Ͳhe great equations of modern physics are a permanent part of scientific knowledge, which may outlast even the beautiful cathedrals of earlier ages.

• Steven Weinberg

🌌 The Great Realm of Physics 🧠

Physics is like solving a cosmic puzzle. Each equation or law is a piece that reveals the hidden rules of the universe. Let’s journey through these mysteries in a way that’s both enlightening and, dare I say, fun!

The universe is under no obligation to make sense to you.
— Neil deGrasse Tyson

🌌 $\color{#00b4d9}Physics$ : The Fundamental Language of Nature

Solving a physics problem is about understanding the hidden relationships and figuring out the physical quantities that connect our universe. Every law and formula we use paints a small part of this cosmic picture.

Solving a physics problem means establishing the unknown relationships and determining the sought physical quantities.

🌍 Position and Motion

Every physical object in space has a position relative to some origin. This is described using coordinates ($P_x$, $P_y$, $P_z$) along the x, y, and z axes.

\color{#00b4d8}
position = P_x · 𝑥̂ + P_y · 𝑦̂ + P_z · 𝑧̂

Motion: Describing Changes in Position

When an object moves, its position changes over time, which gives us velocity and acceleration:

🏎️ 　🚓🚓 🚓 🚓 🚓 🚓 🚓

\color{#caf0f8}
v̅elocity = \frac{∆Position}{∆Time}  ⠀⠀=[m:s] = [m · s̈]⠀= [m·s^{-1}]

The average velocity is how fast the position is changing.

\color{#ade8f4}
a̅cceleration = \frac{∆Velocity}{∆Time}    ⠀⠀⠀=[m:s·s] = [m · s̈²] = [m·s^{-2}]

The average acceleration measures how fast the velocity is changing.

🌌 Newton's Laws of Motion

Isaac Newton laid down the instructions manual for how stuff moves in the universe.

Newton gave us the blueprint for understanding how objects move and interact. His three laws describe motion and forces in any system:

1️⃣ Law of Inertia

Objects in motion stay in motion, and objects at rest stay at rest unless acted upon by a force.

This is the principle of inertia.

\color{#06d6a0}
Momentum = mass \cdot v⃗elocity

Momentum is a conserved quantity in a system, but it can be transferred between objects.

Momentum in a system is conserved, but can be transmitted.

2️⃣ Force = Mass × Acceleration

• “The harder you push, the faster things move.”

If you push a car and a bike with the same force, the bike will zoom off, but the car will inch forward (thanks, mass).

\color{#ffd166}
F⃗orce = mass·a⃗cceleration

\color{#ffd166}
F⃗orce = ∆ Inertia

3️⃣ Every action has an equal and opposite reaction:

For every action, there is an equal and opposite reaction. If you push on a wall, the wall pushes back with an equal force.

Normal Force

\color{#ffd166}
F⃗_N = ⊥F⃗ 

The normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it.

\color{#ffd166}
F⃗_N = ⊥F⃗ \quad \text{(Normal Force)}

$\color{#ffd167}Forces$

Forces: What Moves and Shapes the Universe

A force is any cause capable of modifying the state of rest or motion of a body (dynamic effect), producing a deformation (static effect), or both at the same time.

A force can:

• Change the motion of an object (dynamic effect),
• Deform an object (static effect),
• Or do both at once.

Forces represent mutual interactions between two bodies (external forces) or between parts of the same body (internal forces). These interactions can occur either with direct contact between the bodies (contact forces) or at a distance (action-at-a-distance forces).

Types of Forces

• Contact Forces: Forces between objects in direct contact (e.g., friction, tension).
• Action-at-a-Distance Forces: Forces that act even when objects are not touching (e.g., gravity, electromagnetism).

Forces can be:

• Instantaneous (brief impact) or
• Continuous (like gravity pulling you down all the time).

Forces are measured with a dynamometer, following Hooke’s Law:

F = k \cdot \Delta x

Where $k$ is the stiffness constant, and $\Delta x$ is the displacement.

Forces are vector quantities, represented by vectors. Based on the response of a solid when a force is applied, it can be classified as:

• Deformable: If the solid changes shape.
• Elastic: If the solid returns to its original shape after the force is removed.
• Plastic: If the solid retains its deformed shape.
• Rigid: If the solid does not deform under applied forces.

Measuring Forces

Forces are measured using dynamometers, following Hooke's Law:

F = k \cdot \Delta x

where $F$ is the applied force, $k$ is the stiffness constant of the material, and $\Delta x$ is the deformation (extension or compression).

Resultant of Forces: Combining Multiple Forces

When several forces act on a body simultaneously, we combine them into a resultant force, which has the same effect as all the individual forces.

Concurrent Forces

To find the resultant of concurrent forces, we:

1. Graphically: Use vector addition.
2. Numerically: Break the forces into components along x and y axes.

\vec{R} = \vec{F_x} + \vec{F_y}

Equilibrium, Statics, and Fundamental Principles

A body is said to be in equilibrium when it is either at rest (static equilibrium) or moving with constant velocity (uniform motion or kinetic equilibrium), meaning its acceleration is zero.

Statics is the branch of physics that studies forces in systems where equilibrium exists. For a body to be in equilibrium, the following conditions must hold:

• The resultant force acting on the body must be zero:

  \Sigma \vec{F} = 0
  

• The resultant moment (or torque) of all forces about any point must also be zero:

  \Sigma \vec{\tau} = 0
  

In other words, if nothing is pushing harder than something else, the body is in balance!

Thus, the purpose of statics is to determine the net resultant force and moment acting on a body to establish equilibrium conditions.

Postulates of Statics:

1. A single force acting on a body does not produce equilibrium.
2. Two equal and opposite forces acting along the same line produce equilibrium.
3. In a body in equilibrium, each force is equal and opposite to the resultant of all the other forces.

---

Resultant of a System of Forces

A system of forces is a collection of forces acting simultaneously on a body. Each of these forces is called a component of the system.

The resultant of a system of forces is a single force that has the same effect on the body as all the individual forces combined.

Concurrent Forces

When forces are concurrent (acting at the same point), the resultant can be determined:

1. Graphically: By connecting the origin of the first force to the tip of the next force, forming a vector triangle or polygon.
2. Numerically: By breaking the concurrent forces into their components along the coordinate axes.

• The resultant force is the vector sum of these components:

  \vec{R} = \vec{F_x} + \vec{F_y}
  

Trigonometric Calculation:

To calculate the magnitude of the resultant, we use the Pythagorean theorem:

R = \sqrt{F_x^2 + F_y^2}

To find the direction, we use the angle formed with the x-axis, calculated through:

\theta = \tan^{-1} \left( \frac{F_y}{F_x} \right)

🛠️ Friction

Friction is that annoying force that says, “Nope, you’re not going anywhere without a fight.”

Friction is a resistance that opposes motion between two surfaces in contact. It can stop objects or prevent them from moving.

• Kinetic friction is what happens when two surfaces are rubbing while in motion.
• Static friction is that little resistance that prevents you from moving in the first place.

Uniform Circular Motion

If you’re spinning around in circles, you’re in uniform circular motion. Even though your speed might stay the same, your direction changes constantly, so you still experience acceleration.

The centripetal force ($F_c$) keeps you from flying off:

F_c = \frac{mv^2}{r}

🌍 Newtonian Gravity

Gravity is that universal force that’s trying to keep everything grounded. It’s the reason why apples fall and planets orbit.

Newton’s law of universal gravitation describes the force between two masses:

\color{#ef476f}
F_g = G \cdot \frac{m \cdot M}{r^2}

• $F_g$ = gravitational force
• $G$ = gravitational constant ($6.67 \times 10^{-11}$)
• $m$ and $M$ = masses of two objects
• $r$ = distance between their centers

Where $G$ is the gravitational constant:

\color{#ef476f}
G = 6.67 \cdot 10^{-11} \, \text{[N·m²/kg²]}

In essence, everything pulls on everything. Even you pull on the Earth! But you’re not going anywhere because, well, Earth is huge.

Acceleration over Earth's gravity:

\color{#ef476f}
𝑔 = 9.81 

\color{#ef476f}
F_𝑔 = g·mass [kg·m:s² = Newton]

\color{#ef476f}
F_g = G \frac{m·M}{r²}

From Cavendish's experiment:

\color{#ef476f}
 G = 6.67 ⏨-11 ⠀⠀[N m² : kg²]

💥 Work, Energy, and Power

Work: Energy transferred by a force:

\color{Gold}
Work = Force \cdot Displacement \quad \text{[Joules]}

Work is when a force moves something. Think of it as force spent on a task

Energy: The capacity to do work:

\color{#ffd166}
Energy = \Delta Work \quad \text{[Joules]}

Energy can't be created or destroyed, only moved around.

Power: How fast work is done:

\color{#ffd166}
Power = \frac{Work}{Time} \quad \text{[Watts]}

\color{#ffd166}
Power = Work  : time   ⠀⠀⠀[Watts]

Power is how fast you’re spending energy. More power means you’re getting things done quicker.

$\color{Gold}Work$ is the measure of spent $\color{Gold}Energy$

{\color{Gold}
Work =}
{\color{Red}
 F⃗orce }· 
{\color{Silver}
Displacement }
{\color{Gold}
 ⠀⠀⠀[Jules]}

💥 Collisions

There are two types of collisions:

In collisions, momentum is conserved, and energy can be transferred or transformed into other forms like heat.

• Elastic: No energy is lost; objects bounce off each other like billiard balls
• Inelastic: Some energy is lost, and objects might stick together (like two cars crashing).

Rotation

When objects rotate, they follow some simple rules:

• Angular velocity tells you how fast something’s spinning.
• Torque is like rotational force:

\tau = r \cdot F \cdot \sin(\theta)

🌐 Rotation and Torque

Rotational motion involves objects rotating around a central point. The lever law states:

\color{#d90429}
Weight₁ \cdot Distance₁ = Weight₂ \cdot Distance₂

Lever Law

Two weights that are connected to a rotation apex by a lever follow this rule. $Distance$ refers to the distance from the gravity centre of the object until the apex.

\color{#d90429}
Weight₁ ⨯ Distance₁ = Weight₂ ⨯ Distance₂

Equilibrium: Finding Balance

A system is in equilibrium when the forces and moments (torques) balance perfectly. In equilibrium, objects are either at rest or moving with constant velocity.

Conditions for Equilibrium

• The sum of forces is zero:

  \Sigma \vec{F} = 0
  

• The sum of torques is zero:

  \Sigma \vec{\tau} = 0
  

Postulates of Statics

• A single force cannot create equilibrium.
• Two equal and opposite forces along the same line produce equilibrium.
• For every force, there is an equal and opposite force maintaining balance.

🌊 Fluids: The Physics of Liquids and Gases

Fluids (liquids and gases) flow based on pressure differences.

Fluids move and exert pressure. The study of fluids involves understanding how they flow, resist objects, and transfer forces.

The famous Bernoulli’s principle tells us that as the speed of a fluid increases, its pressure decreases (this explains why airplanes can fly! ✈️).

Trigonometric Calculation:

To calculate the magnitude and direction of the resultant:

R = \sqrt{F_x^2 + F_y^2}
\quad and \quad
\theta = \tan^{-1} \left( \frac{F_y}{F_x} \right)