Classical Mechanics: The Complete Principle Map

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• Why Learn Classical Mechanics?
• Prerequisites
• The Principle Map
• Core Principles
• What’s Next?
• How This Fits in Unisium

Why Learn Classical Mechanics?

Classical mechanics is where physics begins. Every engineering discipline—mechanical, civil, aerospace, biomedical—builds on these foundations. If you want to understand why bridges stay up, how rockets fly, or why cars have crumple zones, you need classical mechanics.

Beyond engineering, classical mechanics teaches a way of thinking: modeling real situations with mathematical equations, identifying what quantities matter, and predicting outcomes before they happen. These skills transfer to every quantitative field.

The core principles in this subdomain aren’t just formulas to memorize—they’re reusable tools. Once you truly understand Newton’s second law ($\sum \vec{F} = m\vec{a}$), you can apply it to any force problem. Once you internalize conservation of energy, complex multi-step problems become tractable.

Prerequisites

Mathematics:

• Algebra (solving equations, manipulating expressions)
• Basic trigonometry (sine, cosine for components)
• Comfort with vectors (direction and magnitude)

Prior Subdomains:

• None—Classical Mechanics is a foundation subdomain

The Principle Map

The principle map organizes Classical Mechanics along two axes:

Columns (Concept Families):

• Kinematics — describing motion (position, velocity, acceleration) without asking why
• Force — forces cause acceleration (Newton’s laws + specific force laws; torque as rotational analog)
• Energy — an alternative to forces: track energy flow to solve problems
• Momentum — track momentum, especially useful for collisions and explosions

Rows (Motion Types):

• Translation — motion of a point or center of mass in a straight line or curve
• Rotation — spinning motion around an axis

Progression numbers (P1, P2, …) indicate learning order. Lower numbers come first. Principles with the same progression number are at the same conceptual level.

Core Principles

Conditions tell you when an equation is valid.
They’re intentionally short here—think of them as the main assumptions for classical mechanics. You’ll refine what they really mean through practice and problem-solving.

Conditions are purely discriminative: they tell you which form you’re using and what would make it invalid. For example, nonrelativistic signals when the Newtonian form breaks down, discrete masses tells you this is the summation form (not the integral), and instantaneous means right-now values (not averaged).

Kinematics (P1–P2)

Principle	Equation	Condition
Kinematics 1 – Algebraic	$x=x_0+v_0 t+\tfrac12 a t^2$	$a=\mathrm{const}$
Kinematics 2 – Algebraic	$v=v_0+a t$	$a=\mathrm{const}$
Kinematics 3 – Velocity–Position	$v^2=v_0^2+2a\Delta x$	$a=\mathrm{const}$
Kinematics 4 – Displacement (avg endpoints)	$\Delta x=\tfrac12 (v_0+v)t$	$a=\mathrm{const}$

Newton’s Laws (P3)

Principle	Equation	Condition
Newton’s First Law (Translation)	$\sum \vec{F}_{\mathrm{ext}}=0 \Rightarrow \vec{a}=0$	inertial
Newton’s Second Law (Translation)	$\sum \vec{F}=m\vec{a}$	inertial; $m=\mathrm{const}$
Newton’s Third Law	$\vec{F}_{AB}=-\vec{F}_{BA}$	Newtonian

Force Laws (P4–P5)

Principle	Equation	Condition
Static Friction	$f_s \le \mu_s F_N$	contact; no slip
Kinetic Friction	$f_k=\mu_k F_N$	contact; slipping
Centripetal Acceleration	$a_c=\frac{v^2}{r}$	curved path; moving
Weight (Near Surface)	$F_g=mg$	$g \approx \mathrm{const}$
Newton’s Law of Gravitation	$F=G\frac{m_1m_2}{r^2}$	point masses
Hooke’s Law	$F=-kx$	linear regime

Work & Energy (P6–P9)

Principle	Equation	Condition
Translational Kinetic Energy	$K=\tfrac12 mv^2$	nonrelativistic
Translational Work	$W=\vec{F}\cdot \vec{d}$	$F=\mathrm{const}$
Work–Energy Theorem	$W_{\mathrm{net}}=\Delta K$	inertial; $m=\mathrm{const}$
Power (Algebraic)	$P=\vec{F}\cdot \vec{v}$	instantaneous
Gravitational Potential (Near Surface)	$U_g=mgh$	$g \approx \mathrm{const}$
Spring Potential Energy	$U_s=\tfrac12 kx^2$	linear spring
Mechanical Energy with $W_{nc}$	$K_1+U_1+W_{nc}=K_2+U_2$	inertial; $m=\mathrm{const}$
Mechanical Energy Conservation	$K_1+U_1=K_2+U_2$	$W_{nc}=0$

Momentum & Impulse (P10–P11)

Principle	Equation	Condition
Linear Momentum (Definition)	$\vec{p}=m\vec{v}$	nonrelativistic
Conservation of Linear Momentum	$\vec{p}_i=\vec{p}_f$	$\sum \vec{F}_{\mathrm{ext}}=0$
Impulse–Momentum Theorem (Algebraic)	$\Delta \vec{p}=\vec{F}_{\mathrm{net}}\Delta t$	$F_{\mathrm{net}}=\mathrm{const}$
Center of Mass (Position)	$\vec{r}_{cm}=\frac{\sum m_i\vec{r}_i}{\sum m_i}$	discrete masses
Center of Mass (Velocity)	$\vec{v}_{cm}=\frac{\sum m_i\vec{v}_i}{\sum m_i}$	discrete masses

Rotational Kinematics (P12–P13)

Principle	Equation	Condition
Rotational Kinematics 1	$\theta=\theta_0+\omega_0 t+\tfrac12 \alpha t^2$	$\alpha=\mathrm{const}$
Rotational Kinematics 2	$\omega=\omega_0+\alpha t$	$\alpha=\mathrm{const}$
Rotational Kinematics 3	$\omega^2=\omega_0^2+2\alpha\Delta\theta$	$\alpha=\mathrm{const}$
Rotational Kinematics 4	$\Delta\theta=\tfrac12(\omega_0+\omega)t$	$\alpha=\mathrm{const}$
Arc Length–Angle	$s=r\theta$	radians
Tangential Speed	$v=r\omega$	$r=\mathrm{const}$
Tangential Acceleration	$a_t=r\alpha$	$r=\mathrm{const}$

Torque & Rotational Newton Laws (P13–P14)

Principle	Equation	Condition
Torque (Definition)	$\vec{\tau}=\vec{r}\times \vec{F}$	about point chosen
Newton’s First Law (Rotation)	$\sum \tau_{\mathrm{ext}}=0 \Leftrightarrow \alpha=0$	fixed axis; $I=\mathrm{const}$
Newton’s Second Law (Rotation)	$\sum \tau_{\mathrm{ext}}=I\alpha$	fixed axis; $I=\mathrm{const}$
Moment of Inertia (Discrete)	$I=\sum m_i r_i^2$	fixed axis; discrete masses
Parallel Axis Theorem	$I_P=I_{cm}+md^2$	axes parallel

Rotational Energy (P15)

Principle	Equation	Condition
Rotational Kinetic Energy	$K_{rot}=\tfrac12 I\omega^2$	rigid body; fixed axis
Rotational Work	$W=\tau\Delta\theta$	fixed axis; $\tau_{\mathrm{net}}=\mathrm{const}$
Power (Rotation)	$P=\vec{\tau}\cdot\vec{\omega}$	fixed axis; instantaneous

Angular Momentum (P16–P17)

Principle	Equation	Condition
Angular Momentum (Particle)	$\vec{L}=\vec{r}\times\vec{p}$	about point chosen
Angular Momentum (Rigid Body)	$\vec{L}=I\vec{\omega}$	fixed axis
Conservation of Angular Momentum	$\vec{L}_i=\vec{L}_f$	$\sum \tau_{\mathrm{ext}}=0$
Angular Impulse–Angular Momentum (Algebraic)	$\Delta\vec{L}=\vec{\tau}_{\mathrm{net}}\Delta t$	$\sum \tau_{\mathrm{ext}}=\mathrm{const}$

Layer 2: Calculus Forms (P18+)

Layer 2 principles use calculus (derivatives and integrals). They unlock problems with non-constant quantities.

Translation Calculus (P18–P22)

Principle	Equation	Condition
Velocity (Derivative)	$v=\frac{dx}{dt}$	$x(t)$; differentiable at t
Acceleration (Derivative)	$a=\frac{dv}{dt}$	$v(t)$; differentiable at t
Displacement (Integral)	$\Delta x=\int_{t_i}^{t_f} v(t)\,dt$	$v(t)$; interval specified
Velocity Change (Integral)	$\Delta v=\int_{t_i}^{t_f} a(t)\,dt$	$a(t)$; interval specified
Newton’s 2nd Law (Momentum Form)	$\sum \vec{F}=\frac{d\vec{p}}{dt}$	inertial; finite forces
Work (Integral)	$W=\int \vec F\cdot d\vec r$	path known; force as function of position
Power (Derivative)	$P=\frac{dW}{dt}$	$W(t)$; differentiable at t
Impulse–Momentum (Integral)	$\Delta \vec{p}=\int_{t_i}^{t_f} \vec{F}_{\mathrm{net}}(t)\,dt$	inertial; $\vec{F}_{\mathrm{net}}(t)$; interval specified

Rotation Calculus (P23–P26)

Principle	Equation	Condition
Angular Velocity (Derivative)	$\omega=\frac{d\theta}{dt}$	$\theta(t)$; differentiable at t
Angular Acceleration (Derivative)	$\alpha=\frac{d\omega}{dt}$	$\omega(t)$; differentiable at t
Angular Displacement (Integral)	$\Delta\theta=\int_{t_i}^{t_f} \omega(t)\,dt$	$\omega(t)$; interval specified
Angular Velocity Change (Integral)	$\Delta\omega=\int_{t_i}^{t_f} \alpha(t)\,dt$	$\alpha(t)$; interval specified
Torque (Angular Momentum Form)	$\sum \vec{\tau}=\frac{d\vec{L}}{dt}$	inertial; finite torque
Angular Impulse–Angular Momentum (Integral)	$\Delta \vec{L}=\int_{t_i}^{t_f} \vec{\tau}_{\mathrm{net}}(t)\,dt$	inertial; $\vec{\tau}_{\mathrm{net}}(t)$; interval specified
Moment of Inertia (Integral)	$I=\int r^2\,dm$	fixed axis; continuous body; density known

Rotation (P12–P17): The rotation row of the principle map mirrors translation: rotational kinematics, torque as the rotational analog of force, angular momentum, and rotational energy.

Enabled Subdomains:

• Gravitation & Orbits — apply these principles to planetary motion
• Waves & Oscillations — simple harmonic motion builds on Hooke’s law and energy
• Thermodynamics — energy concepts extend to heat and work

Suggested Learning Path (Translation Core):

1. Kinematics (P1–P2) — get comfortable with motion description
2. Newton’s Laws (P3) — the core dynamics framework
3. Force Laws (P4–P5) — build your force toolkit
4. Work & Energy (P6–P9) — alternative problem-solving approach
5. Momentum (P10–P11) — complete the translation toolkit

After translation, continue to rotation (P12–P17), which mirrors the same progression.

How This Fits in Unisium

Unisium trains classical mechanics in two complementary ways:

Principles in isolation (fast, targeted)

• Elaborative Encoding: answer short questions to build understanding
• Retrieval Practice: recall the equation + condition

Principles in context (real problem skill)

• Self-Explanation: step through worked solutions and justify each move
• Problem Solving: solve new problems and practice selecting the right principle under uncertainty

The principle map is your navigation layer: it shows what to learn next, and it explains why some problems feel harder (they combine more principles across columns and rows).

Current focus: Layer 1 (algebraic forms) provides the foundation. Layer 2 (calculus forms) extends these principles to handle non-constant forces and time-varying motion.

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