Newton's Laws of Motion

Overview

Newton's three laws form the foundation of classical mechanics, describing the relationship between forces and motion.

Newton's First Law (Law of Inertia)

Statement: 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.

$\text{If } \sum\vec{F} = 0, \text{ then } \vec{v} = \text{constant}$

Inertia

• Property of matter to resist changes in motion
• Mass is the measure of inertia
• Greater mass → greater inertia

Inertial Reference Frames

• Frames in which Newton's first law holds
• Non-accelerating frames
• Earth is approximately inertial for most problems

Newton's Second Law

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

$\sum\vec{F} = m\vec{a}$

Or in component form:

$\sum F_x = ma_x$

$\sum F_y = ma_y$

Key Points

• Force and acceleration are vectors (same direction)
• Mass is a scalar
• SI unit of force: Newton (N) = kg·m/s²
• This is a vector equation (applies to each component)

Newton's Third Law

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

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

Key Points

• Forces always come in pairs
• Action-reaction pairs act on different objects
• The forces are equal in magnitude, opposite in direction
• These pairs never cancel (they act on different objects)

Common Forces

Weight (Gravitational Force)

$W = mg$

Direction: toward Earth's center (downward)

Normal Force ($N$)

• Perpendicular force from a surface
• Adjusts to prevent objects from passing through surfaces
• Not always equal to weight

Tension ($T$)

• Force transmitted through a rope, string, or cable
• Always pulls, never pushes
• Constant throughout an ideal (massless) rope

Friction ($f$)

• Opposes relative motion between surfaces
• Static friction: $f_s \leq \mu_s N$
• Kinetic friction: $f_k = \mu_k N$

Spring Force (Hooke's Law)

$F_s = -kx$

Where $k$ is the spring constant and $x$ is displacement from equilibrium

Free Body Diagrams

Steps to draw:

1. Identify the object of interest
2. Draw the object as a point or simple shape
3. Draw all forces acting ON the object
4. Label each force
5. Choose a coordinate system

Problem-Solving Strategy

1. Draw a free body diagram
2. Choose a convenient coordinate system
3. Resolve forces into components
4. Apply Newton's second law to each direction
5. Solve the resulting equations
6. Check units and reasonableness

Examples

Example 1: Simple Acceleration

A 5 kg box is pushed with a 20 N force on a frictionless surface.

$\sum F = ma$

$20 = 5a$

$a = 4 \text{ m/s}^2$

Example 2: Inclined Plane

A 10 kg block on a 30° frictionless incline. Find acceleration.

Forces parallel to incline:

$\sum F = mg\sin\theta = ma$

$a = g\sin(30°) = 9.8 \times 0.5 = 4.9 \text{ m/s}^2$

Example 3: Atwood Machine

Two masses ($m_1 = 4$ kg, $m_2 = 6$ kg) connected by a rope over a pulley.

$m_2 g - m_1 g = (m_1 + m_2)a$

$a = \frac{(m_2 - m_1)g}{m_1 + m_2} = \frac{(6-4)(9.8)}{6+4} = 1.96 \text{ m/s}^2$

Tension: $T = m_1(g + a) = 4(9.8 + 1.96) = 47$ N

Special Cases

Equilibrium

When $\sum\vec{F} = 0$:

• Object at rest stays at rest
• Object in motion continues with constant velocity

Apparent Weight

In an accelerating elevator:

$N = m(g + a) \quad \text{(accelerating upward)}$

$N = m(g - a) \quad \text{(accelerating downward)}$