viernes, 25 de marzo de 2016

COSMOLOGY

COSMOLOGY


Cosmology describes the largest structure that exists, that is, the universe itself.

* Significantly, physics on a microscopic scale, is a key to determine the nature of the universe at the full scale.
  
Gravitational interactions have a fundamental role in the behavior of the universe on a large scale. Newtonian mechanics and the law of gravitation enabled and facilitated the understanding of the movement of the planets in the solar system. Observations made with powerful optical instruments and the deduction based on mathematical calculations among others, indicates that gravitational interactions also occur in greater astronomical systems,  which include stars, galaxies and nebulae.

 La Galaxie de Florence Porcel(2012, march 23). [Twitter] L'Univers, la voie Lactée, Twitter et moi. Retrieved from http://www.florenceporcel.com/lunivers-la-voie-lactee-twitter-et-moi/

By the 1920s, evidences were given that the universe was not static. Thus, the motions of galaxies can be measured in relation to the Earth by observing shifts in the wavelengths of their spectra. Displacements in the distant galaxies are always to longer wavelengths, so they seem to show that galaxies move away from us and among them.


 Nadia Drake (2015, may 01). National Geographic [Online]. Hubble Revisits an Icon. The Pillars of Creation. Retrieved from http://phenomena.nationalgeographic.com/2015/01/05/hubble-revisits-an-icon-the-pillars-of-creation/



HUBBLE'S LAW

Through various analyzes, Edwin Hubble deduced that: V reverse speed of a galaxy is proportional to its distance r from us. This relationship is now called Hubble's Law as:
                                                                    
    V = Ho.r

Ho: Hubble constant (at a given moment is constant throughout the space)

To determine Ho, it was needed the Hubble Space Telescope, which can measure distances to galaxies with high accuracy.


The value assigned to it nowadays is 2.3x10-18 s-1, with an uncertainty of 5%.
 


This is the most remote galaxy ever seen (Galaxy GN-z11)
 
         Hubble Space Telescope (2016, march 3). heic 1604 - Science Release. Hubble breaks cosmic distance record. Retrieved from: https://www.spacetelescope.org/news/heic1604/


Another important point of Hubble’s observations was that distant galaxies appear to be moving away from us in all directions. This idea is clearly involved in the "Cosmological Principle".




               Credits: NASA, ESA, B. Robertson (University of California, Santa Cruz), A. feild (STScl)
                      Retrieved from: http://www.nasa.gov/feature/goddard/2016/hubble-team-breaks-cosmic-distance-record



 THE BIG BANG


With Hubble's law, among other indicators, it seems evident that all matter in the universe was much more concentrated, to the point that was infinitesimally small (smaller than a proton), and infinitesimally dense; but at any given time, about 13,8 billion years ago, there was a huge explosion, called the “Big Bang”. From there the universe is still expanding, giving the whole matter approximately the speeds we see today.


* The age of the Earth, which is determined by radioactive dating is 4.6 billion (4.6x109) years, indicating that the Universe is older than the Earth.




EXPANDING SPACE


It is often thought that the universe is expanding into the void, but the accumulated evidence shows that our universe is infinite, and therefore has no boundaries or limits, so there's nothing out of it, neither expands into something. The expansion of the universe means that it is increasing the scale factor of the universe, which in other words means that the infinite universe is becoming more infinite.



GRAVITY AND THE UNIVERSE

Significantly, the law of gravitation is not compatible with a static universe, and according to the evidences, we need to analyze the gravity in an expanding universe.

Given the property of gravitational attractions, the initial expansion of the universe should slow down.

In that sense, the universe should expand more and more slowly, taking into account a sufficiently intense gravitational pull, and at some point stop, and begin to contract, to what is called the "Big Crunch" which would be the opposite of the Big Bang.

On the other hand, if the gravitational force were quite weak in relation to the Universe, only would slow down a little the expansion, and thus the universe would continue expanding forever, to which is called the "Big Freeze", which is an expansion to an excessive acceleration of any object in the universe, from galaxies to atoms.


 

DARK MATTER, DARK ENERGY AND THE ACCELERATING  UNIVERSE

Highlighting the fact that the speed of light is the same for any observer and that photons travel more distance to reach the observer because of the remoteness of the source that emits the photons, this result is known as Redshift or shifted toward the red. Thus, if an Astro moves away, the lines move to the area of ​​long wavelengths, which are identified with red color. So, when we talk about  the Redshift of galaxies, the remoteness of these are indicated.

The various observations show that the average density of all matter in the universe is 26% of the critical density, however, the average density of luminous matter accounts for only 4% of the critical density, which means that the most of the matter in the universe is not bright because it does not emit any electromagnetic radiation. In this so-called "Dark Matter", it is credited with the possible explanation of why the universe is accelerating rather than decelerating by the effect of gravity, but it is still controversial and mysterious in the current scientific level.

Some propose that Dark Matter are the WIMPs (Weakly Interacting Massive Particles), which are to be subatomic particles that hypothetically would be much more massive than those that are produced in experiments in particle accelerators.

The other hypothesis for explaining Dark Matter are the MACHO (Massive Compact Halo Objects), which would include objects in the universe such as Black Holes that form halos around galaxies.

Anyway Dark Matter is the most dominant form of matter in the Universe. For example, for every kilogram of ordinary matter, such as stars, planets, molecules, atoms, protons or electrons, there are five and a half kilograms of dark matter.

From a physical analysis of the matter, the average density of matter in the Universe is less than the critical density, then you can draw conclusions that the universe will continue expanding forever, and the gravitational attraction of matter in different parts of the Universe should slow the expansion, but not to the point of stopping the expansion. However, it was not until the 1990s when it was possible to accurately measure distances of extremely distant galaxies for our common understanding, even more it was discovered that very distant galaxies have lower redshifts than what Hubble's Law predicted, showing that the Universe's expansion was slower in the past than now; so the Universe has been accelerating rather than decelerating, which reaches to a very controversial point for astronomers and physicists, that is, in space a kind of energy that has no gravitational effect, is widespread; besides it doesn’t emit electromagnetic radiation, so it works as a kind of "anti-gravity" that would produce a universal revulsion, which today, this invisible and intangible energy is called Dark Energy, that is still poorly understood.

Relating Einstein's equation E=m.c2, the average energy density of matter in the Universe is 0,26ρec2, also considering that the energy density of "dark energy" is almost three times higher than the matter, then the expansion of the universe will continue accelerating.



DECOUPLING OF INTERACTIONS AND THE BEGINNING OF TIME

When the gravitational potential energy increased during the expansion, there were decreases in temperature and average kinetic energy of the particles, while the basic interactions (gravitational, electromagnetic, strong nuclear and weak nuclear force) is progressively uncoupled.

If we talk about the unification of the electromagnetic and the weak nuclear interaction, this occurs at a very large enough energy to make that the mass differences between the various bosons with spin 1, which mediate interactions, are considered insignificant in comparison.

The electromagnetic interaction is defined by the massless photon, and the weak interaction is defined by the weak bosons ω±  y  z0, whose masses are approximately 100 Gev/c2. If we consider that at much lower energies than 100Gev , the two interactions appear to be very different, also is stated that at energies much greater than 100 Gev, the two interactions form a part of a single interaction.

Theories of the Grand Unification state that the strong nuclear interaction is unified with the electroweak interaction at energies of about 1014 Gev, but at lower energies, the two interactions seem very different.

Note that the grand unification theories are more speculative type because  controlled experiments energies on this scale cannot be done, which is higher by a factor of 1011 than the energies that, nowadays, any current particle accelerator have.

With large enough energies and short distances,  the gravity would unify with the other three interactions. Supposedly, the distance at which these events occur is about 10-35m. This distance is called "Planck length" lp, which is determined by the speed of light c and the fundamental constants of quantum mechanics and gravitation h and G.



                     
                                       Where lp is the Planck length

Keep in mind that the Planck time is tp = lp/c, which is the time it takes light to travel the distance tp.



* Currently, there is not an adequate theory that unifies the four interactions, so you can not know how the universe behaved when the time was less than the Planck time, or in any case when its size was less than the Planck length.


miércoles, 17 de febrero de 2016

SIMPLE HARMONIC MOTION


SIMPLE HARMONIC MOTION


CONCEPT

It is the oscillatory and periodic motion; it's made in a straight line caused by a restoring force, which is described mathematically by sine and cosine functions.



Restoring force (F)

It is the inner force that manifest the elastic bodies when they are stretched or compressed, acting on the particle "m" so that it recovers its equilibrium position.

The restoring force is the tension or compression of the springs, which is represented by a vector F acting on the particle, indicating the position of equilibrium.
Hooke's law:


K: Spring constant (N / M)




Elongation (X)


It is the magnitude of the vector and it indicates the position of the particle or body "m" in each instant of time "t" with respect to the equilibrium position.





Amplitude (A)

It is the maximum elongation reached by a moving particle.
The particle begins its movements (t = 0) when X = A and it is established that:

x (t) = A. Cos (ωt)

Relation between Simple Harmonic Motion and Uniform Circular Motion

If a particle moves with Uniform Circular Motion, then their projections on its diameter hold with the requirements of Simple Harmonic Motion.


Where X is the projection of the radius A on the horizontal axis.

The angle "θ" is the angular displacement experienced by the particle with a Uniform Circular Motion, θ = ω.t

X (t) = A Cos θ
X (t) = A Cos (ω.t)



PERIOD OF PARTICLE "m"

We analyze the particle mass "m" dynamically and perform the free body diagram at a given time "t".

 By Newton's second law:
 
FR=m.a
K.X = m(ω2.X)
K = m.ω2


We isolate ω and do the square root to both sides



Replacing values, we obtain






INSTANTANEOUS VELOCITY

It becomes the projection of the tangential velocity on the horizontal axis. The Velocity ​​(V) in the Uniform Circular Motion is V = ωA

V (t) = Vsenθ
V (t) = ωA.Sen (ωt)






INSTANTANEOUS ACCELERATION

It is the projection of the centripetal acceleration on the horizontal axis.

The centripetal acceleration in the Uniform Circular Motion is ω2.A






a(t) = ac . Cosθ
a(t) = ω2 . A. Cos(ωt)

Replacing values

a(t) = ω2 . X



ASSOCIATION OF SPRINGS

The springs tied to masses can be connected in two basic ways: in series and in parallel.

The set of springs may be replaced by a single equivalent spring whose spring constant is called equivalent spring constant (Ke)



SPRINGS IN SERIES

In the graph you can see that the tension that the springs are supporting is the same (T = KX)

In the case of equivalent displacement Xe, it is equal to the sum of the displacements of the each spring.

Xe = X1 + X2 + X3



* In the case of the association of springs in series:




SPRINGS IN PARALLEL

In all the springs the deformation X are the same.

The equivalent tension (Te) is represented by:

Te = T1 + T2 + T3

The tension on each spring is:

T1 = K1.X   ;   T2 = K2.X   ;    T3 = K3.X
→ Ke = K1 + K2 + K3



TOTAL ENERGY SYSTEM
A particle of mass "m" begins its movement at time t = 0, when the elongation is equal to the amplitude (x = A). The restoring force F acting on the particle "m" varies with the distance as shown in the graph.


 

The total energy of the mass-spring system is equal to the area under the line shown in the figure.

ETotal = Área
If:
ETotal = EC + Ep



Where V is the particle velocity in the instant in which "t" is the X distance from the equilibrium position.




THE SIMPLE PENDULUM

CONCEPT:

It is the system consisting of a mass "m" of a small size attached to a string and not extensible with negligible weight, which can oscillate around its equilibrium position, with a movement which is approximately the simple harmonic motion.

In order to identify the cause of oscillations, the free-body diagram of the pendulum mass "m" is performed. Thus, the restoring force is identified in a component of the weight (mg) which is tangent to the circumference.

F = m.g. Sin θ

It is known from the Second Law of Newton:

F = m.a

→ m.g.Sin θ = m.ω2.x

For small amplitudes it is established that (θ in radians)


x = θ.L = Sin θ.L

Replacing values:

g. Sinθ = ω2.Sinθ.L

Whereas:






Illustrative exercises:

1.- A box of mass "M" is on a horizontal table; plus the friction coefficient between the box and the table is equal to μ. Inside the box lies a body of mass "m" that can move without friction on the bottom of the box. This body is attached to the wall within the box as shown in the figure by a spring whose stiffness is K. With what amplitude of the oscillations of the body the box will begin to move on the table?

Solution


The maximum force F that the spring applies to the box is

F = K.A

When the spring is about to move, then the following situation states:

∑ Fx=0
F = fs(max)
F = µ. N;

But considering the system as a whole: N = (m + M) g

K.A. = µ(M+m)g




2.- The period of vibration of the system shown is 0.3 seconds. If block A is removed, the new period is 0.6 seconds. Knowing that the mass of the block A is 22,5 Kg, determine the mass of the block B. Consider that there is no friction.



Solution

It notes that the period of oscillation depends only on the system mass; and the stiffness constant depends on the material and shape of the spring.

- Whereas:


- The following case is also given:




 Dividing both results and simplifying values ​​we have:



Replacing mA=22,5Kg