Conservation of Energy and Power

Conservation of Mechanical EnergyDefining the Law of ConservationMechanical energy (Emech​) is the sum of a system's kinetic energy and its potential energy. In an isolated system—where no external work is done by non-conservative forces like friction or air resistance—the total mechanical energy remains constant over time. This principle allows us to predict the state of a system (velocity, height, or compression) at any point in its motion without needing to calculate the specific forces at every interval.Energy States and System DefinitionsKinetic Energy (K): The energy of motion.K=21​mv2m is mass (kg), v is speed (m/s).Gravitational Potential Energy (Ug​): Energy stored due to an object's position in a gravitational field.Ug​=mghh is the vertical height relative to a defined "zero point" or reference level.Elastic (Spring) Potential Energy (Us​): Energy stored when a spring or elastic material is stretched or compressed.Us​=21​kx2k is the spring constant (N/m), x is the displacement from equilibrium (m).Total Mechanical Energy (E): E=K+Ug​+Us​.Closed vs. Open Systems: Closed System: No energy enters or leaves. Einitial​=Efinal​.Open System: External work (W) is done on the system. ΔE=W.Common Pitfalls:Sign of x: In Us​, x is squared, so whether the spring is compressed or stretched, the potential energy is always positive.Reference Level: Forgetting to define h=0 consistently throughout a problem. Usually, the lowest point in the motion should be h=0.Internal vs. External: If the Earth is not included in your system, gravity is an "external force" doing work, rather than Ug​ being internal energy. In AP Physics 1, always include the Earth in your system to use Ug​.Analyzing Real-World ScenariosScenario 1: The Roller Coaster (Gravitational to Kinetic) A 500 kg coaster car starts from rest at the top of a 40 m hill. How fast is it moving at the top of a loop that is 15 m high? (Assume no friction).Identify initial and final states: $$E_i = U_g$(sincev_i = 0$) $$E_f = K + U_g$$Set Ei​=Ef​: $$mgh_i = \frac{1}{2}mv_f^2 + mgh_f$$Cancel mass (m) and solve for vf​: $$gh_i = \frac{1}{2}v_f^2 + gh_f$$ $$(10)(40) = \frac{1}{2}v_f^2 + (10)(15)$$ $$400 = \frac{1}{2}v_f^2 + 150$$ $$250 = \frac{1}{2}v_f^2$$ $$500 = v_f^2$$ $$v_f \approx 22.36 \text{ m/s}$$Scenario 2: The Horizontal Spring Launcher (Elastic to Kinetic) A 0.2 kg block is pushed against a spring (k=500 N/m), compressing it 0.1 m. The block is released on a frictionless surface. What is the block's speed when it leaves the spring?Set Ei​=Ef​: $$U_s = K$$Substitute equations: $$\frac{1}{2}kx^2 = \frac{1}{2}mv^2$$Cancel the 1/2 and solve for v: $$(500)(0.1)^2 = (0.2)v^2$$ $$(500)(0.01) = 0.2v^2$$ $$5 = 0.2v^2$$ $$25 = v^2$$ $$v = 5 \text{ m/s}$$Work and Non-Conservative ForcesThe Work-Energy TheoremWork (W) is the process of transferring energy into or out of a system via a force applied over a displacement. The Work-Energy Theorem states that the net work done on an object is equal to its change in kinetic energy: Wnet​=ΔK.Components and Subtle NuancesCalculating Work: W=Fdcos(θ)θ is the angle between the Force vector and the Displacement vector.Positive vs. Negative Work:Positive Work: Force and displacement are in the same direction (0∘≤θ<90∘). Energy is added to the system.Negative Work: Force and displacement are in opposite directions (90∘<θ≤180∘). Energy is removed (e.g., friction).Friction and Thermal Energy: When friction acts over a distance d, it does work Wf​=−fk​d. This energy is not "lost" but converted into Internal Energy (Q or Eth​) (heat).Conservation Equation with friction: Ki​+Ui​+Wnc​=Kf​+Uf​ (where Wnc​ is work by non-conservative forces).The "Area Rule": On a Force vs. Position (F vs. x) graph, the work done is the area under the curve.Friction and Work ApplicationProblem: A 2 kg box sliding at 10 m/s hits a rough patch of floor where the coefficient of kinetic friction μk​=0.5. How far does it slide before coming to a stop?Identify Work and Energy change: $$W_{friction} = \Delta K$$Define Wfriction​ and ΔK: $$-f_k d = K_f - K_i$$ $$-(\mu_k mg)d = 0 - \frac{1}{2}mv^2$$Cancel m and solve for d: $$-(0.5)(10)d = -\frac{1}{2}(10)^2$$ $$-5d = -50$$ $$d = 10 \text{ meters}$$Power: The Rate of Energy TransferDefining PowerPower (P) is the rate at which work is done or energy is transformed. In a physical sense, it measures how "fast" a task is completed.Key Equations and VariationsAverage Power: P=ΔtW​=ΔtΔE​Instantaneous Power: P=Fvcos(θ)This is derived from P=tFd​ where td​=v.Units: The Watt (W). 1 W=1 J/s=1 kg⋅m2/s3.Nuance: If an object moves at a constant velocity while a force is applied (like a car driving against air resistance), the power delivered by the engine is exactly P=Fres​v.Power Calculation ExampleAn elevator motor lifts a 1000 kg cabin upward at a constant speed of 2 m/s. What is the power output of the motor?Determine the force required:Since speed is constant, Flift​=Fg​=mg. $$F = (1000)(10) = 10,000 \text{ N}$$Use the constant velocity power formula: $$P = Fv$$ $$P = (10,000 \text{ N})(2 \text{ m/s})$$ $$P = 20,000 \text{ Watts (or 20 kW)}$$Energy Charts and Graphical AnalysisVisualizing Energy TransitionsAP Physics 1 frequently uses Energy Bar Charts (LOL Diagrams) and Energy vs. Position Graphs to test conceptual understanding of energy conservation.Energy Bar Charts (LOL Diagrams)Left Side (L): Represents the initial energy (K,Ug​,Us​).Middle Circle (O): Represents the defined system. Arrows pointing in/out represent Work (Wext​).Right Side (L): Represents the final energy.The Rule: Sum of bars on the left + Work in the circle = Sum of bars on the right.Graphical InterpretationF vs. x Graph: Area = Work. graph[2x][0][10] In the graph above (F=2x), the work done from x=0 to x=10 is the area of the triangle: 21​(10)(20)=100 J.U vs. x Graph (Potential Energy Well): The total energy (Etotal​) is a horizontal line.The distance between the U curve and the Etotal​ line is the Kinetic Energy (K).Where the curve hits the Etotal​ line, K=0; these are "turning points."Diagram Analysis Case StudyThe Energy Chart Problem: A spring-loaded toy is compressed and then fires a ball straight up into the air. System: Ball, Spring, Earth.Initial State: Spring compressed, ball at rest.Bars: Large Us​, zero K, zero Ug​.Intermediate State (as it leaves the spring): Spring at equilibrium, ball moving.Bars: Zero Us​, Large K, small Ug​.Final State (at peak): Ball at highest point.Bars: Zero Us​, zero K, Large Ug​.Key check: In a frictionless system, the total height of the bars in the first "L" must equal the total height of the bars in the last "L".