Although these early engines were crude and inefficient, they attracted the attention of the leading scientists of the time.

The fundamental concepts of heat capacity and latent heat, which were necessary for the development of thermodynamics, were developed by Professor Joseph Black at the University of Glasgow, where James Watt was employed as an instrument maker. Black and Watt performed experiments together, but it was Watt who conceived the idea of the external condenser which resulted in a large increase in steam engine efficiency.^[21] Drawing on all the previous work led Sadi Carnot, the "father of thermodynamics", to publish Reflections on the Motive Power of Fire (1824), a discourse on heat, power, energy and engine efficiency. The paper outlined the basic energetic relations between the Carnot engine, the Carnot cycle, and motive power. It marked the start of thermodynamics as a modern science.^[10]

The first thermodynamic textbook was written in 1859 by William Rankine, originally trained as a physicist and a civil and mechanical engineering professor at the University of Glasgow.^[22] The first and second laws of thermodynamics emerged simultaneously in the 1850s, primarily out of the works of William Rankine, Rudolf Clausius, and William Thomson (Lord Kelvin).

The foundations of statistical thermodynamics were set out by physicists such as James Clerk Maxwell, Ludwig Boltzmann, Max Planck, Rudolf Clausius and J. Willard Gibbs.

During the years 1873-76 the American mathematical physicist Josiah Willard Gibbs published a series of three papers, the most famous being On the Equilibrium of Heterogeneous Substances^[3], in which he showed how thermodynamic processes, including chemical reactions, could be graphically analyzed, by studying the energy, entropy, volume, temperature and pressure of the thermodynamic system in such a manner, one can determine if a process would occur spontaneously.^[23] Also Pierre Duhem in the 19th century wrote about chemical thermodynamics^[4]. During the early 20th century, chemists such as Gilbert N. Lewis, Merle Randall^[5], and E. A. Guggenheim^[6][7] applied the mathematical methods of Gibbs to the analysis of chemical processes.

      The thermodynamicists representative of the original eight founding schools of thermodynamics. The schools with the most-lasting effect in founding the modern versions of thermodynamics are the Berlin school, particularly as established in Rudolf Clausius’s 1865 textbook The Mechanical Theory of Heat, the Vienna school, with the statistical mechanics of Ludwig Boltzmann, and the Gibbsian school at Yale University, American engineer Willard Gibbs' 1876 On the Equilibrium of Heterogeneous Substances launching chemical thermodynamics.^[19]

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Optical Illusions

     What comes in your mind if that is being asked to you?

Look at this picture. Do the lines seem parallel?

 

Well, check with a ruler- because they are!

 

Welcome to Optical Illusions!

An optical illusion is a trick played on the mind using an image that is visually perceived in a way that's different from what it is in objective reality. It uses the fact that our brains interpret and process information from the eye and makes unconscious inferences without you realizing it.

For example, read this fast- can you undrestnad waht this setnenece maeans? Yes, quite easily, because your brain infers the meaning of words based on only a few letter seen by your eyes.

There are various different types of such cognitive illusions:

1. Ambiguous illusions:

Consider the Necker cube. Which side is the front of the cube- the lower left, or the upper right? This type of illusion presents you with 2 different interpretations, both correct. Most often you'll only see one, but after thinking about it you'll begin to recognize the other. This is because the brain relies on classifying images by their surroundings, but once it becomes ambiguous as to which is the object and which is the surroundings, things become weird. Here's another example- Rubin's vase:

Is this a vase, or two people in profile?

2. Depth Distortion illusions:

Which of the yellow lines is longer?

They're both the same length! This image uses parallel and non-parallel lines to distort your perception. That Cafe wall illusion at the beginning of the lesson is another cool example.This is again due to the fact that the mind judges things relative to their background, so different backgrounds lead to confusing perceptions.

3. Color Distortion illusions:

Which is a brighter shade of gray, square A or B?

They're both the same!

This has to do with colour contrast- when the background is dark, the eye perceives an object as lighter, and vice versa. Take a look at another example:

The two central circles are the same.

4. Paradoxical illusions:

These illusions are things that look logical and normal, but in closer inspection make no sense at all. The penrose stairs are a great example:

Try to start at one edge and go up the stairs. What happens? You get stuck in an endless loop. This image uses 2-dimensional distortions in perception to make you think the staircase looks ok, but put it in 3D and it'll fail miserably. The Penrose triangle has a similar problem:	It looks like a nice regular triangle if you don't inspect it too closely… But on second thought, the 3-D shading makes no sense! What's going on?

M.C. Escher (not to be confused with rapper M.C. Hammer) is a big fan of these impossibilities:

His famous drawings revolve around penrose ojects, crazy staircases, and other weird illusions:

Optical illusions can blow your mind away!