Fundamental Laws of Motion

Why This Matters

The fundamental laws of motion aren't just a checklist to memorize—they're the conceptual backbone of everything you'll encounter in AP Physics C: Mechanics. These laws explain why objects accelerate, how energy flows through systems, and what happens when objects collide or rotate. On the exam, you're being tested on your ability to recognize which law governs a given scenario and apply it correctly, whether that's setting up a free-body diagram, writing a conservation equation, or connecting linear and rotational quantities.

What makes these laws powerful is how they interconnect. Newton's Second Law leads directly to the work-energy theorem; conservation of momentum emerges when Newton's Third Law acts within a closed system; Hooke's Law plugs into energy conservation to describe oscillations. The exam loves asking you to bridge these connections—an FRQ might start with forces, shift to energy, and finish with momentum. So don't just memorize $F = ma$—know when each law applies, what conditions must hold, and how to translate between them.

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Newton's Laws: The Foundation of Dynamics

Newton's three laws establish the rules for how forces produce motion. They define what forces are, how they cause acceleration, and why they always come in pairs. Every mechanics problem ultimately traces back to these principles.

Newton's First Law (Law of Inertia)

• Translational equilibrium occurs when $\Sigma \vec{F} = 0$—the object maintains constant velocity, which includes remaining at rest
• Inertia is the property of mass that resists changes in motion; greater mass means greater resistance to acceleration
• Inertial reference frames are required for Newton's laws to hold—accelerating frames introduce fictitious forces that complicate analysis

Newton's Second Law ($F = ma$)

• Net force equals mass times acceleration: $\Sigma \vec{F} = m\vec{a}$—this vector equation applies independently to each component direction
• Acceleration is directly proportional to net force and inversely proportional to mass, making this the quantitative workhorse of dynamics
• This law connects to everything: work-energy theorem, momentum changes, and rotational analogs all derive from $F = ma$

Newton's Third Law (Action-Reaction)

• Forces always occur in pairs—when object A exerts force $\vec{F}_{AB}$ on object B, then B exerts $\vec{F}_{BA} = -\vec{F}_{AB}$ on A
• Action-reaction pairs act on different objects, so they never cancel when analyzing a single object's motion
• This law underlies momentum conservation—internal forces in a system cancel, leaving only external forces to change total momentum

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Conservation Laws: When External Influences Vanish

Conservation laws are powerful shortcuts that bypass force analysis entirely. They apply when specific external influences (forces, torques, or non-conservative work) are absent or negligible. The key is recognizing the conditions that make each conservation law valid.

Conservation of Linear Momentum

• Total momentum $\vec{p} = \Sigma m\vec{v}$ remains constant when $\Sigma \vec{F}_{ext} = 0$—this is the go-to principle for collisions and explosions
• Momentum is a vector quantity, so conservation applies independently to each component direction
• Impulse $\vec{J} = \int \vec{F} \, dt = \Delta \vec{p}$ connects force and momentum change when external forces do act

Conservation of Energy

• Total mechanical energy $E = K + U$ is conserved when only conservative forces do work—gravity and springs qualify; friction does not
• Nonconservative forces like friction dissipate mechanical energy into thermal energy: $\Delta E_{mech} = W_{nc}$
• Energy bookkeeping requires choosing a reference level for potential energy and tracking all forms: $\frac{1}{2}mv^2$, $mgh$, $\frac{1}{2}kx^2$, and $\frac{1}{2}I\omega^2$

Conservation of Angular Momentum

• Total angular momentum $\vec{L} = \Sigma I\vec{\omega}$ remains constant when $\Sigma \vec{\tau}_{ext} = 0$—essential for rotating systems and orbits
• Angular momentum depends on mass distribution; changing $I$ (like a skater pulling in arms) changes $\omega$ to compensate
• This is the rotational analog of linear momentum conservation, with torque playing the role of force

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Energy Transfer: Work and Its Consequences

The work-energy theorem bridges force analysis and energy methods. It tells you how energy enters or leaves a system and connects Newton's Second Law to conservation principles.

Work-Energy Theorem

• Net work equals change in kinetic energy: $W_{net} = \Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$—this follows directly from integrating $F = ma$
• Work by individual forces can be calculated separately: $W = \int \vec{F} \cdot d\vec{r}$, then summed to find net work
• Conservative force work equals negative potential energy change: $W_{conservative} = -\Delta U$, which leads to energy conservation

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Force Laws: Specific Interactions

These laws describe how particular forces behave. They plug into Newton's Second Law or energy expressions to solve specific types of problems.

Hooke's Law for Springs

• Spring force is $\vec{F}_s = -k\Delta \vec{x}$, where $k$ is the spring constant (N/m) and $\Delta x$ is displacement from equilibrium
• The negative sign indicates a restoring force—the spring always pushes or pulls toward the equilibrium position
• Elastic potential energy is $U_s = \frac{1}{2}k(\Delta x)^2$, which combines with kinetic energy to describe oscillations

Universal Law of Gravitation

• Gravitational force is $F_g = \frac{Gm_1m_2}{r^2}$, always attractive and directed along the line connecting the masses
• Gravitational potential energy is $U_g = -\frac{Gm_1m_2}{r}$—the negative sign reflects that work is required to separate masses
• Near Earth's surface, this reduces to $F_g \approx mg$ and $U_g \approx mgh$ when $h \ll R_{Earth}$

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Orbital Motion: Gravity in Action

Kepler's Laws describe what planetary motion looks like; Newton's gravitation explains why it happens. These connect gravitational force to energy and angular momentum conservation.

Kepler's Laws of Planetary Motion

• First Law (Ellipses): Orbits are ellipses with the central mass at one focus—circular orbits are a special case
• Second Law (Equal Areas): A line from planet to sun sweeps equal areas in equal times, which is angular momentum conservation in disguise
• Third Law (Period-Radius): $T^2 \propto a^3$, or more precisely $T^2 = \frac{4\pi^2}{GM}a^3$ for orbits around mass $M$

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Quick Reference Table

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Self-Check Questions

1. A block slides across a rough horizontal surface and comes to rest. Which conservation law is violated, and why? Which law still holds?

2. Two objects collide and stick together. Compare what happens to (a) total momentum and (b) total kinetic energy. Which quantity is conserved, and what condition makes that true?

3. A satellite in elliptical orbit moves fastest at its closest approach to Earth. Explain this using two different fundamental principles from this guide.

4. If you double the mass attached to a spring, what happens to the period of oscillation? What if you double the spring constant instead? Justify using the relevant formula.

5. An astronaut floating in the International Space Station pushes off a wall. Identify which of Newton's three laws explains (a) why the astronaut accelerates, (b) why the wall experiences a force, and (c) why the astronaut continues moving after losing contact with the wall.