AP Physics 1: Dynamics and Newton's Laws

Dynamics is the study of why objects change their motion. In AP Physics 1, that “why” almost always comes down to forces and Newton’s laws. If you can consistently draw a correct free-body diagram and translate it into equations, you can solve most equilibrium and acceleration problems you will see in the course.

This article lays out a systematic approach to force analysis, with clear treatment of Newton’s three laws and the most common forces: weight, normal force, tension, and friction.

The core workflow: from situation to solution

A reliable process prevents mistakes, especially when problems get multi-step.

1) Choose the system

Decide what object (or collection of objects) you are analyzing. The “system” is what your free-body diagram represents. Anything outside the system can exert forces on it.

• If you choose a single block, the rope pulling it is external and tension is a force on your diagram.
• If you choose “block + rope” together, the internal tension may disappear, but forces at the ends remain external.

Good system choices simplify the math.

2) Draw a free-body diagram (FBD)

A free-body diagram is not a picture of the scene. It is a force inventory on the chosen object, drawn as a dot or simple shape with arrows for forces.

Rules that keep FBDs clean:

• Draw only forces acting on the object, not forces the object exerts on others.
• Do not include “motion arrows” or “acceleration arrows” as forces.
• Label forces with meaningful names (such as $T$, $N$, $mg$, $f_k$) and choose clear directions.

3) Choose axes that match the physics

The best axes reduce components. Common smart choices:

• For objects on an incline, choose $x$ along the slope and $y$ perpendicular to it.
• For circular motion, choose radial and tangential directions if appropriate.

4) Apply Newton’s 2nd law in each direction

Newton’s 2nd law is vector-based: $$\sum \vec{F}=m\vec{a}.$$

In component form: $$\sum F_x=m{a}_x,\sum F_y=m{a}_y.$$

Many AP problems are solved by writing these two equations correctly from the FBD and then using constraints (such as “same acceleration” in a rope system).

Newton’s three laws in AP Physics 1

Newton’s First Law (inertia)

If the net force is zero, acceleration is zero. That means constant velocity, which includes rest: $$\sum \vec{F}=0\Rightarrow \vec{a}=0.$$

This is the foundation of equilibrium problems. A common pitfall is assuming “at rest” means no forces exist. Instead, forces balance.

Newton’s Second Law (dynamics)

Net force causes acceleration: $$\sum \vec{F}=m\vec{a}.$$

Key idea: the net force is not one of the individual forces. It is the vector sum of all external forces.

Also, acceleration points in the direction of the net force, not necessarily the direction of motion.

Newton’s Third Law (action-reaction)

For every interaction, forces come in equal and opposite pairs acting on different objects:

• If a rope pulls up on a block with tension $T$, the block pulls down on the rope with tension $T$.
• If a table pushes up on a book with normal force $N$, the book pushes down on the table with normal force $N$.

Third-law pairs never appear on the same free-body diagram because they act on different objects. Confusing this is one of the most common errors.

Common forces you must recognize and model

Weight (gravitational force)

Near Earth’s surface, weight is: $$W=mg.$$

Direction: straight down toward Earth’s center. Weight does not depend on the surface or motion. A moving elevator can change apparent weight (the normal force), but not $mg$.

Normal force

The normal force is the contact force perpendicular to a surface. It adjusts as needed to prevent interpenetration.

Important: $N$ is not automatically equal to $mg$.

• On a horizontal surface with no other vertical forces and no vertical acceleration, $N=mg$.
• On an incline, $N=mg\cos \theta$ if no other forces have perpendicular components.
• If there is vertical acceleration or extra forces (like a person pushing down), $N$ changes.

A strong habit: always solve for $N$ using $\sum F_{\perp }=m{a}_{\perp }$, rather than memorizing special cases.

Tension

Tension is the pulling force transmitted by a string, rope, or cable.

For ideal AP Physics 1 setups, strings are often treated as massless and inextensible, and pulleys as frictionless. Under those assumptions:

• The tension is the same throughout a single continuous rope segment.
• Connected objects share related accelerations based on the rope constraint.

If a rope connects two masses over a pulley, you typically write Newton’s 2nd law for each mass and add the rope constraint that their accelerations have equal magnitude (opposite directions).

Friction (static and kinetic)

Friction acts parallel to the contact surface and opposes relative motion (or the tendency to slip).

• Static friction: adjusts up to a maximum

$$0\le f_s\le {\mu }_{s}N.$$ Direction: opposite the direction the object would slip if there were no friction.

• Kinetic friction: when sliding occurs

$$f_k={\mu }_{k}N.$$ Direction: opposite the direction of slipping.

A key AP skill is deciding whether the object is slipping. If it is not, you cannot set $f_s={\mu }_{s}N$ automatically. You solve for the required friction and check whether it is less than or equal to ${\mu }_{s}N$.

Equilibrium vs. acceleration problems

Equilibrium: $\sum \vec{F}=0$

Equilibrium means no acceleration, not necessarily no motion. You use: $$\sum F_x=0,\sum F_y=0.$$

Typical examples include a block at rest on an incline or an object hanging motionless from a cable. In these, forces balance and unknowns often include $N$, $T$, or $f_s$.

Acceleration: $\sum \vec{F}=m\vec{a}$

When acceleration is nonzero, the net force must be nonzero. The analysis looks similar, but the right-hand side is not zero.

A common pattern:

• Identify direction of acceleration (or assume one and interpret sign).
• Write Newton’s 2nd law along the axis of motion.
• Use constraints (ropes, contact conditions) to link equations.

Inclines: where free-body diagrams pay off

Inclines are a major reason AP Physics emphasizes component analysis.

If the incline angle is $\theta$ and you choose axes parallel and perpendicular to the surface, the weight breaks into:

• Parallel component: $mg\sin \theta$ (down the slope)
• Perpendicular component: $mg\cos \theta$ (into the surface)

Then:

• $\sum F_{\perp }=0$ often gives $N=mg\cos \theta$ (if no other perpendicular forces)
• $\sum F_{\parallel }=ma$ determines the acceleration, with friction and tension included as needed

This setup avoids messy trigonometry later and keeps directions intuitive.

Practical habits that prevent point-losing mistakes

Write the force balance before plugging numbers

Start with symbolic equations like $\sum F_x=m{a}_x$. It helps you catch missing forces and incorrect directions.

Keep sign conventions consistent

If you choose up the incline as positive, then $mg\sin \theta$ is negative. Do not change conventions mid-problem.

Separate “pair forces” from “balanced forces”

• Balanced forces: multiple forces on one object that sum to zero.
• Third-law pairs: forces on different objects that are equal and opposite.

Check limiting cases

If friction coefficients go to zero, does your answer match a frictionless result? If angle $\theta$ goes to $0^{\circ }$, does the incline behave like a flat surface? These checks catch algebra and setup errors.

Why this unit is the critical skill for the course

AP Physics 1 builds repeatedly on the same foundation: identify forces, draw a free-body diagram, apply Newton’s laws, and solve. Later topics like energy and momentum still rely on correct force reasoning to interpret motion and constraints. Mastering dynamics is less about memorizing formulas and more about developing a disciplined, repeatable method. When you can look at a situation and confidently produce the correct FBD, the rest of the problem becomes manageable.