Newton's Laws with Calculus

⚛️ 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). ✅

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Newton's First Law Practice 🧮

1. A book sits on a table with no net force. What is its acceleration $(in m/s^{2})$?

2. A ball rolls at 10 m/s on a frictionless surface. What is its speed (m/s) after 5 seconds?

3. Two forces act on an object: 15 N right and 10 N left. What is the net force magnitude (in N)?

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⚛️ Newton's Second Law with Calculus

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

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

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 🧮

1. A 4 kg object has a net force of 20 N. What is its acceleration $(m/s^{2})$?

2. 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.

3. 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?

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⚛️ Newton's Third Law

Part 3 of 7 — Newton's Third Law

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

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

• 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 🧮

1. You push a desk with 50 N. How many newtons does the desk push back on you?

2. A swimmer pushes water backward with 100 N. What force (N) does the water exert on the swimmer?

3. Object A exerts 20 N on Object B. What force (N) does Object B exert on Object A?

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Practice

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⚛️ Friction Forces

Part 4 of 7 — Friction Forces

• Static friction: $f_s \leq \mu_s N$ (prevents motion)
• Kinetic friction: $f_k = \mu_k N$ (opposes sliding motion)

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 (kinetic, or maximum static): $f = \mu mg\cos\theta$

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 🧮

1. A 10 kg block on a flat surface has $\mu_k = 0.3$. Kinetic friction force (N)? (Use $g = 10$ $m/s^{2}$)

2. A 5 kg block requires 20 N to start sliding on a flat surface. What is $\mu_s$? (Use $g = 10$ $m/s^{2}$)

3. 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 $g = 10$ $m/s^{2}$)

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⚛️ Circular Motion

Part 5 of 7 — Circular Motion

For uniform circular motion:

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

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

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 🧮

1. A car goes around a curve of radius 50 m at 20 m/s. What is the centripetal acceleration $(m/s^{2})$?

2. An object moves in a circle of radius 2 m with centripetal acceleration 8 $m/s^{2}$. What is the speed (m/s)?

3. A 2 kg ball on a 0.5 m string moves at 4 m/s. What is the tension/centripetal force (N)?

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⚛️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Strategy for Newton's Law Problems

1. Draw a free-body diagram (FBD)
2. Choose coordinate axes (often along incline)
3. Write $\sum F_x = ma_x$ and $\sum F_y = ma_y$
4. Solve the system of equations
5. 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. ($g = 10$ $m/s^{2}$)

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

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

$a = 30/5 = 6$ $m/s^{2}$ ✅

Concept Check 🎯

Problem-Solving Workshop 🧮

1. A 5 kg block is pushed with 40 N on a surface ($\mu_k = 0.2$, $g = 10$ $m/s^{2}$). What is the acceleration $(m/s^{2})$?

2. Net force on a 3 kg object is 12 N. What is the acceleration $(m/s^{2})$?

3. A 10 kg object accelerates at 2 $m/s^{2}$ while a 15 N friction force opposes its motion. What horizontal applied force (N) acts on it?

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⚛️ Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

• $\sum F = ma = m\frac{dv}{dt}$
• $f_s \leq \mu_s N$, $f_k = \mu_k N$
• $F_c = mv^2/r$
• Action-reaction: $F_{AB} = -F_{BA}$

Worked Example

An Atwood machine has masses 3 kg and 5 kg. Find the acceleration. ($g = 10$ $m/s^{2}$)

$a = \frac{(m_2 - m_1)g}{m_1 + m_2} = \frac{(5-3)(10)}{3+5} = \frac{20}{8} = 2.5$ $m/s^{2}$ ✅

Concept Check 🎯

Review & Applications 🧮

1. Atwood machine: masses 3 kg and 5 kg. Acceleration $(m/s^{2})$? ($g = 10$ $m/s^{2}$)

2. A 10 kg object hangs from a rope. What is the tension (N)? ($g = 10$ $m/s^{2}$)

3. A 4 kg block is pushed with 30 N. Friction is 10 N. What is the net force (N)?

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Practice

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