Physics: Laws of Motion

Chapter 4: The Laws of Motion

Force

• A force is a push or pull that is required to change an object's motion.
• Force is a vector, possessing both magnitude and direction.
• Forces are measured in Newtons (N).

Types of Forces

• Contact forces require physical contact such as tension, normal force, friction, and applied force.
• Field forces do not require physical contact such as gravity, electric force, and magnetic force.

Newton's First Law (Law of Inertia)

• An object at rest stays at rest, and an object in motion stays in motion with the same speed and direction unless acted upon by a force.

Inertia

• Inertia is the tendency of an object to resist changes in its velocity.
• Mass is a measure of inertia, with more mass indicating more inertia, measured in kilograms (kg).

Inertial Reference Frame

• Newton's Laws are valid only in inertial reference frames, which are non-accelerating reference frames.

Newton's Second Law

• The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
• The formula is expressed as: $\sum \overrightarrow{F} = m\overrightarrow{a}$, where $\sum \overrightarrow{F}$is the net force, m is mass, and $\overrightarrow{a}$ is acceleration.
• The net force in the x and y directions can be expressed as: $\sum F_x = ma_x$ and $\sum F_y = ma_y$.

Units

• Force is measured in Newtons (N), equivalent to $\text{kg * m/s}^2$.

Newton's Third Law

• For every action, there is an equal and opposite reaction.

Action-Reaction Pairs

• Forces exist in pairs, with action and reaction forces acting on different objects.

Examples

• Pushing on a wall results in the wall pushing back.
• A book on a table experiences an upward push from the table.
• A rocket expelling gas downwards is propelled upwards by the gas's reaction.

Weight

• Weight is the force of gravity acting on an object, measured in Newtons.
• The formula for weight is $W = mg$, where m is mass (kg) and g is the acceleration due to gravity ($9.8 \text{ m/s}^2$).

Weight vs. Mass

• Weight depends on gravity, while mass does not.
• Weight varies by location while mass remains constant.

Normal Force

• The normal force is exerted by a surface on an object in contact with it, and it is always perpendicular to the surface.
• The normal force isn't always equal to an object's weight.

Free Body Diagram

• A free body diagram shows all forces acting on an object.

Steps for Drawing a Free Body Diagram

• First, identify the object of interest.
• Represent the object as a point.
• Express each force with its corresponding vector on the object.
• Vectors lengths should correlate with magnitude of the corresponding force.

Examples

• A book rests on a table.
• A box gets pulled across a floor.
• A ball is thrown through the air.

Friction

• Friction opposes motion between two surfaces in contact and is parallel to these surfaces.

Static Friction ($f_s$)

• Static friction prevents an object from starting to move and varies in magnitude depending on the applied force.
• The formula: $f_s \leq \mu_s N$
• $\mu_s$ is static friction coefficient
• N is the normal force

Kinetic Friction ($f_k$)

• Kinetic friction opposes the motion of an object already moving.
• The formula: $f_k = \mu_k N$
• $\mu_k$ is the kinetic friction coefficient.
• N is the normal force.
• $\mu_k < \mu_s$, meaning it's easier to keep something moving than to start it moving.

Research and Development

Research

• Research is an investigation into a specific issue or problem using the scientific method.
• It is the pursuit of knowledge for its own sake.
• Research is defined as a systematic investigation to increase the sum of knowledge.

The Research

• Examine existing literature to identify a problem.
• Formulate testable hypotheses.
• Design an experiment to test the hypotheses.
• Conduct the study and collect data.
• Analyze the data to see if it supports the hypotheses.
• Share results with the scientific community by publication.

Development

• Development uses scientific knowledge to produce new products or processes.
• It is the application of existing knowledge for practical purposes.
• Development systematically uses knowledge directed toward the production of useful materials, devices, systems or methods, including design, development, and improvement of prototypes and new processes to meet specific requirements.

The Development

• Identify needs or opportunities.
• Brainstorm potential solutions.
• Design and build a prototype.
• Test and refine the prototype.
• Manufacture and market the final product.

Primary Differences

Algèbre Linéaire et Analyse Matricielle

Introduction

• Linear algebra is a branch of mathematics that studies vector spaces, linear applications, and linear equation systems.
• Matrices are fundamental tools in linear algebra used to represent and manipulate linear applications efficiently.
• Matrix analysis concentrates on the study of the properties of matrices and their applications in various fields, including engineering, physics, computer science, and economics.

Définitions et Notations

Matrices

• A matrix is a rectangular array of numbers called its elements or coefficients.
• An $m \times n$ matrix has $m$ rows and $n$ columns.
• Matrices are denoted with upper-case letters, while a_{ij} indicates the element in row i and column j.
• The set of all $m \times n$ matrices with coefficients from set K (e.g., $\mathbb R$ or $\mathbb C$) is denoted as $M_{m,n}(K)$.
• If $m=n$, it is a square matrix of order n, and its set is denoted as $M_n(K)$.

Vecteurs

• A vector is a special case of a matrix, with only one column (column vector) or one row (row vector).
• A column vector of size n is an n x 1 matrix.
• A row vector of size n is a 1 x n matrix.

Matrices particulières

• Matrice nulle: Represents a matrix with all its elements equal to zero.
• Matrice identitié: Denotes a square matrix where all the elements of the main diagonal are ones and any other element is zero.
• Matrice diagonale: Indicates a square matrix where all the elements outside the main diagonal are zero.
• Matrice triangulaire supérieure: Represents a square matrix where all the elements below the elements of the main diagonal are zero.
• Matrice triangulaire inéfieure: Denotes a square matrix where all the elements above the main diagonal are zero.

Opérations sur les matrices

Addition et soustraction

• The sum of matrices A and B (having the same size) creates a resultant matrix C, whose elements are the sum of the corresponding elements from A and B.
• Subtraction follows the same principle.

Multiplication par un scalaire

• The product between a scalar and a matrix returns a matrix with the product between each element and the provided scalar.

Multiplication de matrices

• The multiplication of two matrices A and B (where the number of columns of A is equal to the number of rows of B) creates a resultant matrix C, where each element follows this:

$c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$ for all i = 1, ..., m and j = 1, ..., p.

• Important facts about multiplication: the number of columns from Matrix A must be equal to the number of rows from Matrix B in order to multiply them. This operation isn't commutative, but associative and distributive.

Transposition

• The transposition of a matrix involves interchanging its rows and columns.

Properties of transposition:

• $(A^T)^T = A$
• $(A + B)^T = A^T + B^T$
• $(\lambda A)^T = \lambda A^T$
• $(AB)^T = B^T A^T$

Matrices inversibles

• Denotes a square matrix such that its multiplication by another matrix returns the Identity matrix

Properties of invertible matrices

• If A is invertible, its inverse matrix must be unique
• If A and B are invertible, then AB is invertible, and $(AB)^{-1} = B^{-1}A^{-1}$.
• If A is invertible, then $A^T$ is. Also, $(A^T)^{-1} = (A^{-1})^T$.
• A square matrix A is invertible if and only if its determinant is non-zero

Déterminant

• Represents a function that relates a scalar to a square matrix. Written as det(A) or |A|.
• Properties:
  • $\det(I_n) = 1$
  • $\det(A^T) = \det(A)$
  • If all elements in a column or a row are equivalent to zero, then $\det(A) = 0$.
  • If a matrix has two rows or two columns with elements of same value, then $\det(A) = 0$.
  • If B is obtained from A, switching 2 rows of columns, then $\det(B) = -\det(A)$.
  • If B is obtained from A, by the multiplication of one of the rows of columns by a scalar λ, then $\det(B) = \lambda \det(A)$.
  • If B is obtained from A, by the summing of a column (or row) with a multiple of another column (or row), then $\det(B) = \det(A)$.
  • $\det(AB) = \det(A) \det(B)$
• A is invertible if and only if $\det(A) \neq 0$. In that case, $\det(A^{-1}) = \frac{1}{\det(A)}$.

Rang d'une matrice

• The maximum number of lineally independent columns (or rows) from a matrix A.
• Its value is always less or equal to the minimum value between m and n, where A is a matrix of size m x n.
• If A is a square matrix of order n, then A is invertible if and only if $rang(A) = n$.

Partial Differential Equations

Introduction

• PDEs are equations that contain derivatives with respect to more than one independent variable.
• Example: The equation $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ involves partial derivatives with respect to t and x.

Classification

• Order: The order of a PDE is the highest order derivative in the equation.
  • Example: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ (2nd order)
• Linearity: A PDE is linear if the dependent variable/derivatives appear linearly. Otherwise, it's nonlinear.
  • Example:
    • Linear: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$
    • Nonlinear: $u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2}$
• Homogeneity: A linear PDE is homogeneous if every term contains the dependent variable or one of its derivatives. Otherwise, it's nonhomogeneous.
  • Example:
    • Homogeneous: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$
    • Nonhomogeneous: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x, t)$

Common PDEs

• Heat equation: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
  • Describes heat transfer. $\alpha$ is thermal diffusivity.
• Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
  • Describes wave propagation. c is the wave speed.
• Laplace's equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
  • Describes steady-state phenomena.

Machine Learning

• "Machines can learn?"
• Arthur Samuel (1959): Machine learning provides computers the ability to learn without being explicitly programmed.
• Tom Mitchell (1997): Well-posed Learning Problem: A computer program learns from experience E regarding some task T and performance measure P if its performance at T, as measured by P, improves with experience E.

Examples

• Example 1: Spam Filter
  • Task T: Classify emails as spam.
  • Experience E: Watching you label emails as spam.
  • Performance P: Number of emails correctly classified as spam.
• Example 2: playing Checkers
  • Task T: play checkers.
  • Experience E: Playing lots of checkers games.
  • Performance P: Probability of wunning next game.

Types of Machine Learning

• Supervised Learning
• Unsupervised Learning

Supervised Learning

• The learning algorithm is given a training set with right answers.
  • Regression: Predict a number.
    • Example: predict housing prices
  • Classification: predict categories.
    • Exmaple: breast cancer (malignant, benign)

Unsupervised Learning

• "The learning algorithm is not told what the 'right answer' is."
• Clustering:
  • Example 1: Organize computing clusters.
  • Example 2: Social network analysis.
  • Example 3: Market segmentation.
• Non-Clustering:
  • Example: Cocktail party problem

The Electromagnetic Spectrum

• The electromagnetic (EM) spectrum is the range of all types of EM radiation.

Electromagnetic Spectrum

• Illustrates the different radiation types arranged by wavelength/frequency.

Visible Spectrum

• Shows colors of visible light spanning from red (long wavelength) to violet (short wavelength).
  • Red: ~700 nm
  • Orange: ~620 nm
  • Yellow: ~580 nm
  • Green: ~530 nm
  • Blue: ~470 nm
  • Indigo: ~430 nm
  • Violet: ~400 nm

Temperatures of objects emitting black-body radiation

• Humans/Earth (~300 K) emit infrared radiation.
• Light bulbs (~3,000 K) emit visible light.
• The Sun (~6,000 K) emits visible light and ultraviolet radiation.
• X-ray tubes (~$10^7$ K) emit X-rays.
• Gamma-ray sources (~$10^8$ K) emit gamma rays.