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Newton's Laws with Calculus

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Newton's Laws with Calculus - Complete Interactive Lesson

Part 1: Newton's First Law

⚛️ Newton's First Law

Part 1 of 7 — Newton's First Law

An object remains at rest or in uniform motion unless acted upon by a net external force.

Key Ideas

Inertia: the tendency of objects to resist changes in motion
Mass is a measure of inertia
An object in equilibrium has $\sum \vec{F} = 0$
This law defines inertial reference frames

Worked Example

A 5 kg block sits on a frictionless table with two horizontal forces: 20 N to the right and 20 N to the left. Describe its motion.

Step 1: $\sum F = 20 - 20 = 0$ N

Step 2: Net force is zero, so the block is in equilibrium.

Answer: The block remains at rest (or continues at constant velocity if already moving). ✅

Concept Check 🎯

Newton's First Law Practice 🧮

A book sits on a table with no net force. What is its acceleration (in m/s²)?
A ball rolls at 10 m/s on a frictionless surface. What is its speed (m/s) after 5 seconds?
Two forces act on an object: 15 N right and 10 N left. What is the net force magnitude (in N)?

Concept Check 🔍

Practice

#	Problem	Hint
1	If net force is zero, what is the acceleration?	Newton's first law
2	What property measures resistance to acceleration?	Starts with "i"
3	Can an object move at 10 m/s with zero net force?	Think about constant velocity

Challenge Question 📋

Part 2: Newton's Second Law with Calculus

⚛️ Newton's Second Law with Calculus

Part 2 of 7 — Newton's Second Law with Calculus

$\vec{F}_{\text{net}} = m\vec{a} = m\frac{d\vec{v}}{dt} = \frac{d\vec{p}}{dt}$

Part 3: Newton's Third Law

⚛️ Newton's Third Law

Part 3 of 7 — Newton's Third Law

For every action force, there is an equal and opposite reaction force.

$\vec{F}_{A \text{ on } B} = -\vec{F}_{B \text{ on } A}$

Part 4: Friction Forces

⚛️ Friction Forces

Part 4 of 7 — Friction Forces

Static friction: $f_s \leq mu_s N$ (prevents motion)
Kinetic friction: $f_{k} = m_{}$ (opposes sliding motion)

Part 5: Circular Motion

⚛️ Circular Motion

Part 5 of 7 — Circular Motion

For uniform circular motion:

$a_c = \frac{v^2}{r} = \omega^2 r$

Part 6: Problem-Solving Workshop

⚛️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Strategy for Newton's Law Problems

Draw a free-body diagram (FBD)
Choose coordinate axes (often along incline)
Write $\sum F_x = ma_x$ and

Part 7: Review & Applications

⚛️ Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

$\sum F = ma = m\frac{dv}{dt}$

When force varies with time or position, use calculus:

$v(t) = v_0 + int_0^t \frac{F(t')}{m},dt'$
$x(t) = x_0 + int_0^t v(t'),dt'$

Worked Example

A 2 kg object has $F(t) = 6t$ N. Find velocity at $t = 3$ s if $v(0) = 0$ .

$a(t) = \frac{F}{m} = \frac{6t}{2} = 3t$

$v(3) = int_0^3 3t,dt = \frac{3t^2}{2}\Big|_0^3 = \frac{27}{2} = 13.5$ m/s ✅

Concept Check 🎯

Newton's Second Law with Calculus 🧮

A 4 kg object has a net force of 20 N. What is its acceleration (m/s²)?
A 2 kg mass has $F(t) = 6t$ N. What is the velocity (m/s) at $t = 3$ s if starting from rest? (Hint: integrate $a(t) = 3t$ from 0 to 3). Give answer as a decimal.
A constant force of 3 N acts on a 1 kg mass starting from rest. What is the velocity (m/s) at $t = 3$ s?

Concept Check 🔍

Practice

#	Concept	Formula
1	$F = ma$	Constant force
2	$v = \int a,dt$	Variable force
3	$x = \int v,dt$	Position from velocity

Challenge Question 📋

Action-reaction pairs act on different objects
They are equal in magnitude, opposite in direction
They are the same type of force

Worked Example

A person pushes a wall with 50 N. What force does the wall exert on the person?

By Newton's Third Law: The wall exerts 50 N on the person in the opposite direction. ✅

These are an action-reaction pair acting on different objects (person and wall).

Concept Check 🎯

Newton's Third Law 🧮

You push a desk with 50 N. How many newtons does the desk push back on you?
A swimmer pushes water backward with 100 N. What force (N) does the water exert on the swimmer?
Object A exerts 20 N on Object B. What force (N) does Object B exert on Object A?

Concept Check 🔍

Practice

#	Action Force	Reaction Force
1	Earth pulls apple down	Apple pulls Earth up
2	Bat hits ball	Ball hits bat
3	Foot pushes ground back	Ground pushes foot forward

Challenge Question 📋

Where $N$ is the normal force and $\mu$ is the coefficient of friction.

For an object on an incline at angle $\theta$ :

$N = mg\cos\theta$
Friction along the incline: $f = \mu mg\cos\theta$

Worked Example

A 10 kg block on a horizontal surface has $mu_k = 0.3$ . Find the kinetic friction force.

$N = mg = 10 \times 10 = 100$ N

$f_k = mu_k N = 0.3 \times 100 = 30$ N ✅

Concept Check 🎯

Friction Forces 🧮

A 10 kg block on a flat surface has $mu_k = 0.3$ . Kinetic friction force (N)? (Use $g = 10$ m/s²)
A 5 kg block requires 20 N to start sliding on a flat surface. What is $mu_s$ ? (Use $g = 10$ m/s²)
A 10 kg block is on a surface with $mu_k = 0.5$ . What horizontal force (N) is needed to move it at constant velocity? (Use $g = 10$ m/s²)

Concept Check 🔍

Practice

#	Scenario	Key Formula
1	Flat surface	$f = \mu mg$
2	Inclined plane	$f = \mu mg\cos\theta$
3	Threshold of sliding	$f_s = mu_s N$

Challenge Question 📋

$F_c = \frac{mv^2}{r}$

where $\omega = 2\pi f = \frac{2\pi}{T}$ is the angular velocity.

Centripetal force is not a new force — it is the net inward force (tension, gravity, friction, normal force, etc.).

Worked Example

A 2 kg ball on a 0.5 m string moves at 4 m/s in a horizontal circle. Find the centripetal force.

$F_c = \frac{mv^2}{r} = \frac{2(16)}{0.5} = 64$ N ✅

Concept Check 🎯

Circular Motion 🧮

A car goes around a curve of radius 50 m at 20 m/s. What is the centripetal acceleration (m/s²)?
An object moves in a circle of radius 2 m with centripetal acceleration 8 m/s². What is the speed (m/s)?
A 2 kg ball on a 0.5 m string moves at 4 m/s. What is the tension/centripetal force (N)?

Concept Check 🔍

Practice

#	Concept	Formula
1	Centripetal acceleration	$a_c = v^2/r$
2	Centripetal force	$F_c = mv^2/r$
3	Period and frequency	$T = 2\pi r / v$

Challenge Question 📋

\sum F_{y} = m a_{y}

Solve the system of equations

Check units and reasonableness

Worked Example

A 5 kg block is pushed with 40 N along a surface with $mu_k = 0.2$ . Find the acceleration. ( $g = 10$ m/s²)

$N = mg = 50$ N, $f_k = 0.2 \times 50 = 10$ N

$\sum F = 40 - 10 = 30$ N

$a = 30/5 = 6$ m/s² ✅

Concept Check 🎯

Problem-Solving Workshop 🧮

A 5 kg block is pushed with 40 N on a surface ( $mu_k = 0.2$ , $g = 10$ m/s²). What is the acceleration (m/s²)?
Net force on a 3 kg object is 12 N. What is the acceleration (m/s²)?
A 10 kg object accelerates at 2 m/s². The friction force is 15 N. What applied force (in units of 5 N) acts on the object? In other words, what is $(ma + f)/5$ ? (Hint: $F_{app} = ma + f$ , then divide by 5.)

Concept Check 🔍

Practice

#	Problem Type	Key Steps
1	Horizontal surface with friction	FBD → $F_{net} = F_{app} - f_k$
2	Incline problems	Components along and perpendicular
3	Connected objects	Shared acceleration, separate FBDs

Challenge Question 📋

f_s \leq mu_s N

f_k = mu_k N

Action-reaction:

F_{AB} = -F_{BA}

Worked Example

An Atwood machine has masses 3 kg and 5 kg. Find the acceleration. ( $g = 10$ m/s²)

$a = \frac{(m_2 - m_1)g}{m_1 + m_2} = \frac{(5-3)(10)}{3+5} = \frac{20}{8} = 2.5$ m/s² ✅

Concept Check 🎯

Review & Applications 🧮

Atwood machine: masses 3 kg and 5 kg. Acceleration (m/s²)? ( $g = 10$ m/s²)
A 10 kg object hangs from a rope. What is the tension (N)? ( $g = 10$ m/s²)
A 4 kg block is pushed with 30 N. Friction is 10 N. What is the net force (N)?

Concept Check 🔍

Practice

#	Topic	Key Formula
1	Newton's Laws overview	$F = ma$
2	Friction	$f = \mu N$
3	Circular motion	$F_c = mv^2/r$

Challenge Question 📋