Dynamics: Newton's Laws and Forces — AP Physics 1 Phys 1 Study Guide

For: AP Physics 1 candidates sitting AP Physics 1.

Covers: Newton's three laws of motion, free-body diagram construction, common contact and field forces, inclined plane and pulley system problems, and translational equilibrium calculations.

You should already know: Algebra 2, basic trig, no calculus required.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Physics 1 style for educational use. They are not reproductions of past College Board papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official College Board mark schemes for grading conventions.

1. What Is Dynamics: Newton's Laws and Forces?

Dynamics is the branch of classical mechanics that relates forces acting on an object to its resulting motion, formalized by Isaac Newton's three core laws of motion. Unlike kinematics, which only describes how objects move, dynamics explains why their motion changes, making it the foundation for 12-18% of the AP Physics 1 multiple-choice section and 30%+ of free-response prompts, including experimental design and quantitative/qualitative translation questions. It is sometimes referred to as Newtonian mechanics for translational motion, and applies to all non-relativistic objects (speeds far less than the speed of light) that you will encounter on the exam.

2. Newton's 1st, 2nd, 3rd Laws

These three laws form the entire framework for solving dynamics problems on the AP Physics 1 exam, so you must memorize their definitions and applications:

1. Newton's 1st Law (Law of Inertia): An object at rest remains at rest, and an object in motion remains in motion at constant velocity, unless acted on by a net external force. Inertia, the tendency of an object to resist changes in its motion, is directly proportional to its mass: heavier objects are harder to accelerate or decelerate.
2. Newton's 2nd Law: The net external force on an object equals the product of its inertial mass and acceleration, written as the vector equation: $${\vec{F}}_{net}=m\vec{a}$$ For 2D problems, this splits into independent x and y component equations: $\sum F_x=m{a}_x$ and $\sum F_y=m{a}_y$. The acceleration of an object always points in the same direction as the net force acting on it.
3. Newton's 3rd Law: For every action force exerted by object A on object B, there is an equal-magnitude, opposite-direction reaction force exerted by object B on object A. Critically, these force pairs act on different objects, so they never cancel each other out when calculating net force on a single system.

Worked Example

A 2kg textbook is pushed with a 5N rightward applied force, and experiences a 2N leftward kinetic friction force. Calculate its acceleration:

1. Calculate net force in the x-direction: $F_{net,x}=5N-2N=3N$ right
2. Rearrange Newton's 2nd law: $a_x=\frac{F_{net,x}}{m}=\frac{3N}{2kg}=1.5m/s^2$ right

3. Free-body Diagrams

A free-body diagram (FBD) is a simplified sketch of a single system (object or group of objects moving together) that shows all external forces acting on it, with no internal forces or motion labels included. Examiners require correct FBDs for most dynamics free-response questions, and missing a force will lead to automatic deductions for all subsequent calculations. Follow these steps to draw a valid FBD:

1. Define your system clearly (e.g., only the 3kg block, not the incline it rests on)
2. Draw a dot to represent the center of mass of the system
3. Identify all external forces: first field forces (gravity, the only field force on AP Physics 1 dynamics questions), then contact forces (normal, tension, friction, applied force)
4. Draw each force as an arrow starting at the dot, pointing in the direction of the force, with length proportional to its magnitude
5. Label each force with standard notation: $F_g$ for gravity, $F_N$ for normal, $F_T$ for tension, $F_f$ for friction, $F_{app}$ for applied force

Worked Example

Draw an FBD for a sled being pulled rightward at constant acceleration by a rope held at 30 degrees above the horizontal:

• Forces to include: $F_g$ pointing straight down, $F_N$ pointing straight up, $F_T$ pointing at 30 degrees above the positive x-axis, $F_f$ pointing along the negative x-axis
• Do not include an acceleration arrow on the FBD itself; note acceleration separately next to the diagram if needed.

4. Tension, Normal, Friction, Gravity

These four forces appear in 90% of AP Physics 1 dynamics problems, so you must know their definitions, direction rules, and calculation formulas:

1. Gravitational Force ($F_g$): A field force that acts on all objects with mass, pointing straight down toward the center of the Earth. Near the Earth's surface, its magnitude is: $$F_g=mg$$ where $m$ is mass in kg, and $g=9.8m/s^2$ (use this value for all calculations unless told otherwise).
2. Normal Force ($F_N$): A contact force exerted by a surface perpendicular to its face, as a reaction to the force the system exerts on the surface. It is not always equal to $mg$: it changes for inclined planes, accelerating elevators, or objects with vertical applied forces.
3. Tension ($F_T$): A contact force exerted by a taut string, rope, cable, or massless rod, pointing along the length of the string. For AP Physics 1 problems, tension has the same magnitude everywhere along a massless, frictionless string, even if it runs over a pulley.
4. Friction ($F_f$): A contact force parallel to a surface, opposing relative motion between the system and the surface. There are two types:

• Static friction: Acts when the system is not sliding relative to the surface, with magnitude $F_s\le {\mu }_s{F}_N$, where ${\mu }_s$ is the unitless coefficient of static friction. The maximum static friction, $F_{s,max}={\mu }_s{F}_N$, is the force required to start the system sliding.
• Kinetic friction: Acts when the system is sliding relative to the surface, with constant magnitude $F_k={\mu }_k{F}_N$, where ${\mu }_k<{\mu }_s$ is the coefficient of kinetic friction.

Worked Example

A 5kg box rests on a horizontal table with ${\mu }_s=0.4$ and ${\mu }_k=0.25$. Calculate the minimum force needed to start the box sliding, and the friction force acting on it once it is moving:

1. Minimum applied force = maximum static friction: $F_{s,max}={\mu }_{s}mg=0.4\ast 5kg\ast 9.8m/s^2=19.6N$
2. Kinetic friction once sliding: $F_k={\mu }_{k}mg=0.25\ast 5kg\ast 9.8m/s^2=12.25N$

5. Inclined Planes and Pulleys

These two common problem setups rely on coordinate system tricks to simplify calculations, and are frequent free-response question topics:

Inclined Planes

Rotate your coordinate system so the x-axis is parallel to the incline and the y-axis is perpendicular to the incline. This splits the gravitational force into two components aligned with the axes: $$F_{g,x}=mg\sin \theta \text{(down the incline)}$$ $$F_{g,y}=mg\cos \theta \text{(perpendicular into the incline)}$$ where $\theta$ is the angle of the incline above the horizontal. The normal force equals $F_{g,y}$ if there are no other forces perpendicular to the incline: $F_N=mg\cos \theta$.

Inclined Plane Worked Example

A 3kg block rests on a frictionless 30-degree incline. Calculate its acceleration down the incline:

1. Net force along the x-axis = $F_{g,x}=mg\sin 30$
2. Acceleration: $a=\frac{F_{net,x}}{m}=g\sin 30=9.8m/s^2\ast 0.5=4.9m/s^2$ down the incline

Pulleys and Atwood Machines

A massless, frictionless pulley only changes the direction of tension, not its magnitude. For an Atwood machine (two masses hanging over a single pulley), the acceleration of the system is: $$a=\frac{(m_1-m_2)g}{m_1+m_2}$$ where $m_1>m_2$, so the heavier mass accelerates downward and the lighter mass accelerates upward. Always define a single positive direction for the entire system to avoid sign errors.

Pulley Worked Example

An Atwood machine has masses of 4kg and 2kg. Calculate the magnitude of the system's acceleration:

1. $a=\frac{(4kg-2kg)\ast 9.8m/s^2}{4kg+2kg}=\frac{19.6N}{6kg}\approx 3.27m/s^2$

6. Translational Equilibrium

An object is in translational equilibrium if the net external force acting on it is zero, so ${\vec{F}}_{net}=0$. This means its acceleration is zero, so it is either at rest (static equilibrium) or moving at constant velocity (dynamic equilibrium). For 2D problems, equilibrium splits into two independent component rules: $$\sum F_x=0\sum F_y=0$$ You will use these rules to solve for unknown forces (e.g., tension in supporting ropes, normal force on a stationary object) on both multiple-choice and free-response questions.

Worked Example

A 10kg shop sign hangs from two identical ropes, each at 45 degrees to the horizontal. Calculate the tension in each rope:

1. By symmetry, tension $T$ is equal in both ropes
2. Sum forces in the y-direction: $\sum F_y=2T\sin 45-mg=0$
3. Rearrange to solve for T: $T=\frac{mg}{2\sin 45}=\frac{10kg\ast 9.8m/s^2}{2\ast (\sqrt{2}/2)}=\frac{98}{\sqrt{2}}\approx 69.3N$
4. Sum forces in the x-direction: $T\cos 45-T\cos 45=0$, which confirms the system is in equilibrium.

7. Common Pitfalls (and how to avoid them)

These are the most frequent mark-losing mistakes on AP Physics 1 dynamics questions:

• Wrong move: Adding Newton's third law force pairs when calculating net force on a single system. Why it happens: Students assume equal and opposite forces cancel, but they act on different objects. Correct move: Only include forces acting on your defined system in net force calculations; ignore third law pairs acting on other objects.
• Wrong move: Assuming normal force always equals $mg$. Why it happens: Students overgeneralize the horizontal flat surface case. Correct move: Always solve for normal force using the y-component net force equation, especially for inclined planes, accelerating elevators, or objects with vertical applied forces.
• Wrong move: Using kinetic friction for stationary objects, or maximum static friction for objects not on the verge of sliding. Why it happens: Students mix up the two friction types. Correct move: Check if the object is sliding relative to the surface: if yes, use $F_k={\mu }_k{F}_N$; if no, static friction only equals the force it is opposing, up to the ${\mu }_s{F}_N$ maximum.
• Wrong move: Drawing acceleration or velocity vectors on free-body diagrams. Why it happens: Students confuse forces with their resulting motion. Correct move: FBDs only show external forces acting on the system; note acceleration separately next to the diagram if required.
• Wrong move: Assigning different tension values to the same massless string over a frictionless pulley. Why it happens: Students forget standard AP Physics 1 assumptions. Correct move: Unless told the string has mass or the pulley has friction, tension is constant everywhere along the string.

8. Practice Questions (AP Physics 1 Style)

Question 1 (Multiple Choice)

A 7kg block is pulled across a horizontal surface by a 30N force applied at 20 degrees above the horizontal. The coefficient of kinetic friction between the block and surface is 0.2. What is the magnitude of the block's acceleration? A) $1.8m/s^2$ B) $2.4m/s^2$ C) $3.1m/s^2$ D) $4.0m/s^2$

Solution

1. Split the applied force into components: $F_{app,x}=30\cos 20\approx 28.19N$, $F_{app,y}=30\sin 20\approx 10.26N$
2. Calculate normal force: $\sum F_y=F_N+F_{app,y}-mg=0$ → $F_N=7\ast 9.8-10.26=58.34N$
3. Calculate kinetic friction: $F_k=0.2\ast 58.34\approx 11.67N$
4. Net x force: $F_{net,x}=28.19-11.67\approx 16.52N$
5. Acceleration: $a=16.52/7\approx 2.36m/s^2$, so the correct answer is B.

Question 2 (Free Response Part A)

A 2kg block rests on a 25-degree incline, where the coefficient of static friction is 0.5. State if the block slides down the incline, and justify your answer with calculations.

Solution

The block does not slide, justifications below:

1. Component of gravity down the incline: $F_{g,x}=2\ast 9.8\ast \sin 25\approx 8.28N$
2. Normal force: $F_N=2\ast 9.8\ast \cos 25\approx 17.76N$
3. Maximum static friction: $F_{s,max}=0.5\ast 17.76\approx 8.88N$
4. Since $F_{g,x}<F_{s,max}$, static friction can fully oppose the downward gravity component, so the block remains stationary.

Question 3 (Free Response Multi-Part)

An Atwood machine has masses $m_1=5kg$ and $m_2=3kg$, connected by a massless string over a frictionless pulley. (a) Find the magnitude of acceleration of the system. (b) Find the tension in the string.

Solution

(a) Acceleration calculation:

1. Define positive direction as downward for $m_1$, upward for $m_2$
2. Net force on the system: $F_{net}=m_{1}g-m_{2}g=(5-3)\ast 9.8=19.6N$
3. Total system mass: $5+3=8kg$
4. Acceleration: $a=\frac{F_{net}}{M}=19.6/8=2.45m/s^2$

(b) Tension calculation:

1. Use Newton's 2nd law for $m_2$: $\sum F=T-m_{2}g=m_{2}a$
2. Rearrange: $T=m_2(g+a)=3\ast (9.8+2.45)=3\ast 12.25=36.75N$
3. Check with $m_1$: $\sum F=m_{1}g-T=m_{1}a$ → $T=5\ast (9.8-2.45)=36.75N$, which confirms the result.

9. Quick Reference Cheatsheet

10. What's Next

Dynamics is the foundational topic for all subsequent motion units in AP Physics 1: you will use Newton's laws to analyze uniform circular motion, work and energy, momentum and collisions, and even rotational motion where torque replaces force in analogous versions of the rules you learned here. 30% of AP Physics 1 free-response questions will require you to apply force analysis from this guide to experimental design, quantitative/qualitative translation, or multi-step problem solving, so mastering these concepts is non-negotiable for earning a 5 on the exam. You can expect to encounter dynamics concepts combined with later topics, for example using force analysis to find the work done on a sliding object, or the tension in a string swinging a mass in a circle.

If you struggle with any of the subtopics covered here, from free-body diagram construction to inclined plane problem solving, you can ask Ollie for custom practice problems, step-by-step walkthroughs, or simplified explanations tailored to your knowledge gaps at any time. You can also access more AP Physics 1 study resources and full-length practice exams on the homepage to test your mastery of dynamics before moving on to the next unit.