Dynamics — Newton's Laws of Motion

Big Picture

Newton's three laws form a logical chain: First Law — if nothing pushes on an object, its velocity stays the same. Second Law — if something does push, the acceleration is proportional to the net push and inversely proportional to the object's mass. Third Law — all pushes come in pairs: if A pushes B, B pushes A equally hard in the opposite direction. Together they let us predict any motion from known forces.

ELI-10: Explain It Like I'm 10

Imagine a shopping cart on a perfectly smooth floor. If nobody touches it, it sits still. If you give it one push and let go, it keeps rolling — forever — until friction or a wall stops it. If you push twice as hard, it speeds up twice as fast. If the cart is twice as heavy (twice the mass), the same push speeds it up only half as fast. And whenever you push on the cart, the cart pushes back on your hands. That is all three laws.

2.1 Newton's First Law — Inertia

Core Idea

An object at rest stays at rest, and an object in motion stays in motion with constant velocity, unless acted upon by a net external force.

Important Definitions

• Inertia: the tendency of an object to resist changes in its velocity.
• Mass m: a quantitative measure of inertia. SI unit: kilogram (kg). Mass is not weight.
• Equilibrium: state where the net force is zero → acceleration is zero. The object may be at rest or moving with constant velocity.
• Inertial frame: a reference frame that is not accelerating. Newton's laws hold in inertial frames.

ELI-10: Explain It Like I'm 10

A book sitting on a table stays there because no one pushes it. A hockey puck sliding on ice goes much farther than one sliding on rough concrete because ice has less friction. If there were truly zero friction, the puck would slide forever in a straight line without slowing down. Objects do not "need" a force to keep moving — they only need a force to change their motion.

Common Mistakes

• "A moving object must have a force pushing it forward." No — in the absence of friction, an object maintains its velocity without any forward force.
• Thinking equilibrium means "at rest." An object cruising at constant speed is also in equilibrium.

Key Takeaway

Zero net force → constant velocity. This is the law of inertia. No force is needed to sustain motion.

2.2 Newton's Second Law — The Force-Mass-Acceleration Relationship

Core Idea

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The acceleration is in the direction of the net force.

Physics and Mathematics

∑F = ma

Where:

• ∑F = vector sum of all forces (net force), newtons (N)
• m = mass (kg)
• a = acceleration (m/s²)

1 N = 1 kg · m/s²

Critical: ∑F is the vector sum. You must add all forces as vectors before applying the law. This is a vector equation — it applies independently in each direction:

∑F_x = m a_x,   ∑F_y = m a_y

Conceptual Example

If you push a 2 kg block to the right with 10 N of force and friction pushes left with 4 N, the net force is 6 N right. The acceleration is a = 62 = 3 m/s² to the right.

ELI-10: Explain It Like I'm 10

Think of kicking a ball. Kick a soccer ball (light) hard, and it flies away fast. Kick a bowling ball (heavy) with the same kick, and it barely moves. The law says: acceleration equals the total push divided by the heaviness of the object. More push = more acceleration. More mass = less acceleration for the same push.

Why the Squared Term?

F = ma says force is measured in kg · m/s². The m/s² comes from the fact that force causes acceleration, not velocity. It takes a constant force to produce a constant acceleration. This will connect directly to work and energy in Topic 3.

Common Mistakes

• Using individual forces instead of the net force.
• Forgetting that force and acceleration are vectors — direction matters.
• Confusing mass with weight. Mass is intrinsic; weight is the gravitational force mg.

Key Takeaway

∑F = ma. Acceleration is proportional to net force, inversely proportional to mass. Always add forces as vectors first.

2.3 Newton's Third Law — Action-Reaction Pairs

Core Idea

If object A exerts a force on object B, then object B exerts a force of equal magnitude and opposite direction on object A.

F_{A on B} = -F_{B on A}

Important Distinction

These two forces act on different objects. That is why they do not cancel each other out when analyzing the motion of a single object. The force on the object you care about determines its acceleration.

Example: A book resting on a table. The Earth pulls down on the book (gravity). The table pushes up on the book (normal force). These two forces act on the same object (the book) and happen to balance. They are NOT an action-reaction pair. The action-reaction pair is: Earth pulls book down ↔ Book pulls Earth up.

ELI-10: Explain It Like I'm 10

Stand on a skateboard and push against a wall. You roll backward. Why? Because when you push the wall, the wall pushes you back equally hard. The wall does not move much because it is attached to the building, but you move because you are light and on wheels. Every push comes with a push-back. You cannot touch something without it touching you.

Common Mistakes

• Thinking action and reaction forces cancel. They act on different objects — they affect each object's motion separately.
• Confusing balanced forces on one object with action-reaction pairs.

Identifying Action-Reaction Pairs

A reliable method: state the pair as "A exerts a force on B" and "B exerts a force on A." Then check: do the two forces act on DIFFERENT objects? If yes, they are a valid action-reaction pair. If both forces act on the SAME object, they are NOT a third-law pair — they are just two forces that happen to balance.

Example — Book on a table: The Earth pulls down on the book (F_E → B) and the table pushes up on the book (F_T → B). These both act on the book — they are NOT a third-law pair. The actual pairs are:

1. Earth pulls book down ↔ Book pulls Earth up.
2. Table pushes book up ↔ Book pushes table down.

Example — Rocket launch: The rocket pushes exhaust gases downward; the gases push the rocket upward. Both forces are equal in magnitude. The rocket accelerates upward because the force from the gases acts ON the rocket.

Proportional Reasoning for Newton's Second Law

From a = F_net/m:

• Double the net force (same mass) → double the acceleration.
• Double the mass (same net force) → half the acceleration.
• Triple the net force AND triple the mass → acceleration stays the same.

This proportionality is why very different objects can have the same acceleration — a truck and a bicycle both starting from rest with appropriately scaled forces can accelerate identically.

Key Takeaway

Forces always come in equal-and-opposite pairs acting on different objects. Analyze one object at a time.

2.4 Free-Body Diagrams

Core Idea

A free-body diagram (FBD) is a simplified drawing showing all external forces acting on a single object. It is the essential tool for applying Newton's second law.

Common Forces

• Weight W = mg: always downward, toward Earth's center.
• Normal force N: perpendicular to the contact surface, pushing away from the surface.
• Tension T: pulling force along a rope, string, or cable.
• Applied force F_app: a push or pull applied directly.
• Friction f: parallel to the contact surface, opposing relative motion or attempted motion.

How to Draw a Free-Body Diagram

1. Isolate the object — draw it as a dot or simple shape.
2. Draw every force as an arrow pointing in the direction the force acts.
3. Label each force clearly.
4. Choose a coordinate system (usually align one axis with the direction of acceleration).
5. Resolve forces into components along your axes.

ELI-10: Explain It Like I'm 10

Imagine you want to know why a sled moves (or does not move) when you pull it. Draw the sled as a dot. Draw an arrow for your pull, an arrow for friction pulling backward, an arrow for gravity pulling down, and an arrow for the ground pushing up. Add up all the arrows. If the arrows to the right are bigger than the arrows to the left, the sled accelerates right. It really is that simple.

Common Mistakes

• Including forces that act on other objects.
• Forgetting the normal force.
• Drawing the normal force always equal to mg (it is not — see Section 2.6).

Key Takeaway

A free-body diagram converts a physical situation into a solvable vector equation. Never skip the FBD.

2.5 Friction

Core Idea

Friction is a contact force that opposes relative motion between two surfaces. It arises from microscopic interactions at the interface.

Physics and Mathematics

Static friction (surfaces not sliding relative to each other): f_s ≤ μ_s N

• f_s = static friction force (N)
• μ_s = coefficient of static friction (dimensionless)
• N = normal force (N)

The static friction force adjusts to whatever is needed up to its maximum μ_s N.

Kinetic friction (surfaces sliding): f_k = μ_k N

• f_k = kinetic friction force (N)
• μ_k = coefficient of kinetic friction (dimensionless)

Always: μ_s > μ_k for a given pair of surfaces. It is harder to start sliding than to keep sliding.

Direction: Friction always opposes the relative motion (or attempted motion) between the surfaces.

ELI-10: Explain It Like I'm 10

Rub your hands together. The resistance you feel is friction. Push a heavy box across the floor — it is hardest to get it started. Once it is moving, it is easier to keep it going. That is because static friction (not moving yet) is stronger than kinetic friction (already sliding). Friction depends on how hard the two surfaces are pressed together (the normal force) and how rough they are.

Common Mistakes

• Thinking friction always equals μN. Static friction is up to μ_s N, not always equal to it.
• Assuming the normal force always equals mg. It depends on the situation (angle, other vertical forces).

Key Takeaway

Static friction adjusts up to a maximum; kinetic friction is constant. Both depend on the normal force and surface roughness. μ_s > μ_k.

2.6 Applications

Inclined Planes

An object on a frictionless incline of angle θ has acceleration a = gsinθ down the ramp. The normal force is N = mgcosθ.

With friction, the friction force acts up the ramp (opposing the motion down), and the acceleration becomes a = g(sinθ- μ_kcosθ) for sliding.

Worked Example: Block on an Incline

Problem: A 5.0 kg block slides down a 30° incline with μ_k = 0.20. Find its acceleration.

Given: m = 5.0 kg, θ= 30^°, μ_k = 0.20

Find: a

FBD: Weight mg down. Normal N perpendicular to ramp. Kinetic friction f_k up the ramp.

Solution:

• Perpendicular to ramp: ∑F_⊥= N - mgcosθ= 0 → N = (5.0)(9.8)cos30^°= 42.4 N
• Parallel to ramp: ∑F_∥= mgsinθ- f_k = ma
• f_k = μ_k N = (0.20)(42.4) = 8.48 N
• mgsinθ= (5.0)(9.8)sin30^°= 24.5 N
• a = 24.5 - 8.485.0 = 3.2 m/s² down the ramp.

Does it Make Sense? Without friction, a = gsin30^°= 4.9 m/s². With friction, it is lower (3.2 m/s²) — reasonable.

Connected Objects and Atwood Machines

For objects connected by a massless, inextensible string over a frictionless pulley, the magnitude of acceleration is the same for both objects. The tension is the same throughout the string.

Atwood machine (two masses m₁ > m₂ hanging on opposite sides): a = m₁ - m₂m₁ + m₂ g The heavier mass accelerates downward; the lighter mass accelerates upward.

ELI-10: Explain It Like I'm 10

Picture an elevator. When the elevator starts going up, you feel heavier for a moment — the floor pushes up harder than just your weight. When it starts going down, you feel lighter. If the cable snapped (hopefully not!), you would feel weightless because you and the elevator would fall together — the floor would not need to push up on you at all. The normal force is not always equal to mg.

Common Mistakes

• "The normal force always equals mg." It equals mg only on a horizontal surface with no other vertical forces. On an incline, in an accelerating elevator, or with an additional vertical push, it differs.
• Mixing up static and kinetic friction coefficients.

Key Takeaway

FBDs make any dynamics problem solvable. Inclines, pulleys, and banked curves all yield to the same process: isolate the object, draw forces, apply ∑F = ma in each direction.

Topic Summary

• Newton's First Law: Objects maintain constant velocity unless a net external force acts. This is the law of inertia.
• Newton's Second Law: ∑F = ma. Acceleration is proportional to net force and inversely proportional to mass. This is a vector equation.
• Newton's Third Law: Forces come in equal-and-opposite pairs acting on different objects.
• Free-body diagrams: Isolate one object, draw all forces, apply ∑F = ma per direction.
• Friction opposes relative motion. Static friction (f_s ≤ μ_s N) adjusts up to its maximum; kinetic friction (f_k = μ_k N) is constant. μ_s > μ_k.
• Inclined planes: Resolve weight into components. Normal force is mgcosθ, not always mg.
• Connected objects share the same acceleration magnitude; tension is uniform in a massless string.

Essential Equations

Concept Check

1. A book sits on a table. Identify all forces on the book. Identify the Newton's-Third-Law reaction force to each.
2. If you push a crate with 100 N and it does not move, what is the friction force? Why?
3. A block slides down a frictionless incline. Does its acceleration depend on its mass? Why or why not?
4. In an Atwood machine, why is the tension less than the weight of the heavier mass?
5. You are standing on a bathroom scale in an elevator accelerating upward. Does the scale read more, less, or equal to your weight? Explain.

Open Educational References

• OpenStax, College Physics, Chapter 4: Dynamics — Force and Newton's Laws of Motion
• OpenStax, University Physics, Volume 1, Chapters 5–6: Newton's Laws, Applications