Newton and his Laws Issac Newton (1643-1727)
Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light. Alexander Pope.
If anyone can be called the founder of modern science, than it should be Newton. At the age of 25, during 18 months when Cambridge University was closed during an epidemic of plague, Newton developed mathematical calculus; founded optics, the science of light propagation; initiated his work on (what is now called Newtonian) mechanics.
He was made the Lucasian Professor of Mathematics at Trinity College at the age of 26. Three years after that he was elected to the Royal Society of London, and he spent the rest of his scientific career participating in various committees and meetings. His main work is called Philosophiae Naturalis Principia Matematica (The Mathematical Principles of Natural Philosophy), or, in short, Principia. It was published in 1687.
Newton as a Person He was very closed and unapproachable, constantly in fear of competition or of being proven wrong. Never-the-less, his friends remained loyal to him all life long. He rigged a formal resolution of his argument with Leibnitz about who first invented calculus. He published a draft of Flamsteed s book without permission. At the age of 50 he suffered a mental breakdown and spent the rest of his life as a head of British Mint a mere sinecure.
Newton s Laws of Motion He described his three laws of motion in Principia. Newton s laws form the foundation of all physics. Einstein s Theory of Relativity extends Newton s laws to the limit of very high speeds and very strong gravity, but it does not overrule them. Quantum Mechanics extends Newton s laws to the world of elementary particles, but it remains consistent with them.
First Law An object at rest or in a state of uniform motion will remain at rest or in uniform motion, unless acted upon by a net external force This is also known as the law of inertia. Inertial motion is a motion with the constant velocity. Thus, a force always produces a change in velocity, or, in other words, an acceleration.
Second Law The acceleration of an object is proportional to the net force applied to it. The coefficient of proportionality is called the inertial mass. Mathematically, the second law is expressed like this:
Linear Momentum A linear momentum is a very important characteristic of an object in mechanics. It is a product of the object's mass and its velocity: You can think of it as of a measure of inertia.
Linear Momentum II Change in velocity is acceleration: Change in momentum is then the force:
Conservation of Momentum Important case: in the absence of the net external force the linear momentum of an object is conserved. This is called the law of conservation of momentum. It is more general than the law of inertia, because it is a combination of the first and the second Newton's laws.
Question: An empty, freely rolling boxcar is suddenly loaded from the top with a load of coal twice the mass of an empty boxcar. How will the speed of the boxcar change? 2 T 1 T 1 T 3 T v?
A: It will not change. B: It will be half of the original speed. C: It will be third of the original speed. D: It will be twice the original speed.
Newton s Third Law For every action, there is an equal and opposite reaction. Example: If object A exerts a force on object B, then object B exerts an equal and opposite force on object A.
Newton s Third Law Guess the outcome
Newton s Third Law
Newton s Third Law
The Law of Gravity At the time of Newton it was perfectly understood that there existed a force called gravity that made all objects fall to the ground. Newton conjectured that the same force was responsible for the Moon orbiting the Earth and the planets orbiting the Sun.
If that was indeed the case, the acceleration acting on the Moon should be about 3600 weaker than the acceleration of objects falling to the ground. Since the distance to the Moon was about 60 times the size of the Earth, the force of gravity had to obey the inverse square law: Using the inverse square law for the gravitational force, Newton was able to derive all three Kepler's law of planetary motions.
Using additional arguments, Newton finally arrived at the formula that gives the force of gravity between two objects with masses M 1 and M 2 : where R is the distance between two objects, and G is a fundamental constant, i.e. a number that is the same at all times and everywhere in the universe.
If one of the objects is much larger than the other (as, for example, the case of the Sun and a planet), then the mass of the larger object is usually denoted by M, and the mass of the smaller object is denoted by m:
Measuring G Newton s gravitational constant G is by far the worst known fundamental constant: G (6.6726 0.0003) 10 3 m kg s The reason for that is that gravity can be measured very precisely, but it always comes as GM, but it is very hard to measure masses of various objects accurately. We do know GM 8 for the Sun better: 11 GM 8 1.32712440018(8) 10 20 2 m s 2 3
Now we can understand why all the objects fall to the ground with the same acceleration (and, thus, in the same time, if they fall from the same height). From Newton s Second Law: m can be cancelled on both sides, so it disappears.
There is no m any more in this equation, which means that g is independent on the mass of a falling object. At the surface of the Earth Oops! Do we have a problem?
In the equation: G is constant (does not change no matter what), M is the mass of the Earth (does not change no matter what), but R is the distance to the center of the Earth, and it can change. On Skydeck (Sears tower) we are 412 meters further from the center of the Earth, and we therefore should weight less on Skydeck than on the ground. Question: True or false?
Gravitational vs Inertial Mass Recall: all objects fall to the ground with the same acceleration, because the gravity force is proportional to the mass: But who said that two little m(s) are the same?
Inertial mass is a measure of inertia, it enters the Second Law of Newton: Gravitational mass is a measure of how a body reacts to the force of gravity: There is no a priori reason why these two should be the same!
(Weak) Equivalence Principle Equality of inertial and gravitational masses is called a (weak) equivalence principle: inertial and gravitational masses are equivalent. Equivalence principle has been verified experimentally.
Tests of Equivalence Principle 1590, Galileo Galilei: 1 part in 50 1686, Isaac Newton: 1 part in 1,000 1832, Friedrich Bessel: 1 part in 50,000 1908, Baron von Eotvos: 1 part in 100 million 1930, J. Renner: 1 part in 1 billion 1964, Dicke et al: 1 part in 100 billion 1972, Braginsky, Panov: 1 part in 1 trillion 2008, Adelberger et al: 1 part in 30 trillion 2013, Galileo: 1 part in 100 quadrillion
Coordinates Science is based upon observations. We can observe space and time by measuring them. Any spatial position can be characterized by three numbers - coordinates. They are usually denoted by letters x, y, and z. Time is represented by the letter t. Thus, any point in space at every instant in time can be fully described by four numbers: (x,y,z,t).
Frame of Reference Different observers may have different sets of coordinates (x,y,z,t). A set of coordinates specific to a particular observer is called a frame of reference. Not all frames of reference are equal. There is a special subset of all possible frames of reference called inertial reference frames. They are associated with observers that move freely, with no external force acting on them.
Inertial Forces In the non-inertial frame of reference there appear fictitious forces such as centrifugal and Coriolis forces. These forces are called inertial forces. These forces are fictitious in a sense that there is no physical interaction responsible for these forces. However, a person in a non-inertial frame of reference will feel them quite real!
Coriolis Force
Centrifugal Force The Second Law of Newton looks very different in inertial and non-inertial reference frames! In the inertial frame of reference: In the non-inertial frame of reference:
Kepler s Laws Made Easy First Law: Solution of this equation is the ellipse. There is no deep physics there, just math. Second Law: When R is smaller, V is larger planets move faster when they are closer to the Sun.
Kepler s Third Law Period for a circular orbit: Plug this into the last equation from the previous slide, and we get: If we measure P in years and R in AU, then GM 8 2 4
What is important is that the relationship between the size of the system (in this case R) and a measure of how fast objects are moving (in this case the period P) depends on the total mass of the system. Thus, if we know the size of a gravitating system, and how fast objects inside this system are moving, we can apply an appropriately modified form of the Kepler's law to measure the total mass of the system. This is one of only two direct ways to measure masses of astronomical objects!
Strong Equivalence Principle A weak equivalence principle implies that all objects move the same way in an inertial and in a freely falling reference frame. Albert Einstein formulated a strong equivalence principle: All laws of nature are the same in inertial and freely falling reference frames. Einstein s theory of relativity follows from there, but that is another story
Energy Thomas Young (1773 1829) first coined the term energy. He also disproved Newton s corpuscular theory of light. At the time of Newton, the concept of energy existed. Leibnitz called it vis viva. Both, Leibnitz and Newton understood the process of energy conversion for example, the kinetic energy of motion gets transformed into heat by friction.
There is a fact, or if you wish, a law, governing natural phenomena that are known to date. There is no known exception to this law; it is exact, so far we know. The law is called conservation of energy; it states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity, which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same. The Feynman Lectures on Physics
Many Faces of Energy Energy comes in many faces: Kinetic mv 2 /2 Thermal (3/2)kT Gravitational -GM 2 /R Electromagnetic VE 2 /8 Rest energy mc 2 Nuclear Atomic Chemical Dmc 2 E n -E m
Conservation of Energy The conservation of energy implies that the sum of all kinds of energy of a closed (i.e. not interacting with something else) system is always conserved (as long as the system exists). Any particular kind of energy does not have to be conserved. There are no exceptions to this law!!! Never ever!!! Nowhere!!!
Binding Energy Energy has a sign it can be positive or negative. Negative energy is also called binding energy. If an object has binding energy, some other energy needs to be expended to disperse or destroy that object. Gravitational energy is always negative (= binding); nuclear/atomic, or chemical energy can be positive or negative.
Gravitational Energy Gravitational energy is always binding gravity always pulls things together. If an objects gets more massive or smaller, then it binding energy gets more negative (often it is said mathematically incorrectly that its binding energy increases ). That results in production of some other energy.
Escape Velocity Conversely, to take a part of a gravitating object away requires an expenditure of energy. To send a spaceship off the Earth (an asteroid off the solar system, a star off a galaxy, ), the expended energy should be converted into the kinetic energy of motion. The speed that corresponds to that energy is called the escape velocity.
Energy and Force Watch out for the terminology: energetic and forceful are almost synonyms, energy and force are very different things. Energy is part of an object state: objects have energy (aka people have money). Force is (an actor in) the process of changing object states: objects are acted upon by forces (aka people spend/earn money in a transaction).
Galilean Relativity
Galilean Relativity A physical quality is said to be: invariant, if different inertial observers would obtain the same result from a measurement of this quantity. Example: the mass of an object. relative, if different inertial observers would obtain different result from a measurement of this quantity. Example: the speed of an object.
Galilean Relativity The principle of Galilean relativity states that Newton's laws of motion are the same in all inertial frames of reference. Until about 1860s, the Newtonian physics was considered completed, giving the ultimate and final description of the universe. (Recall, what Aristotle thought about his teaching?)
Layover: Entropy The concept of entropy has been understood since XVIII century, but the mathematical formulation and name was invented by Rudolf Clausius in 1865. Rudolf Clausius (1822-1888) Entropy is related to the concept of a state for some matter (gas, fluid, solid, anything). One state is a specific distribution of molecules in some volume, plus the values of their velocities.
Number of States The number of states for any measurable volume of matter is humongous (there are molecules in our classroom). Entropy is a measure of the number of states of matter under given conditions (density, temperature, pressure, etc). Since that number is so huge, it is actually a logarithm of that number:
Ludwig Boltzmann (1844-1906) Here Boltzmann constant. Several other important things in physics are named after him. Was so famous at the end of his life, the Austrian Emperor send him a personal invitation. His life story emphasizes a danger of using pink slips. is the
Laws of Thermodynamics The first law of thermodynamics = conservation of energy. The second law of thermodynamics states that the entropy of a closed system is never decreasing. It is extremely unlikely for air molecules in the classroom to randomly assemble in its front half. Corollary: since the entropy increases in some places, the universe cannot be infinitely old.
More on Conservation Laws A major insight into the reason for the existence of conservation laws was made by Emily Noether (1882-1935) one of the first great female scientists. Was an Assistant Professor (equivalent) at Groningen University in Germany since 1915; was never promoted to the full professor. Emigrated to the US in 1933. Taught at Bryn Mawr college.
More on Conservation Laws Noether proved that each conservation law is connected to a particular symmetry of space: Linear momentum is conserved when physical laws are invariant wrt translation in space. Energy is conserved when physical laws are invariant wrt translation in time. Angular momentum is conserved when physical laws are invariant wrt rotation in space. Hence, conservation laws are the results of the fundamental symmetries of space and time.
More on Conservation Laws In particle physics there are other conservations laws; they are the results of other fundamental symmetries: conservation of electric charge: gauge invariance (don t ask me what it is). CPT symmetry (particle-antiparticle symmetry): Lorenz invariance (translational symmetry of space-time in Einstein s theory of relativity). Conservation of color charge.. Conservation of lepton & baryon numbers