Laws of Motion

Chapter 5: Laws of Motion

Dynamics asks · why · objects move the way they do. The answer lies in Newton's three laws of motion, which form the foundation of classical mechanics.

Newton's First Law (Law of Inertia)

"Every object continues in its state of rest or uniform motion in a straight line unless acted upon by an external net force."

Inertia is the tendency of an object to resist changes in its state of motion. Mass is the measure of inertia — heavier objects are harder to accelerate.

A force is any push or pull that can change the state of motion. When the net force is zero, the object is in equilibrium (either at rest or moving at constant velocity).

Newton's Second Law

"The rate of change of momentum of an object is proportional to the applied net force, and takes place in the direction of the force."

F_net = m × a

Where:
F_net is net force (N = kg m/s²)
m is mass (kg)
a is acceleration (m/s²)

Momentum (p) = m × v (vector, unit: kg m/s)
Newton's 2nd law in terms of momentum: F = dp/dt

This form is more general — it works even when mass changes (like rockets).

Newton's Third Law

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

If object A exerts force F on object B, then B exerts force -F on A. These action-reaction forces:
Are equal in magnitude
Opposite in direction
Act on different objects (never on the same object)
Cannot cancel each other

Common Forces

Weight (W = mg): gravitational force downward
Normal force (N): perpendicular to a surface, prevents objects from passing through it
Tension (T): force in a string/rope, directed away from the object along the string
Friction (f): opposes relative motion between surfaces
Static friction: f_s ≤ mu_s × N (prevents motion)
Kinetic friction: f_k = mu_k × N (acts during sliding; mu_k < mu_s)

Free Body Diagrams (FBD)

1.An FBD shows all forces acting on one object. Steps:
2.Isolate the object
3.Draw all external forces as vectors
4.Apply Newton's 2nd law: ΣF = ma

Applications

Atwood Machine: Two masses m1 and m2 connected by a string over a frictionless pulley.
Acceleration: a = (m1 - m2)g / (m1 + m2)
Tension: T = 2 m1 m2 g / (m1 + m2)

Inclined Plane: For mass m on a smooth incline of angle theta:
Component along incline: mg sin(theta) — causes acceleration
Component perpendicular to incline: mg cos(theta) = Normal force N

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Example 1

A 10 kg box is pushed with a net force of 30 N. Find acceleration.
a = F/m = 30/10 = 3 m/s²

Example 2

A 5 kg object on a surface has coefficient of kinetic friction 0.3 (g = 10 m/s²). Find friction force and net acceleration if pushed with 20 N.
N = mg = 50 N; f_k = 0.3 × 50 = 15 N; F_net = 20 - 15 = 5 N; a = 5/5 = 1 m/s²

Example 3

In an Atwood machine, m1 = 6 kg and m2 = 4 kg. Find acceleration (g = 10 m/s²).
a = (6 - 4) × 10 / (6 + 4) = 20/10 = 2 m/s²

Example 4

Identify the action-reaction pair when a book rests on a table.
The book pulls the Earth (gravitational) and Earth pulls the book. The book pushes down on the table; the table pushes up on the book. Note: the weight and normal force on the book are NOT action-reaction pairs — they are different forces on the same object.

Example 5

A 2 kg block slides down a frictionless 30° incline. Find acceleration.
a = g sin(30°) = 10 × 0.5 = 5 m/s² down the incline.

Example 6

A rocket of mass 1000 kg ejects gas at 500 kg/s with speed 800 m/s. Find thrust force.
F = rate of change of momentum = 500 × 800 = 400,000 N = 4 × 10⁵ N

Example 7

Two blocks of 3 kg and 5 kg are connected by a string on a smooth surface; a force of 16 N pulls the 5 kg block. Find tension in string.
Total mass = 8 kg; a = 16/8 = 2 m/s². Tension pulls 3 kg block: T = 3 × 2 = 6 N.

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Common mistakes

Confusing weight (force, in N) with mass (in kg): W = mg.
Including reaction forces in the free body diagram of a single object — an FBD shows only forces ON that object, not forces it exerts on others.
Thinking friction always opposes motion — static friction opposes the · tendency · of motion, not actual motion.

Summary

Newton's three laws provide the foundation of mechanics. The first law defines inertia and equilibrium. The second law (F = ma) quantifies how force causes acceleration. The third law states that forces always come in equal-and-opposite pairs on different bodies. Friction, normal force, tension, and weight are the common forces analyzed using free body diagrams.