⚖️ Newton's First Law and Inertia

Part 1 of 7 — Newton's First and Second Laws

For centuries, people believed that objects naturally come to rest — that you need a force to keep things moving. Galileo challenged this, and Newton formalized it into his First Law.

Newton's First Law tells us what happens when forces are balanced (or absent). It's more profound than it seems — it defines the very framework in which physics works.

Newton's First Law (The Law of Inertia)

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.

Breaking It Down

Key Insight

No force is needed to maintain motion — only to change it. A hockey puck sliding on frictionless ice would slide forever at constant velocity.

Common Misconception

❌ "An object in motion will eventually stop."

✅ Objects stop because of friction, air resistance, or other forces — not because motion naturally "wears out."

Inertia

Inertia is an object's tendency to resist changes in its state of motion.

Mass as a Measure of Inertia

• Mass ($m$) quantifies inertia
• Greater mass → greater inertia → harder to accelerate
• Mass is a scalar quantity measured in kilograms (kg)
• Mass is not the same as weight (weight depends on gravity)

Everyday Examples of Inertia

Mass vs. Weight

Inertial Reference Frames

Newton's First Law doesn't work in every reference frame. It works in inertial reference frames.

What Is a Reference Frame?

A reference frame is a coordinate system attached to an observer. Different observers can describe the same event differently.

Inertial vs. Non-Inertial

Why It Matters

In a non-inertial frame, objects appear to accelerate without any real force. For example:

• In a turning car, you feel "pushed" outward — but there's no outward force
• This "fictitious force" is called the centrifugal force

AP Physics 1 focuses on inertial reference frames, where Newton's laws apply directly.

Newton's First Law Concept Check 🎯

Inertia Calculations 🧮

1. A 1500 kg car and a 75 kg person both experience the same net force. The ratio of the car's acceleration to the person's acceleration is $a_{\text{car}}/a_{\text{person}}$ = ? (express as a decimal)

2. On the Moon, $g_{\text{Moon}} = 1.6$ m/s². What is the weight (in N) of a 60 kg astronaut on the Moon?

3. An object weighs 490 N on Earth ($g = 9.8$ m/s²). What is its mass in kg?

Round all answers to 3 significant figures.

Classify and Identify 🔍

Exit Quiz — Newton's First Law ✅

🚀 Newton's Second Law

Part 2 of 7 — Newton's First and Second Laws

Newton's First Law tells us what happens when there's no net force. Newton's Second Law tells us what happens when there is a net force — it's the quantitative heart of mechanics.

$\vec{F}_{\text{net}} = m\vec{a}$

This single equation lets us predict the motion of everything from baseballs to planets.

Newton's Second Law

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

$\vec{F}_{\text{net}} = m\vec{a}$

Or equivalently:

$\vec{a} = \frac{\vec{F}_{\text{net}}}{m}$

What This Tells Us

Units

$1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$

A newton is the force needed to accelerate a 1 kg mass at 1 m/s².

Important Clarifications

• $F_{\text{net}}$ is the vector sum of ALL forces, not just one force
• If $F_{\text{net}} = 0$, then $a = 0$ (recovers Newton's First Law!)
• The law applies instantaneously — the acceleration at any moment equals the net force at that moment divided by mass

Finding the Net Force

The net force is the vector sum of all individual forces acting on an object:

$\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \cdots = \sum \vec{F}$

One Dimension

In 1D, assign positive/negative directions and add algebraically:

$F_{\text{net}} = F_1 + F_2 + \cdots$

Example: Tug of War

Two people pull a box along the x-axis:

• Person A pulls right: $F_A = +40$ N
• Person B pulls left: $F_B = -25$ N

$F_{\text{net}} = 40 + (-25) = +15 \text{ N (to the right)}$

If the box has mass $m = 5$ kg:

$a = \frac{F_{\text{net}}}{m} = \frac{15}{5} = 3 \text{ m/s}^2 \text{ (to the right)}$

Two Dimensions

In 2D, break forces into components and sum each direction:

$F_{\text{net},x} = \sum F_x, \quad F_{\text{net},y} = \sum F_y$

$a_x = \frac{F_{\text{net},x}}{m}, \quad a_y = \frac{F_{\text{net},y}}{m}$

Proportional Reasoning with $F = ma$

Many AP problems test your ability to reason about how changes in force or mass affect acceleration — without plugging in numbers.

Doubling / Halving Problems

Example

A force $F$ gives a mass $m$ an acceleration of $4$ m/s².

What acceleration does a force $3F$ give to a mass $2m$?

$a' = \frac{3F}{2m} = \frac{3}{2} \cdot \frac{F}{m} = \frac{3}{2} \cdot 4 = 6 \text{ m/s}^2$

Newton's Second Law Concept Check 🎯

Newton's Second Law Calculations 🧮

1. What net force (in N) is needed to accelerate a 1200 kg car at 2.5 m/s²?

2. A 0.50 kg ball experiences a net force of 4.0 N. What is its acceleration (in m/s²)?

3. An object accelerates at 6 m/s² under a net force of 18 N. What is its mass (in kg)?

Proportional Reasoning Practice 🔍

Exit Quiz — Newton's Second Law ✅

📐 Free Body Diagrams

Part 3 of 7 — Newton's First and Second Laws

A Free Body Diagram (FBD) is the single most important tool in mechanics. It's a picture that shows all the forces acting on a single object, represented as arrows pointing away from the object.

Every force problem on the AP exam starts with drawing a correct FBD. Master this skill and you'll master dynamics.

What Is a Free Body Diagram?

A free body diagram:

1. Isolates a single object (the "free body")
2. Represents the object as a dot or simple shape
3. Draws all external forces as arrows starting at the object
4. Labels each force with its name and/or magnitude
5. Shows a coordinate system (x-y axes)

Steps to Draw an FBD

1. Identify the object you're analyzing
2. List all forces acting ON that object (not forces the object exerts on others)
3. Draw each force as an arrow in the correct direction
4. Label each arrow
5. Choose axes — usually align one axis with the direction of acceleration

Common Mistake

❌ Including forces that the object exerts on other objects

✅ Only include forces that act on the object you're analyzing

Common Forces in AP Physics 1

Weight

$W = mg$

where $g = 9.8$ m/s² (or $\approx 10$ m/s² for quick estimates).

Weight always points straight down, regardless of the surface orientation.

FBD Examples

Example 1: Book on a Table

Forces on the book:

• $\vec{W}$ (weight) pointing down: $W = mg$
• $\vec{N}$ (normal force) pointing up

Since the book is in equilibrium: $N = W = mg$

Example 2: Block Pulled by a Rope on a Rough Surface

Forces on the block:

• $\vec{W}$ pointing down
• $\vec{N}$ pointing up
• $\vec{T}$ pointing in the direction of the rope
• $\vec{f}$ (friction) pointing opposite to motion/tendency of motion

Example 3: Object in Free Fall

Forces on the object:

• $\vec{W}$ pointing down

That's it! No normal force, no tension — just gravity.

Tip: If the object isn't touching a surface, there's no normal force. If it's not connected to a rope, there's no tension.

Identify Forces on FBDs 🎯

FBD Force Analysis 🧮

Consider a 5 kg block resting on a horizontal surface.

1. What is the magnitude of the weight force (in N)? Use $g = 9.8$ m/s².

2. The block is in equilibrium on the surface. What is the magnitude of the normal force (in N)?

3. A person pushes the block to the right with 20 N on a frictionless surface. What is the acceleration (in m/s²)?

Force Direction Practice 🔍

Exit Quiz — Free Body Diagrams ✅

🧮 Applying F = ma — Single Object Problems

Part 4 of 7 — Newton's First and Second Laws

Now we combine free body diagrams with Newton's Second Law to solve real problems. The strategy is always the same:

1. Draw the FBD
2. Choose coordinate axes
3. Write $\sum F_x = ma_x$ and $\sum F_y = ma_y$
4. Solve for unknowns

The Problem-Solving Strategy

Step-by-Step Method

1. Draw a picture and identify the object
2. Draw the FBD — all forces labeled
3. Choose axes — one axis along the direction of acceleration
4. Decompose forces into x and y components
5. Apply Newton's Second Law in each direction:
   • $\sum F_x = ma_x$
   • $\sum F_y = ma_y$
6. Solve the resulting equations

Key Insight

If the object doesn't accelerate in a particular direction, then $\sum F = 0$ in that direction. This is the equilibrium condition for that axis.

Example: Horizontal Push

A person pushes a 20 kg box across a frictionless floor with a horizontal force of 60 N.

FBD forces: Weight down ($W = 196$ N), normal up ($N$), push right ($F = 60$ N)

x-direction: $\sum F_x = F = ma_x$ $60 = 20 \cdot a_x \Rightarrow a_x = 3 \text{ m/s}^2$

y-direction: $\sum F_y = N - W = 0$ $N = W = 196 \text{ N}$

Vertical Acceleration Problems

Elevator Problems

An elevator is a classic AP scenario. A person of mass $m$ stands on a scale in an elevator.

The scale reads the normal force $N$ — what we call the apparent weight.

Applying Newton's Second Law (taking up as positive):

$N - mg = ma$

$N = m(g + a)$

Example

A 70 kg person stands on a scale in an elevator accelerating upward at 2 m/s².

$N = m(g + a) = 70(9.8 + 2) = 70 \times 11.8 = 826 \text{ N}$

True weight: $W = 70 \times 9.8 = 686$ N. The scale reads 826 N — the person feels heavier!

Applied Force at an Angle

When a force is applied at an angle $\theta$ to the horizontal, you must decompose it:

$F_x = F\cos\theta, \quad F_y = F\sin\theta$

Example: Pulling a Sled

A person pulls a 15 kg sled with a 40 N force at 30° above the horizontal on a frictionless surface.

x-direction: $\sum F_x = F\cos\theta = ma_x$ $40\cos 30° = 15 \cdot a_x$ $34.6 = 15 \cdot a_x$ $a_x = 2.31 \text{ m/s}^2$

y-direction: $\sum F_y = N + F\sin\theta - mg = 0$ $N = mg - F\sin\theta = 15(9.8) - 40\sin 30° = 147 - 20 = 127 \text{ N}$

Notice: Pulling upward at an angle reduces the normal force! This will matter when we study friction.

Application Concept Check 🎯

F = ma Problem Solving 🧮

1. A 60 kg person stands on a scale in an elevator accelerating downward at 3 m/s². What does the scale read (in N)? Use $g = 9.8$ m/s².

2. A horizontal force of 50 N pushes a 10 kg box across a frictionless floor. What is the acceleration (in m/s²)?

3. A person pulls a box with 100 N at 37° above horizontal (frictionless surface). The box has mass 25 kg. What is the horizontal acceleration (in m/s²)? Use $\cos 37° = 0.80$.

Round all answers to 3 significant figures.

Scenario Analysis 🔍

Exit Quiz — Applying F = ma ✅

⚖️ Weight and Normal Force

Part 5 of 7 — Newton's First and Second Laws

Weight and normal force are the two most common forces in mechanics. Understanding their relationship — when they're equal, when they're not — is essential for solving nearly every dynamics problem.

Weight: The Gravitational Force

Weight is the gravitational force exerted by the Earth on an object:

$W = mg$

Weight on Different Planets

Since $g$ varies by location:

Mass stays the same everywhere — weight changes with $g$.

Normal Force

The normal force ($\vec{N}$ or $\vec{F}_N$) is the contact force a surface exerts on an object, perpendicular to the surface.

Key Properties

• Direction: Perpendicular to the contact surface, away from the surface
• It's a contact force — only exists when objects touch
• It's a response force — adjusts its magnitude to prevent objects from passing through each other
• It does NOT always equal $mg$!

When Does $N = mg$?

Only on a horizontal surface with no other vertical forces and no vertical acceleration:

$\sum F_y = N - mg = 0 \quad \Rightarrow \quad N = mg$

When Does $N \neq mg$?

Apparent Weight

What a scale reads is the normal force, not the true weight. We call this the apparent weight.

$W_{\text{apparent}} = N$

Why It Changes

In an accelerating elevator (taking up as positive):

$N - mg = ma$ $N = m(g + a)$

• Accelerating up ($a > 0$): $N > mg$ → feel heavier
• Accelerating down ($a < 0$): $N < mg$ → feel lighter
• Free fall ($a = -g$): $N = 0$ → weightlessness!

Weightlessness

Astronauts in orbit are NOT outside Earth's gravity — they're in free fall around the Earth. Since the ISS and everything inside it falls together, the normal force between the astronaut and the floor is zero.

$N = m(g - g) = 0$

This is why they float — not because there's no gravity, but because there's no normal force.

Weight and Normal Force Concepts 🎯

Weight and Normal Force Calculations 🧮

1. What is the weight of a 25 kg object on Earth (in N)? Use $g = 9.8$ m/s².

2. A 50 kg person stands on a scale in an elevator accelerating upward at 2 m/s². What does the scale read (in N)?

3. A 40 kg child stands on a scale in an elevator in free fall. What does the scale read (in N)?

Normal Force Scenarios 🔍

Exit Quiz — Weight and Normal Force ✅

🛠️ Problem-Solving Workshop

Part 6 of 7 — Newton's First and Second Laws

This workshop pulls together everything from Parts 1–5. We'll work through increasingly challenging problems using the systematic FBD → Newton's Second Law → Solve approach.

Problem-Solving Framework Review

The 5-Step Process

1. Read & Sketch — Draw the physical situation
2. FBD — Isolate the object; draw ALL forces
3. Axes — Choose a coordinate system (align with acceleration)
4. Newton's Second Law — Write $\sum F_x = ma_x$ and $\sum F_y = ma_y$
5. Solve — Algebra to find unknowns

Common Pitfalls

Worked Example 1: Two Horizontal Forces

A 12 kg box on a frictionless surface has two forces applied:

• $F_1 = 50$ N to the right
• $F_2 = 20$ N to the left

Step 1: FBD — Weight down, normal up, $F_1$ right, $F_2$ left

Step 2: x-direction $\sum F_x = F_1 - F_2 = ma_x$ $50 - 20 = 12a_x$ $a_x = 30/12 = 2.5 \text{ m/s}^2 \text{ (to the right)}$

Step 3: y-direction $\sum F_y = N - mg = 0 \Rightarrow N = 12(9.8) = 117.6 \text{ N}$

Worked Example 2: Vertical Tension

Two blocks hang vertically from strings. Block A (3 kg) hangs from the ceiling. Block B (2 kg) hangs from block A.

FBD of Block B: $T_2 - m_Bg = 0 \Rightarrow T_2 = 2(9.8) = 19.6 \text{ N}$

FBD of Block A: $T_1 - T_2 - m_Ag = 0$ $T_1 = T_2 + m_Ag = 19.6 + 3(9.8) = 19.6 + 29.4 = 49 \text{ N}$

Key insight: The upper string supports BOTH blocks, so $T_1 = (m_A + m_B)g = 5(9.8) = 49$ N.

Multi-Step Problem Practice 🎯

Workshop Calculations 🧮

1. A 8 kg block on a frictionless floor is pushed with 56 N horizontally. What is the acceleration (in m/s²)?

2. The same block from #1 starts from rest. What is its speed (in m/s) after 4 seconds?

3. A 5 kg mass hangs from a string. The string is pulled upward so the mass accelerates upward at 2 m/s². What is the tension in the string (in N)? Use $g = 9.8$ m/s².

Quick Reasoning Checks 🔍

Exit Quiz — Problem-Solving Workshop ✅

🎓 Synthesis & AP Review

Part 7 of 7 — Newton's First and Second Laws

This final part brings together every concept from the topic. You'll face AP-style questions that require combining multiple ideas: inertia, $F_{\text{net}} = ma$, FBDs, weight, normal force, and multi-step problem solving.

Concept Summary

Newton's First Law (Inertia)

• No net force → no change in velocity
• Inertia is quantified by mass
• Valid in inertial reference frames

Newton's Second Law

$\vec{F}_{\text{net}} = m\vec{a}$

• Net force = vector sum of all forces
• Acceleration is in the direction of the net force
• $a \propto F$, $a \propto 1/m$

Free Body Diagrams

• Show ALL forces ON a single object
• Common forces: $W$, $N$, $T$, $f$, $F_{\text{app}}$
• Normal force ⊥ surface, weight always down

Weight and Normal Force

• $W = mg$ (always downward)
• $N$ depends on situation — NOT always $mg$
• Apparent weight = normal force (what a scale reads)

Key Equations

AP-Style Multiple Choice — Set 1 🎯

AP-Style Free Response 🧮

A 3 kg block sits on a frictionless table. A string runs horizontally from the block, over a frictionless pulley at the table's edge, and down to a hanging 2 kg block.

1. What is the acceleration of the system (in m/s²)? Round to 3 significant figures. Use $g = 9.8$ m/s².

2. What is the tension in the string (in N)? Round to 3 significant figures.

3. How far does the 2 kg block fall from rest in 2 seconds (in m)? Round to 3 significant figures.

Synthesis Quick Check 🔍

AP-Style Multiple Choice — Set 2 🎯

Final Exit Quiz — Newton's 1st & 2nd Laws ✅