Newton's laws of motion Study Guide

📖 Core Concepts Newton’s Laws – Foundation of classical mechanics: Inertia – Motion unchanged without net external force. \( \mathbf{F}{\text{net}} = m\mathbf{a} \) – Force produces acceleration. Action–Reaction – Forces occur in equal‑and‑opposite pairs. Inertial Frame – Reference frame with no acceleration; Newton’s laws hold exactly. Point‑Mass Approximation – Treat an object as a mass located at a single point when its size ≪ separation from other bodies. Vectors – Quantities with magnitude and direction (position r, velocity v, acceleration a, force F). Vector addition follows tip‑to‑tail rule. Momentum – \(\displaystyle \mathbf{p}=m\mathbf{v}\); conserved when net external force is zero. Free‑Body Diagram (FBD) – Sketch showing all external forces acting on a chosen body; essential for applying \(\mathbf{F}=m\mathbf{a}\). Work–Energy Theorem – Net work equals change in kinetic energy: \(\displaystyle W{\text{net}}=\Delta K\). Conservative vs. Non‑Conservative Forces – Conservative forces have a potential \(U\) with \(\mathbf{F}=-\nabla U\); work around a closed loop is zero. Friction is non‑conservative. Rotational Analogues – Moment of inertia \(I\) ↔ mass, torque \(\boldsymbol{\tau}\) ↔ force, angular momentum \(\mathbf{L}=\mathbf{r}\times\mathbf{p}\) ↔ linear momentum. Universal Gravitation – \(\displaystyle \mathbf{F}g = G\frac{M1M2}{r^{2}}\hat{\mathbf r}\). Simple Harmonic Motion (SHM) – Restoring force \( \mathbf{F}= -k\mathbf{x}\) leads to sinusoidal motion with \(\omega=\sqrt{k/m}\). --- 📌 Must Remember First Law – No net force → constant velocity (including rest). Second Law (constant mass) – \(\displaystyle \mathbf{F}{\text{net}} = m\mathbf{a}\). Momentum Definition – \(\displaystyle \mathbf{p}=m\mathbf{v}\). Third Law – \(\displaystyle \mathbf{F}{AB} = -\mathbf{F}{BA}\). Centripetal Acceleration – \(\displaystyle ac = \frac{v^{2}}{r}\); required force \(\displaystyle \mathbf{F}c = m\frac{v^{2}}{r}\hat{\mathbf r}\). Kinetic Energy – \(\displaystyle K=\frac{1}{2}mv^{2}\). Potential Energy (near Earth) – \(\displaystyle U=mgh\). Work‑Energy – \(\displaystyle W{\text{net}} = \Delta K\). Gravitational Force – \(\displaystyle \mathbf{F}g = G\frac{M{\earth}m}{r^{2}}\hat{\mathbf r}\); acceleration \(g\approx9.81\ \text{m/s}^2\). Angular Momentum Conservation – \(\displaystyle \sum\boldsymbol{\tau}=0 \;\Rightarrow\; \mathbf{L} = \text{const}\). SHM Frequency – \(\displaystyle \omega = \sqrt{k/m}\); period \(T = 2\pi/\omega\). --- 🔄 Key Processes Solve a Linear‑Motion Problem Draw the FBD → list all forces. Resolve forces into components. Apply \(\displaystyle \sum Fi = mai\) for each axis. Integrate (or use kinematic formulas) to obtain \(v(t)\) and \(x(t)\). Apply the Work‑Energy Theorem Identify all forces and decide which are conservative. Compute net work: \(W{\text{net}} = \int \mathbf{F}\cdot d\mathbf{r}\). Set \(W{\text{net}} = \Delta K\) → solve for unknown speed or displacement. Conserve Momentum in Collisions Write \(\displaystyle \sum \mathbf{p}{\text{initial}} = \sum \mathbf{p}{\text{final}}\). Use any additional constraints (elastic → kinetic energy conserved). Analyze Rotational Motion Compute moment of inertia \(I\) about the axis. Write torque equation \(\displaystyle \sum \tau = I\alpha\). Relate \(\alpha\) to linear acceleration if needed (\(a = r\alpha\)). Derive Motion of a Rocket (Variable Mass) Start with momentum balance: \(m\frac{d\mathbf{v}}{dt} = -\dot m\,\mathbf{v}e + \mathbf{F}{\text{ext}}\). Integrate for \(\mathbf{v}(t)\) given \(\dot m\) and exhaust velocity \(\mathbf{v}e\). Use Lagrangian/Hamiltonian When Constraints Are Complex Write \(L = T - V\). Apply Euler–Lagrange: \(\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot q}\right)-\frac{\partial L}{\partial q}=0\). For Hamiltonian \(H\), use \(\dot q = \partial H/\partial p\), \(\dot p = -\partial H/\partial q\). --- 🔍 Key Comparisons Force vs. Power – Force is a vector push/pull (\(N\)); power is the rate of doing work (\(W = \mathbf{F}\cdot\mathbf{v}\), units \(J/s\)). Mass vs. Weight – Mass \(m\) is intrinsic (kg); weight \(W = mg\) depends on local gravity. Conservative vs. Non‑Conservative Forces – Conservative: path‑independent work, has potential \(U\). Non‑conservative (e.g., friction): dissipates mechanical energy, no scalar potential. Newtonian vs. Relativistic Mechanics – Newtonian: \( \mathbf{F}=m\mathbf{a}\) with constant \(m\). Relativistic: momentum \(\mathbf{p}= \gamma m\mathbf{v}\), force‑acceleration relation altered by \(\gamma\). Inertial vs. Non‑Inertial Frame – Inertial: no fictitious forces; Newton’s laws hold directly. Non‑inertial: must introduce pseudo‑forces (e.g., centrifugal). --- ⚠️ Common Misunderstandings “First law means objects like to stay at rest.” – It means stay in whatever uniform straight‑line motion they already have unless a net force acts. Action–Reaction act on the same body. – They act on different bodies; the pair does not cancel on a single object. Zero net force ⇒ object is “not moving.” – Zero net force ⇒ no acceleration; the object can still move at constant velocity. Mass cancels in gravity, so weight is the same for all objects. – Mass cancels in the acceleration (all fall with \(g\)), but weight \(W=mg\) still depends on mass. Friction is a “force of contact” → can be written as \(-\nabla U\). – Friction is non‑conservative; it cannot be derived from a potential. --- 🧠 Mental Models / Intuition Vector Addition as Arrow Chaining – Place the tail of the next arrow at the head of the previous; the resulting arrow gives the sum. Free‑Body Diagram = “What pushes on me?” – List every external push/pull; ignore internal forces. Inertia = “Resistance to change.” – Larger mass → larger \(\mathbf{F}\) needed for same \(\mathbf{a}\). Energy = “Capacity to do work.” – Kinetic ↔ motion, potential ↔ position; total mechanical energy stays constant when only conservative forces act. Centripetal Force = “Pull toward center,” not a new kind of force. It’s simply the net radial component of the existing forces. --- 🚩 Exceptions & Edge Cases Relativistic Speeds – Newton’s laws break down; use \(\mathbf{p}= \gamma m\mathbf{v}\). Variable‑Mass Systems – Rocket equation; \( \mathbf{F}=m\mathbf{a}\) not directly applicable. Non‑Inertial Frames – Must add fictitious forces (e.g., Coriolis, centrifugal). Three‑Body Problem – No closed‑form solution; rely on numerical integration. Strong Gravitational Fields – General relativity replaces \(\mathbf{F}=G M m / r^2\) with spacetime curvature. --- 📍 When to Use Which \( \mathbf{F}=m\mathbf{a}\) – Straightforward problems with constant mass and explicit forces. Momentum Form \(\displaystyle \mathbf{F}{\text{net}} = \frac{d\mathbf{p}}{dt}\) – Variable‑mass or impulse problems. Work‑Energy – When forces vary with position or when only initial/final speeds are needed. Conservation of Momentum – Isolated collisions or explosions, especially in 2‑D. Torque & Rotational Equations – Rigid bodies rotating about a fixed axis. Lagrangian – Systems with constraints (e.g., pendulum, rolling without slipping) where forces are cumbersome to list. Hamiltonian – Phase‑space analysis, canonical transformations, or when energy is the natural conserved quantity. --- 👀 Patterns to Recognize Constant acceleration → linear \(v(t)\) and quadratic \(x(t)\) equations. Projectile motion → horizontal \(vx = \text{const}\); vertical \(ay = -g\); parabolic trajectory. Circular motion → required radial force \(Fr = m v^2 / r\). SHM → sinusoidal \(x(t)=A\cos(\omega t+\phi)\) with \(\omega=\sqrt{k/m}\). Resonance → amplitude peaks when driving frequency ≈ natural frequency. Zero net torque → angular momentum constant; look for symmetry about the rotation axis. Conservative force field → closed‑loop work = 0; can define a scalar potential. --- 🗂️ Exam Traps Choosing \(mg\) for “force of gravity” on a satellite. – In orbit, the net acceleration is centripetal, not \(g\) (value changes with altitude). Sign errors in centripetal force direction. – Always point toward the center of the circle. Assuming action–reaction forces cancel on a single object. – They act on different bodies; the net force on the object is the sum of forces acting on it only. Using \(F=ma\) for rockets without accounting for \(-\dot m ve\). – Leads to over‑estimation of acceleration. Treating friction as a conservative force. – Will incorrectly allow a potential energy function for friction. Mixing up mass and weight in energy problems. – \(K=\frac12 mv^2\) uses mass; potential energy near Earth uses \(U=mgh\). Neglecting pseudo‑forces in accelerating frames. – A block on a car accelerating forward feels a backward fictitious force; forgetting it gives the wrong net force. ---