Newton's Laws of Motion

Why This Matters

Newton's Laws aren't just three statements you memorize for the exam—they're the foundation for everything in mechanics. When you see problems about objects on ramps, cars rounding curves, collisions, or rockets launching, you're really being tested on whether you understand how forces cause (or don't cause) changes in motion. The AP Physics 1 exam loves to probe your conceptual understanding: Can you draw a correct free-body diagram? Do you know which forces form Newton's Third Law pairs? Can you explain why an object accelerates or stays in equilibrium?

These laws connect directly to Unit 2: Force and Translational Dynamics and Unit 4: Linear Momentum, showing up in multiple-choice questions and nearly every mechanics FRQ. You'll need to apply them to inclined planes, circular motion, collisions, and systems of objects. Don't just memorize "$F = ma$"—know what each law tells you about the relationship between forces and motion, and practice identifying which law applies in different scenarios.

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The Three Laws: Foundations of Dynamics

Each of Newton's Laws addresses a different aspect of how forces relate to motion—from what happens when forces are absent, to how forces cause acceleration, to how objects interact with each other.

Newton's First Law (Law of Inertia)

• An object maintains its velocity—whether at rest or moving—unless a net external force acts on it
• Inertia is the property that quantifies an object's resistance to changes in motion; more mass means more inertia
• Zero net force means zero acceleration, not zero motion—a common exam misconception to avoid

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

• Net force equals mass times acceleration ($\Sigma \vec{F} = m\vec{a}$)—this is the quantitative heart of dynamics
• Acceleration is directly proportional to force and inversely proportional to mass; doubling mass halves acceleration for the same force
• This law defines the newton: $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$, connecting force units to fundamental quantities

Newton's Third Law (Action-Reaction)

• Forces always come in pairs—if object A pushes on B, then B pushes on A with equal magnitude and opposite direction
• Third Law pairs act on different objects, so they never cancel each other in a free-body diagram
• Common exam trap: Normal force and weight are not a Third Law pair—they act on the same object

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Analyzing Forces: Tools and Techniques

Before applying Newton's Laws, you need to correctly identify and represent all forces acting on an object. Free-body diagrams are your most important problem-solving tool.

Free-Body Diagrams

• Show only forces acting ON the object—never include forces the object exerts on other things
• Each arrow's direction shows force direction, and length should roughly indicate relative magnitude
• Start every dynamics problem here; the exam awards points for correct FBDs even if your final answer is wrong

Common Force Types

• Gravitational force (weight): $\vec{F}_g = m\vec{g}$, always points toward Earth's center (straight down near the surface)
• Normal force: Perpendicular to the contact surface, adjusts magnitude to prevent objects from passing through each other
• Friction: Opposes relative motion (or attempted motion) between surfaces; $f_s \leq \mu_s N$ for static, $f_k = \mu_k N$ for kinetic
• Tension: Pulls along a rope or string; for a massless rope, tension is the same throughout

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Mass, Weight, and Equilibrium

Understanding the distinction between mass and weight—and knowing when forces balance—is essential for solving statics problems and understanding apparent weight.

Mass vs. Weight

• Mass ($m$) measures the amount of matter in kilograms; it's an intrinsic property that doesn't change with location
• Weight ($F_g = mg$) is the gravitational force on an object, measured in newtons
• Apparent weight equals normal force, which can differ from true weight in accelerating systems (elevators, free fall)

Equilibrium and Net Force

• Static equilibrium means $\Sigma \vec{F} = 0$ and $\vec{v} = 0$; the object is at rest and stays at rest
• Dynamic equilibrium means $\Sigma \vec{F} = 0$ but $\vec{v} \neq 0$; the object moves at constant velocity
• For equilibrium problems, set up equations where forces in each direction sum to zero: $\Sigma F_x = 0$ and $\Sigma F_y = 0$

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Applications: Inclined Planes and Circular Motion

These two scenarios appear constantly on AP Physics 1 because they require you to decompose forces, choose coordinate systems wisely, and apply Newton's Second Law in component form.

Objects on Inclined Planes

• Tilt your coordinate system so x is parallel to the incline and y is perpendicular; this simplifies force decomposition
• Weight components: $F_{g,\parallel} = mg\sin\theta$ (causes acceleration down the ramp), $F_{g,\perp} = mg\cos\theta$ (balanced by normal force)
• Normal force equals $mg\cos\theta$ on a simple incline, which is less than the object's full weight

Centripetal Force and Circular Motion

• Centripetal force is not a new force—it's the net force directed toward the center of the circular path
• Magnitude: $F_c = \frac{mv^2}{r} = m\omega^2 r$, where $v$ is tangential speed and $r$ is radius
• Identify what provides centripetal force in each scenario: tension (ball on string), friction (car on curve), gravity (satellite in orbit), normal force (loop-the-loop)

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Momentum and Impulse: Extending Newton's Laws

Newton's Second Law can be rewritten in terms of momentum, revealing the impulse-momentum theorem and conservation principles that dominate Unit 4.

Momentum

• Momentum is $\vec{p} = m\vec{v}$, a vector quantity measured in $\text{kg} \cdot \text{m/s}$
• Newton's Second Law in momentum form: $\Sigma \vec{F} = \frac{d\vec{p}}{dt}$; force equals the rate of change of momentum
• Total momentum is conserved in isolated systems (no external forces), making this essential for collision problems

Impulse

• Impulse equals change in momentum: $\vec{J} = \Delta \vec{p} = \vec{F}_{avg} \Delta t$
• Impulse-momentum theorem connects force, time, and motion change; longer contact time means smaller average force for the same momentum change
• Graphically, impulse equals the area under a force vs. time graph—a common AP exam question type

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Quick Reference Table

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Self-Check Questions

1. A book sits motionless on a table. Identify the forces acting on the book and explain why the normal force and the book's weight are not a Newton's Third Law pair.

2. Two objects experience the same net force, but object A has twice the mass of object B. Compare their accelerations and explain your reasoning using Newton's Second Law.

3. A car rounds a flat curve at constant speed. What force provides the centripetal acceleration, and what would happen if this force suddenly disappeared?

4. An object slides down a frictionless incline at angle $\theta$. Derive an expression for its acceleration and explain why the acceleration is independent of the object's mass.

5. Compare and contrast momentum and kinetic energy: Which is conserved in all collisions? Which can be negative? How does each quantity depend on velocity?