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}

.

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 boundar 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.

    transmissaoonda

  2. Reflection: all of the wave’s energy propagates in the first medium, but with opposite direction.

    reflexaoonda3

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.

interferenciaconstrutivapulsoondas

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.

interferenciadestrutivapulsoondas

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

InterferenciaOndas

— 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:

PadraoDifraccao

 

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Newtonian Mechanics 01

— 2. Initial Considerations —

Without being too far away from the truth one can say that modern Fundamental Physics rests on these three conceptions

  1. The concept of field
  2. The theory of relativity
  3. Quantum Physics

The concept of field is essential to all of our future discussion hence we’ll define it right away:

Definition 1 Field is an mathematical structure with a defined value in a given set of points.
Definition 2 A field is said to be a vectoral field when it assumes vector values.
Definition 3 A field is said to be a scalar field when it assumes scalar values.

The field equations that we are going to use always represent linear interactions. Hence one can consider each interaction resulting from a field as being independent of all other interactions being analyzed and the resulting interaction is just the sum of all interactions.

Associated to the concept of field we have the concept of potential energy. This energy arises as a result of the result of the interaction of the particle with the field {\vec{A}} and in general it is proportional to {\displaystyle\int_a^b\vec{A}\cdot d\vec{s}} where {d\vec{s}} represents an infinitesimal displacement.

— 3. Mechanics —

This section will be a very brief and shallow introduction to the results and triumphs of the first modern physical thepry. Nevertheless one hopes that its internal elegance and depth can be glimpsed through.

— 3.1. Basic concepts and preliminary definitions —

All mechanical quantities can be expressed in units that derive from the units of the following quantities:

  • Length {L}.
  • Time {T}.
  • Mass {M}. In classical mechanics the mass of a body its a measure of its resistance to alter its state of movement. This characteristic has the name of inertia.

The units that one uses to express the previous quantities are totally conventional. In the following we’ll use the Internation System of Units:

  • {\left[ L \right] =m}
  • {\left[ T \right] =s}
  • {\left[ M \right] = \mathrm{Kg}}
Definition 4

A frame is a set of axes that represent the degrees of freedom of the system that is being studied and an arbitrary point that serves has its origin.

Definition 5 A frame is said to be inertial when it possesses the following properties:

  • Space is homogeneous (all points are equivalent) and isotropic (there is no special orientation)
  • Time is homogeneous (all time instants are equivalent)
Definition 6 Position is the geometric localization of a particle in a frame.
Definition 7 Trajectory is the geometric place of the sucessive positions of a particle in a frame in a given time interval.
Definition 8 Displacement it’s the difference between the final position and the initial position of a particle. The displacement is represented by the symbol {\Delta \vec{x}}.

We know by everyday experience that bodies move by different displacements during different time intervals. The concept that expresses how a particle position changes in a given time interval it’s called velocity.

Definition 9 Average velocity: vectorial quantity that expresses that rate of change in a particles position for a given time interval:

\displaystyle \vec{v}_m=\dfrac{\Delta \vec{x}}{\Delta t} \ \ \ \ \ (4)

Definition 10 Velocity: Vectorial quantity that expresses a particles velocity in a given time instant:

\displaystyle \vec{v}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta \vec{x}}{\Delta t}=\dfrac{d\vec{x}}{dt} \ \ \ \ \ (5)

Since a particle’s velocity varies in time one can introduce the concept of acceleration.

Definition 11 Average acceleration: Vectorial quantity that expresses the rate of change of velocity for a given time interval.

\displaystyle \vec{a}_m=\dfrac{\Delta \vec{v}}{\Delta t} \ \ \ \ \ (6)

Definition 12 Acceleration: vectorial quantity that expresses the change in velocity in a given time instant.

\displaystyle \vec{a}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta \vec{v}}{\Delta t}=\dfrac{d\vec{v}}{dt} \ \ \ \ \ (7)

Definition 13 Kinetic energy is the energy associated with the movement of a particle and is defined by:

\displaystyle K=\dfrac{1}{2}m\vec{v}\cdot\vec{v}=\dfrac{1}{2}mv^2=\dfrac{1}{2}m\left( \dfrac{d\vec{x}}{dt}\right)^2 \ \ \ \ \ (8)

Definition 14 Linear momentum : vectorial quantity that is associated with the movement of a particle

\displaystyle \vec{p}=m \vec{v}=m \dfrac{d\vec{x}}{dt} \ \ \ \ \ (9)

Definition 15

The mechanical state of a particle is specified by the simultaneous determination (and with infinite precision) of its coordinates and linear momentum.

— 3.2. Newton’s Axioms —

Thus far we have defined the actors in our stage play but we haven’t defined by which rules do they interact. These rules are given by Newton’s Axioms.

Axiom 1 There is an inertial frame where the linear momentum of a free particle (a particle that experiences no interactions) is constant

Our definition 5 of an inertial frame is, strictly speaking, a mathematical definition. As such nothing in it implies that such a frame exists in our world. As such the function of Newton’s first Axiom is to guarantee that in inertial frame exists in our world.

One important detail is that Axiom 5 only posits the existance of one inertial frame, but we can conclude that there exists an infinite number of inertial frames.

Since an inertial frame space is homogenous and isotropic the point one picks as the origin is arbitrary. Hence one can make a translation of the first origin point and consider the final origin point as being the origin of this new frame, which has to be a new inertial frame.

Moreover one can apply a rotation to our firs inertial frame and obtain a new frame (the time the axes have different orientations). Since space is isotropic this orientation can’t change the nature of the frame and we can conclude that the final frame is also an inertial one.

Additionally one can consider and inertial that moves with constant velocity relatively to an inertial frame. Since space is homegeneous this operation can’t change the nature of the the frame in question and once again it still has to be an inertial frame.

Likewise, since time time is homogeneous one can consider a frame that differs of an inertial frame only in the instant of time that was taken as its origin. Since time is homogeneous this second frame also has to be an inertial one.

Finally let us just notice that any finite or infinite composition of these transformations also produces an inertial frame.

Axiom 2

An in inertial frame the change of linear momentum of a particle is caused by the action of a force {\vec{F}}.

{\vec{F}= \dfrac{d\vec{p}}{dt}}.

When the mass of the particle being considered is constant this axiom becomes {\vec{F}=m\vec{a}}.

In what follows the mass of a particle is to be considered constant unless stated otherwise.

Axiom 3

When two particles interact the force {\vec{F}_{12}} (force that object {1} exerts on object {2}) has an equal intensity as {\vec{F}_{21}} but opposite direction{\vec{F}_{12}=-\vec{F}_{21}}

— 3.3. Kinematics and Dynamics —

On this section we’ll introduce very briefly the concepts that describe (kinematics) and explain (dynamics) the movement of a particle.

— 3.3.1. Equations of movement —

From the definitions of velocity and acceleration that we introduced in section 3 it follows that

\displaystyle d\vec{v}= \vec{a}dt \Rightarrow \int_{t_0}^t d\vec{v}= \int_{t_0}^t \vec{a}dt \Rightarrow \vec{v}(t)-\vec{v}(t_0)=\int_{t_0}^t \vec{a}dt \ \ \ \ \ (10)

 

Since the functional relationship between the acceleration and time is unknown the right hand side of the equality can not be calculated.

It also is

\displaystyle d\vec{x}= \vec{v}dt \Rightarrow \int_{t_0}^t d\vec{x}= \int_{t_0}^t \vec{v}dt \Rightarrow \vec{x}(t)-\vec{x}(t_0)=\int_{t_0}^t \vec{v}dt \ \ \ \ \ (11)

 

Since {\vec{v}(t)} is also an unknown function the calculation has to stop.

If we assume that {\vec{a}} is constant in time (uniformly accelerated motion) we can solve equation 10, { \vec{v}=\vec{v}_0+\vec{a}(t-t_0)}. After substituting in equation 11 it follows

\displaystyle \vec{x}(t)=\vec{x}_0+\vec{v}_0(t-t_0)+\frac{1}{2}\vec{a}(t-t_0)^2 \ \ \ \ \ (12)

 

For the special case {\vec{a}=\vec{0}} the motion is said to be uniform.

— 3.3.2. Galileu Transformations —

After axiom 1 we concluded that there is an infinite number of inertial frames. Hence it makes sense to ask how can calculate the coordinates (velocity) of particle in an inertial frame when knowing its coordinates (velocity) in a first inertial frame.

Let {S} and {S'} be two inertial frames whose origins coincide at {t=0}. Moreover {S'} moves with velocity {\vec{v}_0} relative to {S}.

TransformacaoGalileu

By simple vector addition it is {\vec{v}_0 t+\vec{r}'=\vec{r}} which we can write in component form:

{\begin{aligned} x' & = & x-v_{0x}t\\ y' & = & y-v_{0y}t\\ z' & = & z-v_{0z}t \end{aligned}}

Deriving in order to {t} it is

{\begin{aligned} v'_x & = & v_x-v_{0x}\\ v'_y & = & v_y-v_{0y}\\ v'_z & = & v_z-v_{0z} \end{aligned}}

Galileu transformations are equivalent to the affirmation that the form of the equations of mechanics don’t depend on the inertial frame that one chooses to study the motion of a particle.

 

Matrices, Scalars, Vectors and Vector Calculus 3

After introducing some mathematical machinery with our first and second posts it is now time for us to look into some Newtonian Physics, after a brief look into vector integration.

— 1. Vector Integration —

When dealing with vectors and the mathematical operation we have three basic options:

  • Volume integration
  • Surface integration
  • Line (contour) integration

The result of integrating a vector, {\vec{A}=\vec{A}(x_i)}, over a volume is also a vector and the result is given by the following expression:

\displaystyle \int_V \vec{A}dv = \left( \int_V A_1 dv, \int_V A_2 dv, \int_V A_3 dv \right) \ \ \ \ \ (1)

 

Hence, the result of vector integration is just three separate integration operations (one of each spatial dimension).

The result of integrating the projection of a vector {\vec{A}=\vec{A}(x_1)} over an area is what is called surface integration.

Surface integration is always done with the normal component of {\vec{A}} over the surface {S} in question. Thus what we need to define first is the normal of a surface at a given point. {d\vec{a}=\vec{n}da} will be this normal. We still have the ambiguity of having two possible directions for the normal at any given point, but this is taken care of by defining the normal to be on the outward direction of a closed surface.

Hence the quantity of interest is {\vec{A}\cdot d\vec{a}=\vec{A}\cdot \vec{n} da} ({da_1=dx_2dx_3} for example) with

\displaystyle \int_S \vec{A}\cdot d\vec{a} = \int_S \vec{A}\cdot \vec{n}da \ \ \ \ \ (2)

 

As for the line integral it is define along the path between two points {B} and {C}. Again we have to consider the normal of {\vec{A}=\vec{A}(x_i)}, but this time the quantity of interest is:

\displaystyle \int_{BC} \vec{A}\cdot d\vec{s} \ \ \ \ \ (3)

 

The quantity {d\vec{s}} is an element of length along {BC} and is taken to be positive along the direction in which the path is defined.