Force Study Guide

📖 Core Concepts Force – a push or pull; vector quantity with magnitude (N) and direction. Newton’s Laws – (1) Inertia, (2) \(F = ma\) (or \(F = dp/dt\)), (3) action–reaction pairs. Free‑Body Diagram (FBD) – schematic showing all forces acting on a chosen body; essential for vector addition. Equilibrium – net force and net torque are zero (static: no motion; dynamic: constant velocity). Torque – \(\boldsymbol{\tau}= \mathbf{r}\times\mathbf{F}\); causes rotational acceleration via \(\tau = I\alpha\). Work & Energy – \(W = \int \mathbf{F}\cdot d\mathbf{s}\); \( \Delta K = W\); mechanical energy \(E{\text{mech}} = K+U\) conserved only with conservative forces. Potential Energy – \( \mathbf{F}= -\nabla U\); conservative if \(\oint\mathbf{F}\cdot d\mathbf{s}=0\). Impulse – \(\mathbf{J}= \int \mathbf{F}\,dt = \Delta \mathbf{p}\). Friction – opposes relative motion; static \(fs\le \mus N\), kinetic \(fk = \muk N\). 📌 Must Remember SI unit of force: newton (N); \(1\;{\rm N}=1\;{\rm kg\,m/s^{2}}\). Newton’s 2nd law (constant mass): \(F = ma\). Gravitational weight near Earth: \(W = mg\) with \(g \approx 9.81\;{\rm m/s^{2}}\). Coulomb’s law: \(F = k\frac{q1q2}{r^{2}}\). Spring (Hooke’s) law: \(F = -k x\). Centripetal force magnitude: \(Fc = \frac{mv^{2}}{r}\). Static equilibrium conditions: \(\sum Fx = 0,\; \sum Fy = 0,\; \sum \tau = 0\). Dynamic equilibrium: \(\sum \mathbf{F}=0\) while velocity ≠ 0. Work–energy theorem: \(W{\text{net}} = \Delta K\). Conservative force test: closed‑loop line integral = 0 or can be written \(\mathbf{F}= -\nabla U\). 🔄 Key Processes Solve a static‑equilibrium problem Draw a clear FBD. Resolve each force into components (choose convenient axes). Write \(\sum Fx = 0,\; \sum Fy = 0\). Write \(\sum \tau = 0\) about a point that eliminates unknown forces. Solve the linear equations for unknown magnitudes. Find the resultant of multiple forces Convert each force to components: \(Fx = F\cos\theta,\; Fy = F\sin\theta\). Sum all \(Fx\) and all \(Fy\). Resultant magnitude \(R = \sqrt{(\Sigma Fx)^2+(\Sigma Fy)^2}\). Direction \(\phi = \tan^{-1}(\Sigma Fy/\Sigma Fx)\). Work done by a variable force Identify the force as a function of position \(F(x)\). Integrate: \(W = \int{xi}^{xf} F(x)\,dx\). Impulse–momentum calculation If force is constant: \(J = F\Delta t\). Use \(J = \Delta p = m\Delta v\) to find final speed or required force. Energy method for a spring‑mass system Set mechanical energy constant: \(\frac12 mv^{2} + \frac12 kx^{2}= \text{const}\). Solve for unknown \(v\) or \(x\). 🔍 Key Comparisons Static friction vs. kinetic friction \(fs \le \mus N\) (can be less than the maximum). \(fk = \muk N\) (fixed value, usually \(\muk < \mus\)). Weight vs. mass Mass \(m\): intrinsic amount of matter (kg). Weight \(W = mg\): force due to gravity (N). Normal force vs. tension Normal: perpendicular to contact surface, arises from deformation. Tension: along a massless, inextensible string; same magnitude throughout an ideal pulley. Centripetal force vs. centrifugal “force” Centripetal: real force directed toward centre, required for circular motion. Centrifugal: fictitious force felt in rotating frame, directed outward. Conservative vs. nonconservative forces Conservative: path‑independent work, associated with a potential \(U\). Nonconservative: dissipative (e.g., friction), path‑dependent, converts mechanical energy to heat. ⚠️ Common Misunderstandings “Force = mass × velocity” – confusing momentum; correct is \(F = ma\) (or \(dp/dt\)). Treating friction as always equal to \(\mu N\) – static friction adjusts up to \(\mus N\); only kinetic friction is fixed. Assuming net force is zero in any constant‑velocity motion – true, but remember torque must also be zero for rotational equilibrium. Using \(F = kx\) instead of \(F = -kx\) – sign indicates restoring direction toward equilibrium. Believing normal force always equals weight – only true on horizontal surfaces without additional vertical forces. 🧠 Mental Models / Intuition Force as a vector arrow: length = magnitude, direction = line of action; adding forces = “tip‑to‑tail” of arrows (parallelogram rule). Equilibrium as a balanced scale: forces/torques on each side must perfectly balance; any imbalance tips the system. Energy conservation as a bank account: conservative forces are “interest‑free transfers” (no loss), nonconservative forces are “fees” that drain the account. Spring force as a rubber band: pull it further, the tighter it pulls back – linear relation until the limit. 🚩 Exceptions & Edge Cases Variable mass systems (e.g., rockets): use \(F = dp/dt\) rather than \(F = ma\). High‑speed (relativistic) motion: replace \(m\) with \(\gamma m0\); force needed grows dramatically as \(v \to c\). Non‑inertial frames: add fictitious forces (centrifugal, Coriolis) to apply Newton’s laws. Fluid drag at high Reynolds number: quadratic drag \(Fd = \frac12 Cd \rho A v^{2}\) replaces linear Stokes drag. 📍 When to Use Which Free‑body diagram + component resolution – whenever multiple forces act at angles. Torque equation \(\tau = I\alpha\) – when dealing with rotational acceleration of rigid bodies. Work–energy theorem – for problems where forces vary or the path is complicated but start/end states are known. Impulse–momentum – for short‑duration impacts or when force vs. time is given. Conservation of mechanical energy – only if all forces are conservative (no friction, air resistance, etc.). Fictitious forces – when analyzing motion from a rotating or accelerating reference frame. 👀 Patterns to Recognize “Zero net force → constant velocity” appears in many dynamics questions. “Zero net torque → no angular acceleration” signals rotational equilibrium. Quadratic dependence on distance (e.g., gravitation, electrostatics) → look for \(1/r^{2}\) form. Linear dependence on displacement → Hooke’s law situation. Proportional to speed → kinetic friction or Stokes drag; proportional to \(v^{2}\) → high‑speed aerodynamic drag. 🗂️ Exam Traps Choosing the wrong pivot point – picking a point that does not eliminate unknown forces can lead to extra equations and algebra errors. Sign errors in torque – remember \(\tau = rF\sin\theta\); clockwise vs. counter‑clockwise must be consistently defined. Assuming normal force = \(mg\) on an incline – actually \(N = mg\cos\theta\). Mixing up static vs. kinetic friction coefficients – static coefficient is used only to determine if motion starts. Forgetting to include the weight component along an incline – weight parallel to plane is \(mg\sin\theta\). Using \(F = ma\) for a system losing mass – must use \(F = dp/dt\). Treating centripetal force as a separate “new” force – it is the net radial component of existing forces (tension, gravity, normal, etc.). --- Study tip: Sketch an FBD first, label all knowns/unknowns, write the three equilibrium equations (∑Fx, ∑Fy, ∑τ), and then solve. If the problem involves motion, decide whether a force‑based, energy‑based, or impulse‑based approach is simpler. Good luck!