Newtonian Mechanics 02

— 3.4. Gravitational Field —

Stating the Law of Universal Gravitation in usual terms one usually says that all particles in the Universe attract eachother with a force that is directly proportional to the masses of the particles and inversely proportional to the square of the distance between them. Stated in these terms it seems like the force of gravitation is an instantaneous one. Hence one has to resort to field theoretical language in order to describe the action of a gravitational field in terms that are acceptable to us.

Definition 16 Gravitational field: vectorial field, $ {\vec{g}}$, created by a body of mass $ {m_1}$ in all points of space (except on the point where the particle is situated) which is responsible for the gravitational interaction. $ \displaystyle \vec{g}=G\frac{m_1}{r^2}\hat{r} \ \ \ \ \ (13)$

If a particle of mass $ {m_2}$ interacts with a gravitic field $ {\vec{g}}$ the particle experiences the force $ {\vec{F}_g}$:

$ \displaystyle \vec{F}_g=\vec{g}m_2=G \frac{m_1 m_2}{r^2}\hat{r} \ \ \ \ \ (14)$

  $ {\hat{r}}$ is a unit vector whose direction points from the position of $ {m_2}$ to the position of $ {m_1}$. For the particular case of a body of mass $ {m}$ which is suspended from height $ {h}$ and interacts with the gravitational field of the Earth the gravitational force is

$ \displaystyle F_g=G \frac{M_T m}{(R_T+h)^2}$

Since $ {\vec{F}=m\vec{a}}$ holds for body of constant mass, one can write for the intensity of gravitational acceleration:

$ \displaystyle g=G\frac{M_T}{(R_T+h)^2}$

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Definition 17 When two bodies of mass $ {m_1}$ and mass $ {m_2}$ interact via gravity an energy which derives from the gravitational field is established between them. This energy has the name of gravitational potential energy: $ \displaystyle U=-G \frac{m_1 m_2}{r} \ \ \ \ \ (15)$

— 4. Waves and Oscillations —

Definition 18 Period is the minimum time interval necessary for two points in the same oscillatory phenomenon to be i the same mechanical state. It is represented by $ {T}$.

Definition 19 Frequency is the number of cycles of an oscillatory phenomenon that occur in a second. It is represented by $ {f}$ and can be calculated by $ {f=1/T}$.

Definition 20 Angular frequency is $ { \omega = 2\pi/T=2\pi f }$

— 4.1. Oscillations —

In this subsection one will study the harmonic motion. This is an important kind of movement since that in first approximation one can always use this model to study oscillatory m otions. Let us suppose that a particle moves along a straight line under the effect of a force $ {F}$.

Definition 21 A motion is said to be harmonic when in an oscillatory movement the force is proportional to the displacement (initial, also called equilibrium, position is taken as the origin) and as the opposite direction of the movement. $ \displaystyle F=-k x $

Using Newton's Axiom 2 and introducing $ {k/m=\omega^2}$ one can write:

$ \displaystyle \frac{\partial ^2 x}{\partial t^2}=-\omega ^2 x \ \ \ \ \ (16)$

Solutions to this equation can be of the form $ {x(t)=A\cos (\omega t + \theta)}$ where $ {A}$ is the maximum displacement relative to the equilibrium position and $ {\theta}$ is the specific phase which identifies the particle's initial position. In the case of harmonic motion the definitions 18 and e 19 Can be written in the form $ {T=2\pi \sqrt{m/k} }$ e $ {f=1/(2\pi) \sqrt{k/m} }$. For an harmonic motion the kinetic and the potential energy are:

• $ {K=\dfrac{1}{2} m \omega^2 A^2 \sin^2( \omega t + \theta ) }$
• $ {U=\dfrac{1}{2} k A^2 \cos^2( \omega t + \theta ) }$

Thus the total energy of the system is $ {E=\dfrac{1}{2}kA^2}$

— 4.2. Waves —

Definition 22 A wave is a propagation in a medium that propagates itself transporting energy.

Definition 23 Wavelength, $ {\lambda}$ is the minimum distance between two points in a wave that are in the same mechanical state.

Definition 24 The speed of a wave of wavelength $ {\lambda}$ and period $ {T}$ is $ {c=\lambda/T=\lambda f}$

Definition 25 The wavenumber of a wave is é $ {k=2\pi/\lambda}$

It is possible to demonstrate mathematically that the equation that governs the propagation of a perturbation that moves with constant speed $ {c}$:

$ \displaystyle \frac{\partial ^2 \phi}{\partial x^2}=\frac{1}{c^2}\frac{\partial ^2 \phi}{\partial t^2} \ \ \ \ \ (17)$

With the previous definitions it is trivial to see that equations of the form $ {f_1=A\sin(kx \pm \omega t)}$ and $ {f_2=A\cos(kx \pm \omega t)}$ are solutions of equation 17. These functions are called trignometric functions, $ {A}$ is the amplitude and it represents the maximum displacement of the entity that is vibrating. In general one can say that a progressive wave that propagates to the right is always of the form $ {f=f(x-ct)}$, while a wave that moves to the left is always of the form $ {g=g(x+ct)}$. Since the wave equation is a linear partial derivative equation any linear combination of solutions of equation 17 is still a solution of the wave equation. In order for our solutions to have physical sense one has to impose certain conditions that the equations must follow in certain regions of space (and time).

These conditions are called boundary conditions and its effect is to restrict the set of values that the solutions might take.

The solutions of the wave equation that follow from boundary conditions are said to be normal modes of vibration. When is propagating and it encounters the boundary between two different media two things can happen:

1. Transmission: some of the energy of the wave propagates into the second medium.
   
2. Reflection: all of the wave's energy propagates in the first medium, but with opposite direction.
   

When two trigonometric waves of the same amplitude and frequency propagate in the same medium with opposite directions interact thet create a resulting wave whose mathematical expression is given by $ {f=2A\sin kx \cos \omega t}$. This is the mathematical expression of a stationary wave.

— 4.3. Interference —

When two waves of equal wavelength and constant phase difference interact they are said to interfere. If the two waves are in the same region of space and are of equal phase the interference is said to be constructive and the amplitude of the resulting wave equals the sum of the individual amplitudes of each original wave.

If the two waves are in the same region of space in phase opposition the interference is said to be destructive and the amplitude of the resulting wave is equal to the subtraction of the amplitudes of the two original waves.

The following diagram is a schematic representation of an experimental realization of an interference pattern:

— 4.4. Diffraction —

When light of a well defined wavelength passes through a barrier with a slit of width $ {d}$ the phenomenom that occurs is called diffraction. Each portion of the slit acts as if it is an independent source of a propagation and as a consequence waves coming from different portions of the slit have different phases. From this interaction one can observe destructive or constructive interference. The following diagram shows a schematic representation of an experimental realization of the phenomenon of diffraction: