Work & Energy Intro

System Analysis & Energy Classifications(Essential for Multiple Choice & Ranking Tasks)Definitions & System BoundariesSystem: A defined collection of objects. The choice of system determines how energy transfer is classified (Work vs. Potential Energy).Closed System: No mass or energy enters or leaves. Total Energy is conserved (ΔEtotal​=0).Open System: External forces perform Work on the system, changing its total energy (Wext​=ΔE).Kinetic Energy (K): Energy of motion. K=21​mv2. Scalar quantity (always positive).Gravitational Potential Energy (Ug​): Energy stored due to position in a gravitational field. Ug​=mgh (near Earth). Requires the Earth to be part of the system.Elastic Potential Energy (Us​): Energy stored in a deformed elastic object (spring). Us​=21​kx2.Key Relationships & Proportionality (Ranking Task Logic)Velocity vs. Kinetic Energy: Since K∝v2, doubling velocity quadruples kinetic energy.Ranking Tip: When ranking objects by energy, check the square of the velocity first; it has a larger impact than mass.Height vs. Gravitational Energy: Ug​∝h. Linear relationship.Mechanical Energy (ME): The sum of kinetic and potential energies (ME=K+U). Conserved only if no non-conservative forces (friction, air resistance) do work.Conceptual Application: Dropped ObjectScenario: A block of mass m is dropped from height h.System = Block only: Gravity is an external force. Gravity does positive work on the block, increasing its Kinetic Energy.System = Block + Earth: Gravity is an internal force. Potential Energy (Ug​) transforms into Kinetic Energy (K). Total Mechanical Energy remains constant.The Work-Energy Theorem(Crucial for FRQs and Calculation Problems)Force-Displacement DefinitionWork (W): The transfer of energy via a force acting over a distance.W=Fdcos(θ)F: Magnitude of Force.d: Magnitude of Displacement.θ: Angle between the Force vector and Displacement vector (tail-to-tail).Work-Energy Theorem: The net work done on an object equals its change in kinetic energy.Wnet​=ΔK=Kf​−Ki​Nuances & Graphical AnalysisSign Convention:Positive Work (0∘≤θ<90∘): Force adds energy; object speeds up (if Fnet​).Negative Work (90∘<θ≤180∘): Force removes energy; object slows down (e.g., Friction).Zero Work (θ=90∘): Force is perpendicular to motion (e.g., Normal Force on a flat surface, Tension in a pendulum).Force vs. Position Graphs:The area under the curve of an F vs. x graph represents Work done.graph[x][0][10] (Imagine a linear force here; the area under the line is the work).Worked Example: Box on a Rough SurfaceProblem: A 10 kg box is pulled 5 meters across a rough floor by a tension force of 50 N at an angle of 37∘ above horizontal. The coefficient of kinetic friction μk​ is 0.2. Calculate the final velocity if it starts from rest.Calculate Vertical Forces (to find Normal Force FN​):∑Fy​=0⇒FN​+Tsin(37∘)−mg=0FN​=mg−Tsin(37∘)FN​=(10)(9.8)−50(0.6)FN​=98−30=68 NCalculate Friction Force (fk​):fk​=μk​FN​fk​=0.2(68)=13.6 NCalculate Work done by individual forces:Wtension​=Tdcos(37∘)=50(5)(0.8)=200 JWfriction​=fk​dcos(180∘)=13.6(5)(−1)=−68 JWgravity​=0 J and Wnormal​=0 J (Perpendicular to motion).Apply Work-Energy Theorem:Wnet​=Wtension​+Wfriction​=200−68=132 JWnet​=ΔK=21​mvf2​−21​mvi2​132=21​(10)vf2​−026.4=vf2​vf​≈5.14 m/sHooke's Law & Spring Potential Energy(Focus of Derivations and Relationships)Hooke's Law (Restoring Force)Definition: The force exerted by a spring is proportional to its displacement from equilibrium and acts in the opposite direction.Fs​=−kxk: Spring Constant (N/m). Measure of stiffness (higher k = stiffer spring).x: Displacement from equilibrium position (x=0).The negative sign indicates it is a restoring force.Linear Relationship:On a Force (y-axis) vs. Displacement (x-axis) graph, the relationship is linear.Slope = Spring Constant (k).graph[2x][-5][5] (Example of linear Hooke's Law relationship).Elastic Potential Energy (Us​)Derivation from Graph:Work is the area under the Force vs. Displacement graph.For a spring, the graph is a triangle with base x and height kx.Area=21​(base)(height)=21​(x)(kx).Formula: Us​=21​kx2Parabolic Relationship:Energy is proportional to the square of displacement.Doubling the stretch distance (x) quadruples the stored energy (Us​).graph[x^2][-5][5] (Example of parabolic Energy vs. Displacement).Worked Example: The Pinball Launcher (FRQ Style)Problem: A spring with constant k=400 N/m is compressed by 0.1 m. A ball of mass 0.05 kg is placed against it. The spring is released, launching the ball across a frictionless horizontal surface.State Conservation Principle:Einitial​=Efinal​ (No non-conservative work).Us,i​+Ki​=Us,f​+Kf​Substitute Variables:Initial: Spring compressed (xi​), Ball at rest (vi​=0).Final: Spring relaxed (xf​=0), Ball moving (vf​).21​kx2+0=0+21​mv2Solve for Velocity:kx2=mv2v=mkx2​​v=xmk​​v=0.10.05400​​v=0.18000​v=0.1(89.44)v≈8.94 m/sCommon Derivation/Ranking TrapScenario: Two different springs (Stiff k1​ and Loose k2​) are compressed by the same force F. Which stores more energy?Analysis:F=kx⇒x=F/kUs​=21​kx2=21​k(kF​)2=2kF2​Result: Energy is inversely proportional to k when Force is constant. The looser spring (smaller k) stores more energy because it displaces further for the same force.