Important Energy Concepts

Why This Matters

Energy is the currency of physics—every mechanical system you'll analyze on the AP Physics C exam involves energy transformations, transfers, or conservation. The exam tests whether you understand why energy behaves the way it does: why some forces "store" energy while others dissipate it, why certain collisions preserve kinetic energy while others don't, and how to track energy flow through complex systems. These concepts connect directly to work, momentum, and even rotational dynamics, making energy the thread that ties Units 3, 4, and 6 together.

You're being tested on your ability to apply energy principles, not just recall formulas. Can you identify when mechanical energy is conserved versus when you need to account for nonconservative work? Can you extract information from a potential energy curve? Can you set up an integral for work done by a variable force? Don't just memorize $KE = \frac{1}{2}mv^2$—know when to use energy methods instead of force methods, and understand what each term in the conservation equation represents physically.

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The Work-Energy Connection

Work is the mechanism by which energy enters or leaves a system. Understanding work—both conceptually and mathematically—is essential for tracking energy changes. The work-energy theorem provides the bridge between force analysis and energy analysis.

Work-Energy Theorem

• Net work equals change in kinetic energy—mathematically, $W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$
• All forces contribute to net work, including gravity, friction, applied forces, and spring forces acting on the object
• Powerful problem-solving tool when you care about speed changes but not the details of acceleration over time

Work Done by Variable Forces

• Integral calculus required—when force varies with position, $W = \int F(x)\,dx$ gives the work as the area under the force-displacement curve
• Spring force is the classic example—since $F_s = -kx$, work done by the spring is $W_s = -\frac{1}{2}k\Delta x^2$ (negative when stretching)
• Graphical interpretation matters—the AP exam frequently asks you to find work from $F$ vs. $x$ graphs using area calculations

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Conservative vs. Nonconservative Forces

The distinction between these force types determines whether you can use conservation of mechanical energy or must account for energy dissipation. Conservative forces allow energy to be "stored" and recovered; nonconservative forces convert mechanical energy into other forms.

Conservative Forces

• Path-independent work—the work done depends only on initial and final positions, not the route taken between them
• Associated potential energy functions exist for gravity ($U_g = mgh$) and springs ($U_s = \frac{1}{2}kx^2$), allowing energy storage
• Work done equals negative change in potential energy—mathematically, $W_{conservative} = -\Delta U$, which is why potential energy decreases when these forces do positive work

Nonconservative Forces

• Path-dependent work—friction and air resistance do different amounts of work depending on the path length and direction
• Energy dissipation occurs—mechanical energy converts to thermal energy, sound, or deformation that cannot be recovered
• Must be tracked separately in energy equations: $\Delta E_{mech} = W_{nc}$, where $W_{nc}$ is typically negative (energy leaves the system)

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Forms of Mechanical Energy

Mechanical energy comes in two fundamental forms: kinetic (motion) and potential (configuration). The total mechanical energy of a system is $E = KE + U$, and tracking how energy shifts between these forms is central to problem-solving.

Kinetic Energy

• Energy of motion defined as $KE = \frac{1}{2}mv^2$ for translational motion (and $KE_{rot} = \frac{1}{2}I\omega^2$ for rotation)
• Quadratic dependence on speed—doubling velocity quadruples kinetic energy, making high-speed collisions dramatically more energetic
• Always positive or zero—a system's kinetic energy cannot be negative, which constrains allowed motions in energy analysis

Gravitational Potential Energy

• Near Earth's surface: $U_g = mgh$, where $h$ is height above a chosen reference level (you pick where $U = 0$)
• Reference level is arbitrary—only changes in potential energy have physical meaning, so choose a convenient zero point
• Converts to/from kinetic energy as objects rise and fall, enabling analysis of projectiles, pendulums, and roller coasters

Elastic Potential Energy

• Stored in deformed springs: $U_s = \frac{1}{2}kx^2$, where $x$ is displacement from the spring's relaxed length
• Quadratic in displacement—compressing or stretching a spring twice as far stores four times the energy
• Drives simple harmonic motion—energy oscillates between $U_s$ and $KE$ in mass-spring systems with period $T = 2\pi\sqrt{m/k}$

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Conservation and Transfer Principles

Conservation of energy is one of the most powerful problem-solving tools in mechanics. When mechanical energy is conserved, you can relate initial and final states without knowing the details of the motion in between.

Conservation of Mechanical Energy

• Applies when only conservative forces do work—in this case, $KE_i + U_i = KE_f + U_f$ (total mechanical energy is constant)
• System selection matters—include all interacting objects (Earth, springs, etc.) so that internal forces become conservative potential energies
• Bypasses kinematics—you can find final speeds without solving differential equations or tracking acceleration

Energy Accounting with Nonconservative Forces

• Modified conservation equation: $KE_i + U_i + W_{nc} = KE_f + U_f$, where $W_{nc}$ accounts for energy added or removed
• Friction always removes energy—$W_{friction} = -f_k \cdot d$ (negative), reducing the system's mechanical energy
• Energy bar charts help—visual representations of initial energy, work done, and final energy prevent sign errors

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Power and Energy Rate

Power measures how quickly energy is transferred or transformed. While energy tells you "how much," power tells you "how fast"—a critical distinction in real-world applications.

Power

• Rate of energy transfer: $P = \frac{dW}{dt} = \frac{dE}{dt}$, measured in watts ($1 \text{ W} = 1 \text{ J/s}$)
• Force-velocity relationship: $P = \vec{F} \cdot \vec{v} = Fv\cos\theta$, useful when force and velocity are known directly
• Average vs. instantaneous—average power is $P_{avg} = \frac{W}{\Delta t}$, while instantaneous power uses derivatives

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Energy Diagrams and Equilibrium

Potential energy diagrams provide a powerful visual tool for analyzing motion without solving equations. The shape of the $U(x)$ curve tells you about forces, equilibrium points, and allowed regions of motion.

Energy Diagrams

• Force from slope: $F_x = -\frac{dU}{dx}$—the force points "downhill" on the potential energy curve, toward lower $U$
• Total energy line (horizontal) shows $E = KE + U$; the gap between $E$ and $U(x)$ equals kinetic energy at that position
• Turning points occur where $E = U(x)$—kinetic energy is zero, and the object reverses direction

Equilibrium and Stability

• Equilibrium at $\frac{dU}{dx} = 0$—where the slope is zero, the force is zero, and the object can remain at rest
• Stable equilibrium occurs at potential energy minima (object returns when displaced); unstable at maxima (object accelerates away)
• Bounded vs. unbounded motion—if $E$ is below surrounding peaks, motion is confined between turning points (like SHM)

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Collisions and Energy in Multi-Body Systems

Collisions test your ability to combine energy and momentum principles. Momentum is always conserved in collisions (assuming no external impulse), but kinetic energy conservation depends on the collision type.

Elastic and Inelastic Collisions

• Elastic collisions conserve both momentum and kinetic energy—use both conservation laws to solve for two unknowns
• Inelastic collisions conserve momentum only—kinetic energy decreases as some converts to deformation, heat, or sound
• Perfectly inelastic (objects stick together) loses the maximum kinetic energy while still conserving momentum

Center of Mass and System Energy

• Center of mass velocity $v_{cm} = \frac{p_{total}}{M_{total}}$ remains constant in isolated collisions
• Kinetic energy splits into center-of-mass motion plus motion relative to center of mass; only the relative part can be lost in collisions
• Multi-body problems often simplify when you track total momentum and energy rather than individual objects

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Quick Reference Table

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Self-Check Questions

1. A block slides down a rough incline. Which quantities are conserved: momentum, kinetic energy, mechanical energy, or total energy? Explain why for each.

2. Two springs have constants $k_1$ and $k_2$. If compressed the same distance, which stores more elastic potential energy? What if they're stretched by the same force instead?

3. On a potential energy diagram, how can you identify (a) where the force is zero, (b) where the force is maximum, and (c) where motion is forbidden for a given total energy?

4. Compare and contrast: A ball dropped from height $h$ vs. a ball launched horizontally from the same height. How do their speeds at ground level compare, and why does energy analysis give this result so easily?

5. In a perfectly inelastic collision between a moving cart and a stationary cart of equal mass, what fraction of the initial kinetic energy is lost? Show how you'd set up this calculation using both momentum and energy conservation.