Energy Conservation Principles

Why This Matters

Energy conservation isn't just one topic—it's the backbone of nearly every problem you'll encounter in AP Physics 1. Whether you're analyzing a roller coaster, a collision, or a spring launcher, you're being tested on your ability to track where energy goes and why. The College Board wants you to recognize that conservation laws are powerful problem-solving tools: when you can identify a closed system and account for all energy transfers, you can predict outcomes without knowing every detail of the motion in between.

This guide organizes energy concepts by how they function in problem-solving: defining energy types, understanding transfers through work, applying conservation in different scenarios, and recognizing when energy "leaves" a system. Don't just memorize formulas—know which principle applies to which situation, and practice identifying the mechanism behind each energy change. That's what earns full credit on FRQs.

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Defining Energy Types

Before you can conserve energy, you need to identify what forms exist in your system. Each energy type has a specific formula and depends on particular variables—mass, velocity, position, or deformation.

Kinetic Energy

• Translational kinetic energy is the energy of motion, calculated as $KE = \frac{1}{2}mv^2$—note that velocity is squared, so doubling speed quadruples kinetic energy
• Rotational kinetic energy adds $KE_{rot} = \frac{1}{2}I\omega^2$ for spinning objects, where $I$ is moment of inertia and $\omega$ is angular velocity
• Total kinetic energy for rolling objects combines both: $K_{tot} = K_{trans} + K_{rot}$, a key equation for incline and rolling problems

Gravitational Potential Energy

• Position-based energy stored in a gravitational field, calculated as $PE_g = mgh$, where $h$ is height relative to a chosen reference point
• Reference point choice is arbitrary but must stay consistent throughout a problem—the zero level is wherever you define it
• Converts to kinetic energy as objects fall, making this the primary energy type in projectile and free-fall analysis

Elastic Potential Energy

• Stored energy in deformed materials like springs, calculated as $PE_s = \frac{1}{2}kx^2$, where $k$ is spring constant and $x$ is displacement from equilibrium
• Quadratic dependence on displacement means compressing a spring twice as far stores four times the energy
• Releases as kinetic energy when the spring returns to equilibrium, powering launchers and oscillating systems

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Energy Transfer Through Work

Work is the mechanism by which energy enters or leaves a system. Understanding the work-energy theorem lets you connect forces to energy changes without tracking every instant of motion.

Work-Energy Theorem

• Work equals change in kinetic energy: $W_{net} = \Delta KE$, making this your go-to equation when forces act over a distance
• Connects force and energy by defining work as $W = Fd\cos\theta$, where $\theta$ is the angle between force and displacement
• Applies to any force—gravity, springs, friction, applied forces—allowing you to calculate energy changes from force information

Power

• Rate of energy transfer defined as $P = \frac{W}{t}$ or equivalently $P = Fv$ for constant force and velocity
• Measured in watts (W), where $1 \text{ W} = 1 \text{ J/s}$—this unit conversion appears frequently on the exam
• Indicates system performance by showing how quickly energy is converted, crucial for comparing machines or analyzing time-dependent problems

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Conservation in Closed Systems

The real power of energy conservation emerges when you can treat a system as closed—no net external work done. In these cases, total mechanical energy stays constant, and you can relate initial and final states directly.

Law of Conservation of Energy

• Energy cannot be created or destroyed—only transformed between forms or transferred between objects
• Total energy of an isolated system remains constant, making this the most fundamental principle in physics
• Applies universally to all physical processes, from atomic interactions to planetary motion

Mechanical Energy Conservation

• In absence of non-conservative forces, total mechanical energy $E_{mech} = KE + PE$ remains constant throughout motion
• Enables shortcut calculations by setting $KE_i + PE_i = KE_f + PE_f$ without analyzing forces at every point
• Classic applications include pendulums, roller coasters, and objects on frictionless inclines—identify these scenarios quickly on the exam

Rolling Without Slipping

• Combines translational and rotational energy with the constraint $v_{cm} = r\omega$, linking linear and angular motion
• Static friction does no work in pure rolling because the contact point has zero instantaneous velocity—this is why mechanical energy is conserved
• Acceleration down an incline becomes $a = \frac{g\sin\theta}{1 + I/(MR^2)}$, showing that objects with larger moments of inertia accelerate more slowly

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System Boundaries and Energy Flow

Choosing your system boundary determines what counts as "internal" vs. "external" and whether energy is conserved or transferred. This is where many students lose points—always define your system explicitly.

Closed and Open Systems

• Closed systems exchange energy but not matter with surroundings—conservation of energy applies to the system's total energy
• Open systems allow both energy and matter transfer, requiring you to track what crosses the boundary
• System choice affects analysis: including friction as internal means mechanical energy isn't conserved; treating the surface as external means friction does work on your system

Energy Transformations

• Energy changes form during processes—gravitational PE becomes kinetic during free fall, kinetic becomes elastic in a spring collision
• Track all forms present at initial and final states to apply conservation correctly
• Real-world systems involve multiple transformations: a bouncing ball converts KE → elastic PE → KE → gravitational PE repeatedly

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When Energy "Leaves": Non-Conservative Forces

Not all forces conserve mechanical energy. Friction, air resistance, and other non-conservative forces convert mechanical energy to thermal energy, which typically leaves your system.

Efficiency and Energy Loss

• Efficiency equals useful energy output divided by total energy input: $\eta = \frac{E_{out}}{E_{in}} \times 100\%$
• Energy "losses" to friction and heat don't violate conservation—the energy still exists, just in forms not useful for mechanical work
• Practical applications require efficiency analysis: engines, machines, and biological systems all operate below 100% efficiency

Collisions and Energy Conservation

• Elastic collisions conserve both momentum and kinetic energy—rare in real life but common on exams
• Inelastic collisions conserve momentum but not kinetic energy; some KE converts to thermal energy, sound, or deformation
• Perfectly inelastic collisions (objects stick together) lose maximum kinetic energy while still conserving momentum—use this to find final velocity, then calculate KE loss

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Quick Reference Table

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Self-Check Questions

1. A solid sphere and a hollow sphere of equal mass and radius roll down the same frictionless incline from rest. Which reaches the bottom with greater translational speed, and why does mechanical energy conservation still apply to both?

2. Compare the energy transformations in a vertically oscillating mass on a spring versus a simple pendulum. What forms of potential energy are involved in each, and at what points is kinetic energy maximum?

3. Two carts collide on a frictionless track: one collision is elastic, the other is perfectly inelastic. Both conserve momentum—what quantity would you calculate to determine which collision occurred?

4. An FRQ shows a block sliding down a rough incline onto a frictionless surface, then compressing a spring. Identify which segments conserve mechanical energy and which require a work-energy approach accounting for friction.

5. A motor lifts a 50 kg mass 10 meters in 5 seconds, while another motor lifts the same mass the same height in 10 seconds. Compare the work done and power output of each motor—which quantities are equal and which differ?