Any diagonal vector can be resolved into unique x and y components.
You're in a car. The driver slams the brakes. Your body lurches forward - but WHY? You didn't push yourself. Nobody shoved you. Yet your body flew toward the dashboard as if possessed. The answer lives in three laws written by a young genius during a plague lockdown in 1665. These laws - Newton's Laws of Motion - explain everything from why seatbelts save lives to why the International Space Station doesn't fall out of the sky. Master them, and you hold the key to predicting the motion of everything in the universe.
Learning Outcome: PH.1.03 - Predict an object's motion based on the forces acting on it.
Essential Questions:
Big Idea: Every change in motion - speeding up, slowing down, turning, orbiting - is caused by forces. Newton's three laws give us the complete mathematical framework to predict and control that motion.
Before we can predict motion or state the laws that govern the universe, we must first understand what a force is - its magnitude and direction - and the perspective (reference frame) from which we measure it.
A push or pull acting on an object that can cause it to accelerate, decelerate, or change direction. Force is measured in Newtons (N), where 1 N = 1 kg·m/s². Force is a vector quantity - it has both magnitude AND direction.
Forces come in two fundamental categories:
To solve force problems, you must classify physical quantities into two categories:
| Scalar Quantities (كميات قياسية) | Vector Quantities (كميات متجهة) |
|---|---|
| Have magnitude only | Have magnitude + direction |
| Mass [kg] | Weight [N] ↓ |
| Time [s] | Force [N] → |
| Distance [m] | Displacement [m] → |
| Speed [m/s] | Velocity [m/s] → |
| Energy [J] | Acceleration [m/s²] → |
| Temperature [°C] | Momentum [kg·m/s] → |
COMMON MISCONCEPTION - Mass ≠ Weight
Students frequently use "mass" and "weight" interchangeably, but they are entirely different quantities:
Because forces are vectors, you cannot simply add them like regular numbers if they point in different directions. For example, 5 N North + 5 N East ≠ 10 N. Instead, forces must be resolved into an x–y coordinate system.
Draw standard x (horizontal) and y (vertical) axes. For inclined planes, tilt the axes so x runs parallel to the surface.
Use trigonometry: Fₓ = F cos θ (horizontal) and Fᵧ = F sin θ (vertical). The component adjacent to the angle gets cosine; the component opposite gets sine.
Add all horizontal components: ΣFₓ (Right = positive, Left = negative).
Add all vertical components: ΣFᵧ (Up = positive, Down = negative).
Use the Pythagorean theorem: R = √(ΣFₓ² + ΣFᵧ²)
θ = tan⁻¹(|ΣFᵧ| / |ΣFₓ|) - then determine the quadrant based on the signs of ΣFₓ and ΣFᵧ.
COMMON MISCONCEPTION - Trig Functions
Students often assume the x-component always uses cosine and y always uses sine. This is only true if θ is measured from the horizontal x-axis. If the angle is given from the vertical, the functions flip. Always draw your right triangle first and identify which side is adjacent (cos) and which is opposite (sin) to the given angle.
Scenario: A box is subjected to two forces: F₁ = 45 N pushing horizontally to the right (+x), and F₂ = 60 N acting vertically downward (−y). Find the resultant net force vector and its angle.
Step 1 - Components: Both forces lie on the axes, so no decomposition needed. ΣFₓ = +45 N, ΣFᵧ = −60 N.
Step 2 - Magnitude: R = √(45² + 60²) = √(2025 + 3600) = √5625 = 75 N
Step 3 - Angle: θ = tan⁻¹(60/45) = 53.1° below the positive x-axis (Standard position: 360° − 53.1° = 306.9°)
The perspective from which an observer measures position, velocity, and acceleration. An object can be "moving" in one frame and "at rest" in another - motion is relative.
V_Object - the velocity of the thing you're observing.
V_Frame - the velocity of YOUR observation point.
e.g., East = positive (+), West = negative (−).
V_Relative = V_Object − V_Frame
Scenario: Car A moves East at +20 m/s. Car B moves West at −70 m/s. Find the speed of Car A relative to Car B.
V_Relative = V_A − V_B = (+20) − (−70) = 20 + 70 = 90 m/s
From Car B's perspective, Car A approaches at 90 m/s.
A reference frame that is at rest or moving at constant velocity (zero acceleration). Newton's Laws are only valid in inertial reference frames. A frame accelerating or rotating is non-inertial - fictitious forces appear.
REAL-WORLD ANCHOR - The Accelerating Elevator
A moving bus at constant speed is a valid inertial frame. But the moment it accelerates or brakes, it becomes non-inertial - which is why you lurch!
If you can accurately identify forces, draw a Free-Body Diagram (FBD), and calculate the net force, applying Newton's Laws becomes straightforward. This is your most important tool.
| Force | Symbol | Direction | Formula | When It Appears |
|---|---|---|---|---|
| Weight / Gravity | W or Fg | Always straight down | W = mg | Always (every object has weight) |
| Normal Force | N or Fₙ | Perpendicular to the contact surface, outward | N = mg (flat); N = mg cos θ (incline) | Object rests on or is pressed against a surface |
| Applied Force | Fₐₚₚ | Direction of the push/pull | Given in problem | Someone/something pushes or pulls |
| Friction | f or Ff | Opposite to direction of motion (or intended motion) | f = μN | Rough surfaces in contact |
| Tension | T | Along the rope/string, away from the object | From FBD analysis | Ropes, strings, cables |
| Centripetal | Fᴄ | Toward the center of circular path | Fᴄ = mv²/r | Object moves in a circle |
A pictorial tool used to analyze forces acting on a body. The object is isolated from its environment and replaced by force arrows showing every external force. It is the first step in solving any force problem.
Draw a simple box or dot to represent the object. Remove everything else - no ground, no ropes, no other objects. Just the body.
List every force acting on the body: Weight? Normal? Tension? Applied? Friction? Do NOT include internal forces or forces the object exerts on other things.
Draw an arrow for each force pointing away from the center of the object. Arrow length ≈ relative magnitude. Arrowhead points in the exact direction of the force.
Superimpose x–y axes. On a ramp, tilt the axes so the x-axis runs parallel to the surface and the y-axis is perpendicular to it.
Label each arrow with its symbol (Fg, Fₙ, T, Fₐₚₚ, f) and any given values or angles.
COMMON MISCONCEPTION - Normal Force ≠ mg Always
Students frequently assume Fₙ = mg in every scenario. This is only true on a perfectly flat, horizontal surface with no other vertical forces acting. Here's what actually happens:
Always calculate the normal force from ΣFᵧ = 0 (or ΣFᵧ = ma if accelerating vertically).
The vector sum of all individual forces acting on a body. It is the single equivalent force that produces the same effect as all the individual forces combined. Also called the resultant force.
F_net = √( (ΣFₓ)² + (ΣFᵧ)² )
If all forces are along one axis: F_net = ΣF (just add/subtract with signs).
Scenario: A sled is tied to a tree on a frictionless hill (θ = 30.0°). The sled weighs 77.0 N. It is in static equilibrium. Find the tension (T) and normal force (N).
Step 1 - FBD: Tilt axes so x is parallel to the hill. Three forces act: Tension (up the incline, +x), Normal (perpendicular to hill, +y), Weight (straight down - must be decomposed).
Step 2 - Weight Components:
Step 3 - ΣFₓ = 0: T − W sin 30° = 0 → T = 38.5 N
Step 4 - ΣFᵧ = 0: N − W cos 30° = 0 → N = 66.7 N
We start our journey into Newton's Laws with the simplest state: when the net force equals zero (ΣF = 0). Understanding inertia and equilibrium is essential before we introduce acceleration.
An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction, unless acted upon by an unbalanced (net) force. Objects tend to "keep on doing what they're doing."
Mathematical form: If ΣF = 0, then a = 0 (velocity is constant or zero).
The natural tendency of an object to resist changes in its state of motion. An object resists any attempt to speed it up, slow it down, or change its direction. Inertia is directly proportional to mass - the more massive an object, the harder it is to change its motion.
REAL-WORLD ANCHOR - The Tablecloth Trick
Place a heavy coin on a piece of paper over a cup. Yank the paper quickly - the coin drops straight into the cup instead of flying with the paper. Why? The coin's large mass gives it enormous inertia. The friction force acts for too little time (impulse J = F·Δt ≈ 0) to overcome that inertia. The coin stays stationary while the paper slides out beneath it. This is Newton's First Law in action - and it's also the answer to our essential question about dragging paper from beneath a heavy body!
When all forces perfectly balance (ΣF = 0), the object does not accelerate - this state is called Equilibrium.
COMMON MISCONCEPTION - "Motion Requires a Constant Force"
Many students assume that if an object is moving to the right, there must be a force pushing it to the right. WRONG. Newton's First Law says a force is NOT needed to keep a moving object moving. A book on a frictionless surface slides forever. It only slows in the real world because of unbalanced friction - not because "motion needs force."
When an object is in equilibrium, you automatically know ΣFₓ = 0 and ΣFᵧ = 0. This gives you two powerful equations to solve for unknown forces.
Verify the object is at rest or moving at constant velocity (a = 0 m/s²). Keywords: "at rest," "constant speed," "uniform velocity," "equilibrium."
Represent the object as a point or box. Draw ALL force vectors with correct directions and labels.
Break any diagonal forces into x and y components using trigonometry (F cos θ, F sin θ).
Write: ΣFₓ = 0 and ΣFᵧ = 0
Use algebra to isolate the missing variable - tension, normal force, friction, applied force, etc.
CAPSTONE CONNECTION - Designing Wind-Resistant Structures
To ensure a tall solar tower (15,000 kg) doesn't blow over in a 45,000 N wind, engineers apply ΣFₓ = 0: F_wind − F_friction = 0 → F_friction = 45,000 N. The foundation must provide exactly 45,000 N of shear resistance to maintain static equilibrium. A low center of gravity and wide base of support ensure the structure won't topple. This is Newton's 1st Law applied to real engineering!
In Part 3 we established the rules for equilibrium (ΣF = 0). Now we break the equilibrium - what happens when the net force is NOT zero?
The acceleration of an object is directly proportional to the net force acting on it, in the same direction as the net force, and inversely proportional to the object's mass.
ΣF = m × a → a = F_net / m
ΣF = ma
Where:
COMMON MISCONCEPTION - "Use Any Force"
Students often plug a single applied force directly into F = ma. Newton's 2nd Law requires the NET force (F_net = ΣF) - the vector sum of ALL forces. You must subtract opposing forces (like friction) before calculating acceleration!
| Law | Name | Statement | Equation | Condition |
|---|---|---|---|---|
| 1st | Law of Inertia | An object at rest stays at rest; in motion stays in motion, unless acted on by an unbalanced force | ΣF = 0 → a = 0 | Balanced forces |
| 2nd | Law of Acceleration | Acceleration is proportional to net force and inversely proportional to mass | ΣF = ma | Unbalanced forces |
| 3rd | Law of Action-Reaction | For every action force, there is an equal and opposite reaction force | F₁₂ = −F₂₁ | All interactions |
From a = F_net / m (with mass constant):
Graph: A straight line through the origin when plotting a vs F (at constant mass).
W = mg
Where g = 9.8 m/s² near Earth's surface. Weight is a force [N], not a mass [kg].
Fg = G · m₁ · m₂ / r²
Where G = 6.67 × 10⁻¹¹ N·m²/kg². Every object with mass attracts every other object with mass.
This method works for ANY force problem - equilibrium or accelerating, flat or inclined, simple or complex. Master these 5 steps and you can solve any question the exam throws at you.
Represent the object as a dot or box. Draw every external force as an arrow. Label each with its symbol and given values.
Resolve any diagonal forces: Fₓ = F cos θ (parallel to surface) and Fᵧ = F sin θ (perpendicular to surface).
Create a clean version of your FBD showing only the pure x and y vectors. This prevents calculation errors.
Add/subtract all x-forces (Right = +, Left = −) and all y-forces (Up = +, Down = −). Get ΣFₓ and ΣFᵧ.
For acceleration: ΣF = ma → a = ΣF/m. For equilibrium: ΣF = 0 → solve for the unknown force.
Scenario: A 50 N applied force drags an 8.16 kg log to the right. A 40.0 N friction force resists the motion. What is the acceleration?
Step 1 - Forces: Applied (+50 N right), Friction (−40 N left), Weight (down), Normal (up).
Step 2 - Components: All forces lie on the axes - no trig needed.
Step 3 - ΣFₓ: 50 − 40 = +10 N (net rightward)
Step 4 - Apply F = ma: 10 = 8.16 × a → a = 10/8.16 = 1.23 m/s² to the right.
Scenario: An airboat (mass 350 kg) has an engine producing a net horizontal force of 770 N. (a) Find the acceleration. (b) Starting from rest, how long to reach 12.0 m/s?
(a) a = F_net / m = 770 / 350 = 2.20 m/s²
(b) Using kinematics: v = v₀ + at → 12.0 = 0 + (2.20)t → t = 5.45 s
The real world does not occur in isolated textbook problems. Here we apply Newton's Laws to multi-body systems and action-reaction pairs.
For every action force, there is an equal and opposite reaction force. These forces act on DIFFERENT objects.
F₁₂ = −F₂₁ (equal magnitudes, opposite directions, different objects)
Key Insight: It is critical to distinguish between balanced forces and action-reaction pairs:
COMMON MISCONCEPTION - Normal Force is NOT the Reaction to Gravity
Students often think that the normal force on an object is the "reaction" to gravity because they are equal and opposite. Impossible - they both act on the same object! The true reaction to Earth's gravity pulling the cantaloupe down is the cantaloupe's gravity pulling Earth up. They act on different objects (cantaloupe → Earth, Earth → cantaloupe).
When multiple objects are connected (by ropes, pulleys, or contact), they accelerate together as a single system.
| System | Setup | Acceleration Formula |
|---|---|---|
| Vertical Atwood Machine | Two masses over a frictionless pulley; m₂ > m₁ | a = (m₂ − m₁)g / (m₁ + m₂) |
| Horizontal Table System | m₁ on smooth table, m₂ hanging off the edge | a = m₂g / (m₁ + m₂) |
| Inclined Plane + Hanging Mass | m₁ on incline (angle θ), m₂ hanging; m₂ > m₁ sin θ | a = (m₂ − m₁ sin θ)g / (m₁ + m₂) |
| Blocks in Contact | Two blocks pushed by external force F on a smooth surface | a = F / (m₁ + m₂) |
Additional Useful Formulas:
T = 2m₁m₂g / (m₁ + m₂)
F_contact = m₂F / (m₁ + m₂)
Where F is the applied force on block 1, and F_contact is the force block 1 exerts on block 2.
The theoretical point where all the mass of an object is considered concentrated. An object remains in stable equilibrium as long as the center of gravity falls vertically within its base of support.
COMMON MISCONCEPTION - "Heavier = More Stable"
Adding mass to the TOP of a structure raises its center of gravity, making it LESS stable. True stability comes from a LOW center of gravity and a WIDE base of support, not just total mass.
REAL-WORLD ANCHOR - The Leaning Tower of Pisa
Despite its severe tilt, the tower remains standing because its center of gravity still falls vertically within its wide foundation base. Engineers also added lead counterweights to the base and removed soil from one side to correct the lean - all applications of center-of-mass physics.
Up to now, we treated acceleration primarily as a change in speed. But because velocity is a vector, a change in direction ALSO constitutes acceleration. Objects can move at constant speed while continuously accelerating.
Motion in a circle at constant speed. Although speed is constant, the velocity continuously changes direction - therefore the object is always accelerating toward the center.
Three key vectors in circular motion:
| Vector | Direction | Symbol |
|---|---|---|
| Instantaneous Velocity (v) | Always TANGENT to the circular path (pointing "outward") | v |
| Centripetal Acceleration (aᴄ) | Always TOWARD the center (perpendicular to v) | aᴄ |
| Centripetal Force (Fᴄ) | Always TOWARD the center (same direction as aᴄ) | Fᴄ |
WHY A CIRCLE REQUIRES FORCE
Newton's 1st Law says an object moves in a straight line unless a net force acts on it. If you release a rock on a string, it flies off in a straight tangent - not a circle. Therefore, to maintain circular motion, a continuous inward force must constantly pull the object away from that straight-line path. That inward force is the centripetal force.
v = 2πr / T
Where r = radius [m] and T = period (time for one complete revolution) [s].
aᴄ = v² / r = 4π²r / T²
Always directed toward the center of the circular path.
Fᴄ = m·aᴄ = m·v² / r
This is NOT a new type of force - it is the net inward force (could be tension, gravity, friction, normal force, or a combination) that causes circular motion.
COMMON MISCONCEPTION - "Centrifugal Force"
When riding in a turning car, you feel "pushed" outward. There is NO force pushing you out. Your body's inertia (Newton's 1st Law) wants to continue in a straight line while the car turns away from you. NEVER draw a "centrifugal force" on a free-body diagram. Centripetal force is strictly an inward net force.
Scenario: A 0.150 kg rock is whirled in a horizontal circle (radius = 0.600 m) with period T = 0.500 s. What is the tension in the string?
Step 1 - Find Speed: v = 2πr/T = 2π(0.600)/0.500 = 7.54 m/s
Step 2 - Identify Radial Force: Only the string's tension acts horizontally toward the center.
Step 3 - Apply ΣFᵣ = mv²/r: F_T = (0.150)(7.54²)/(0.600) = (0.150)(56.85)/(0.600) = 14.2 N
Conclusion: The string exerts 14.2 N of inward centripetal force to maintain the circular orbit.
GRAND CHALLENGE CONNECTION
Exponential Population Growth: As cities grow taller and denser, every high-rise must be engineered to remain in static equilibrium under wind loads, seismic forces, and occupant weight. Newton's Laws - especially ΣF = 0 for stability and ΣF = ma for dynamic loading - are the core tools.
Efficient Use of Arid Land: Wind turbine blades spinning at high speed are governed by centripetal force (Fᴄ = mv²/r). Solar arrays in deserts must withstand forces exceeding 60,000 N in sandstorms - equilibrium analysis determines the minimum anchor force.
| Formula | Equation | Units |
|---|---|---|
| Newton's 2nd Law | ΣF = ma → a = F_net / m | N, kg, m/s² |
| Weight | W = mg | N |
| Vector Components | Fₓ = F cos θ, Fᵧ = F sin θ | N |
| Resultant Magnitude | R = √(Rₓ² + Rᵧ²) | N |
| Resultant Angle | θ = tan⁻¹(|Rᵧ| / |Rₓ|) | degrees |
| Relative Velocity | V_BA = V_B − V_A | m/s |
| Centripetal Acceleration | aᴄ = v² / r = 4π²r / T² | m/s² |
| Centripetal Force | Fᴄ = mv² / r | N |
| Linear Speed (circular) | v = 2πr / T | m/s |
| Universal Gravitation | Fg = Gm₁m₂ / r² | N |
| Friction | f = μN | N |
| Normal Force (incline) | N = mg cos θ | N |
| Atwood Acceleration | a = (m₂−m₁)g / (m₁+m₂) | m/s² |
| Atwood Tension | T = 2m₁m₂g / (m₁+m₂) | N |
| Contact Force | F_contact = m₂F / (m₁+m₂) | N |
| Kinematics (from rest) | d = ½at², v = at | m, m/s |
102 practice questions sourced from past STEM exams and reference materials. For questions that include diagrams, the image is shown within the question card.
After studying this material, you've learned to predict the motion of anything in the universe using just three laws and a handful of equations. Newton published the Principia at age 44 - you're learning this at 15. That's a head start. Use it wisely.
"If I have seen further, it is by standing on the shoulders of giants." - Isaac Newton