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 probes your conceptual understanding hard: 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: what happens when forces are absent, how forces cause acceleration, and how objects interact with each other.

Newton's First Law (Law of Inertia)

An object keeps doing whatever it's already doing (sitting still or moving at constant velocity) unless a net external force acts on it. This means zero net force produces zero acceleration, not zero motion. That distinction is a classic exam trap.

• Inertia is an object's resistance to changes in motion. More mass means more inertia, which means it's harder to speed up, slow down, or change direction.
• A hockey puck sliding on frictionless ice at 5 m/s will keep sliding at 5 m/s forever. No force is needed to maintain constant velocity.

Newton's Second Law ($\Sigma \vec{F} = m\vec{a}$)

This is the quantitative heart of dynamics. The net force on an object equals its mass times its acceleration:

$\Sigma \vec{F} = m\vec{a}$

Acceleration is directly proportional to net force and inversely proportional to mass. Doubling the net force doubles the acceleration, while doubling the mass halves the acceleration for the same force.

• This law also defines the unit of force: $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$
• On the exam, always use the net force, not just any single force. A common mistake is plugging in only one force (like gravity) when friction or a normal force also acts on the object.
• This is a vector equation. You apply it independently along each axis: $\Sigma F_x = ma_x$ and $\Sigma F_y = ma_y$. Many problems require you to work with both component equations simultaneously.

Newton's Third Law (Action-Reaction)

Forces always come in pairs. If object A pushes on object B, then B pushes back on A with equal magnitude and opposite direction. These are called Third Law pairs (or action-reaction pairs).

• Third Law pairs act on different objects, so they never cancel each other on a single free-body diagram.
• Common exam trap: The normal force on a book sitting on a table and the book's weight are not a Third Law pair. Both of those forces act on the book. The actual Third Law partner of the book's weight is the gravitational pull the book exerts on the Earth. Similarly, the Third Law partner of the normal force from the table on the book is the contact force the book exerts downward on the table.

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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

A free-body diagram (FBD) isolates a single object and shows every force acting on that object as an arrow. Never include forces the object exerts on other things.

Steps to draw one:

1. Identify the object you're analyzing and draw it as a simple dot or box.
2. Draw an arrow for each force acting on the object. The direction of the arrow shows the force's direction, and the length should roughly indicate relative magnitude.
3. Label every arrow ($F_g$, $F_N$, $f$, $T$, etc.).
4. Choose a coordinate system (often tilted for inclines) and resolve forces into components.

The exam awards points for correct FBDs even if your final numerical answer is wrong, so always draw one.

Common Force Types

• Gravitational force (weight): $\vec{F}_g = m\vec{g}$, always points straight down toward Earth's center. On Earth, $g \approx 9.8 \text{ m/s}^2$ (the exam often uses $10 \text{ m/s}^2$ for quick calculations).
• Normal force ($F_N$): Acts perpendicular to the contact surface. It adjusts its magnitude to prevent objects from passing through each other. It's not always equal to $mg$. On an incline, the normal force equals $mg\cos\theta$. In an accelerating elevator, it differs from $mg$ by $ma$.
• Friction: Opposes relative motion (or attempted motion) between surfaces. Static friction can range from zero up to a maximum: $f_s \leq \mu_s F_N$. Kinetic friction has a fixed value: $f_k = \mu_k F_N$. Note that $\mu_s > \mu_k$ for the same pair of surfaces, which is why it takes more force to start an object sliding than to keep it sliding.
• Tension ($T$): Pulls along a rope or string, always directed away from the object along the rope. For a massless, non-stretching (ideal) rope, tension is the same throughout its length.
• Applied force: Any push or pull from an external agent, like a person pushing a box or a motor pulling a cable.

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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. It depends on the local gravitational field strength $g$. A 70 kg person weighs about 686 N on Earth ($70 \times 9.8$) but only about 114 N on the Moon ($70 \times 1.6$).
• Apparent weight is the normal force you feel. In an elevator accelerating upward, the normal force exceeds your true weight, so you feel heavier. In free fall, the normal force is zero, so you feel weightless even though gravity still acts on you.

Equilibrium and Net Force

An object is in equilibrium whenever the net force on it is zero ($\Sigma \vec{F} = 0$). There are two types:

• Static equilibrium: $\Sigma \vec{F} = 0$ and $\vec{v} = 0$. The object is at rest and stays at rest. Example: a hanging sign supported by two cables.
• Dynamic equilibrium: $\Sigma \vec{F} = 0$ but $\vec{v} \neq 0$. The object moves at constant velocity. Example: a car cruising at a steady 60 km/h on a flat road with driving force exactly balancing air resistance and friction.

For equilibrium problems, set up component equations: $\Sigma F_x = 0$ and $\Sigma F_y = 0$, then solve for unknown forces.

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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

The key move is to tilt your coordinate system so the x-axis runs parallel to the incline and the y-axis runs perpendicular to it. This way, acceleration points purely along x, and the y-direction stays in equilibrium.

Weight decomposes into two components:

• Parallel to the incline: $F_{g,\parallel} = mg\sin\theta$ (pulls the object down the ramp)
• Perpendicular to the incline: $F_{g,\perp} = mg\cos\theta$ (balanced by the normal force)

A quick way to remember which trig function goes where: the component along the incline uses $\sin\theta$, and the component into the surface uses $\cos\theta$. As a sanity check, when $\theta = 0$ (flat surface), $\sin 0 = 0$ (no component pulling along the surface) and $\cos 0 = 1$ (full weight pushes into the surface). That confirms the assignments.

On a frictionless incline, the normal force equals $mg\cos\theta$, which is less than the object's full weight. The acceleration down the ramp is $a = g\sin\theta$, which notably doesn't depend on mass. On a frictionless ramp, a bowling ball and a marble accelerate at the same rate.

If friction is present, the net force along the ramp becomes $mg\sin\theta - \mu_k mg\cos\theta$ (for an object sliding down), giving $a = g(\sin\theta - \mu_k \cos\theta)$.

Centripetal Force and Circular Motion

Centripetal force is not a new type of force. It's the label for whatever net force points toward the center of a circular path. You need to identify which real force (or combination of forces) provides it in each situation.

$F_c = \frac{mv^2}{r}$

where $v$ is the tangential speed and $r$ is the radius of the circular path. The direction of centripetal acceleration is always toward the center, even though the object's velocity is tangent to the circle.

Examples of what provides centripetal force:

• Ball on a string: tension
• Car on a flat curve: static friction
• Satellite in orbit: gravity
• Car on a banked curve: a component of the normal force (plus friction, if present)

If the centripetal force were suddenly removed, the object wouldn't fly outward. It would continue in a straight line tangent to the circle, following Newton's First Law. "Centrifugal force" is not a real force in an inertial reference frame, and you should avoid using that term on the AP exam.

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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 defined as $\vec{p} = m\vec{v}$. It's a vector quantity measured in $\text{kg} \cdot \text{m/s}$, and it points in the same direction as velocity.

Newton's Second Law in momentum form is:

$\Sigma \vec{F} = \frac{\Delta \vec{p}}{\Delta t}$

Force equals the rate of change of momentum. This form is actually more general than $F = ma$ because it also applies to situations where mass changes (like a rocket expelling fuel), though AP Physics 1 mostly sticks to constant-mass problems.

Conservation of momentum: In an isolated system (no net external force), total momentum is conserved. This is the principle behind every collision and explosion problem on the exam.

$m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

Impulse

Impulse equals the change in momentum:

$\vec{J} = \Delta \vec{p} = \vec{F}_{avg} \Delta t$

The impulse-momentum theorem connects force, time, and the resulting change in motion. For the same momentum change, a longer contact time means a smaller average force. This is why airbags work: they increase the time over which your momentum drops to zero during a crash, reducing the peak force on your body.

Graphically, impulse equals the area under a force vs. time graph. This shows up frequently on the AP exam. For rectangular or triangular force profiles, calculate the area using basic geometry. For irregular shapes, estimate by counting grid squares or breaking the area into simpler shapes.

Collision Types

The AP exam expects you to distinguish between collision types:

• Elastic collision: Both momentum and kinetic energy are conserved. Objects bounce off each other. Rare in everyday life but common in exam problems (e.g., billiard balls as an approximation).
• Inelastic collision: Momentum is conserved, but kinetic energy is not (some converts to heat, sound, or deformation). Most real-world collisions are inelastic.
• Perfectly inelastic collision: The objects stick together after colliding. Momentum is still conserved, but this type loses the maximum amount of kinetic energy. Use $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f$.

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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?