Chapter 4: The Laws of Motion – Classical Mechanics and Newton’s Laws

Classical Mechanics

Overview

Classical mechanics describes the relationship between the motion of objects in our everyday world and the forces acting on them. It is the foundation for understanding how and why objects move.

• Applicability: Classical mechanics applies to macroscopic objects moving at speeds much less than the speed of light.

• Limitations: It does not apply to very tiny objects (smaller than atomic sizes) or objects moving near the speed of light.

Forces

Definition and Types

A force is usually thought of as a push or pull. It is a vector quantity, meaning it has both magnitude and direction.

• Contact Force: Results from physical contact between two objects (e.g., friction, tension, normal force).

• Field Force: Acts between disconnected objects (e.g., gravity, electromagnetic force). Also called "action at a distance."

Contact and Field Forces (Illustrative Table)

Contact Forces	Field Forces
Hand pushing a spring	Gravitational attraction between planets
Pulling a wagon	Electrostatic force between charges
Ice skater pushing off the wall	Magnetic force between magnets

Fundamental Forces

Types and Characteristics

• Strong Nuclear Force: Holds atomic nuclei together; strongest force but acts over very short distances.

• Electromagnetic Force: Acts between charged particles; infinite range.

• Weak Nuclear Force: Responsible for radioactive decay; short range.

• Gravity: Attraction between masses; weakest but infinite range.

All fundamental forces are field forces. In mechanics, only gravity and electromagnetic forces are typically considered.

Newton’s Laws of Motion

Newton’s First Law (Law of Inertia)

A body remains at rest or moves with constant velocity and direction if the vector sum of the forces acting on it is zero.

• Net Force: The vector sum of all external forces exerted on the object.

• Inertia: The tendency of an object to continue in its original motion.

External and Internal Forces

• External Force: Originates from the interaction between the object and its environment.

• Internal Force: Originates within the object itself; cannot change the object's velocity.

Force and Motion

• Motion can occur even in the absence of forces.

• Forces cause changes in motion (acceleration).

Mass

• Definition: Mass is a measure of the resistance of an object to changes in its motion due to a force.

• Scalar Quantity: Has magnitude but no direction.

• SI Unit: Kilogram (kg).

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.

• Equation:

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

• Both force and acceleration are vectors.

• Can be applied in three dimensions.

Units of Force

• SI Unit: Newton (N)

• Definition: $1\,\text{N} = 1\,\frac{\text{kg} \cdot \text{m}}{\text{s}^2}$

• US Customary Unit: Pound (lb); $1\,\text{N} = 0.225\,\text{lb}$

Units of Mass, Acceleration, and Force (Table)

System	Mass	Acceleration	Force
SI	kg	m/s2	N = kg·m/s2
U.S. customary	slug	ft/s2	lb = slug·ft/s2

Newton’s Third Law

If object 1 and object 2 interact, the force exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force exerted by object 2 on object 1.

$\vec{F}_{12} = -\vec{F}_{21}$

• Action and reaction forces act on different objects.

• A single isolated force cannot exist.

Action-Reaction Pairs (Examples)

• Normal Force: The force a table exerts on a TV is equal and opposite to the force the TV exerts on the table.

• Gravitational Force: The force the Earth exerts on an object is equal and opposite to the force the object exerts on the Earth.

Applications of Newton’s Laws

Assumptions

• Objects behave as particles (rotational motion can be ignored for now).

• Masses of strings or ropes are negligible.

• Only forces acting on the object are considered; reaction forces can be neglected.

Free Body Diagrams

• Identify all forces acting on the object of interest.

• Choose an appropriate coordinate system.

• If the free body diagram is incorrect, the solution will likely be incorrect.

Example: For a box being pulled by a rope, the forces include tension (T), gravitational force ($F_g$), and normal force (n).

Gravitational Force and Weight

Newton’s Law of Universal Gravitation

The mutual force of attraction between any two objects is given by:

$F_g = G \frac{m_1 m_2}{r^2}$ $G = 6.67 \times 10^{-11}\,\text{N} \cdot \text{m}^2/\text{kg}^2$

Weight

• The magnitude of the gravitational force acting on an object of mass m near the Earth's surface is called the weight w of the object.

• Special case: $w = m g$ (where $g$ is the acceleration due to gravity).

• $g$ can also be found from the Law of Universal Gravitation: $g = G \frac{M_e}{r^2}$

• Weight is not an inherent property of an object; it depends on location (e.g., $g_{sun} = 274\,\text{m/s}^2$, $g_{moon} = 1.62\,\text{m/s}^2$).

Problem Solving Strategies

• Read the problem carefully and identify the object of interest.

• Draw a free body diagram, labeling all forces.

• Choose a coordinate system and resolve forces into components if necessary.

• Apply Newton’s laws to solve for unknowns.

Sample Problems and Applications

• Calculating acceleration and forces for objects in one and two dimensions (e.g., airboat, horses pulling a barge).

• Understanding the effect of mass and radius on weight on different planets.

• Illustrating Newton’s third law with action-reaction pairs (e.g., ice skaters pushing off each other).

Summary Table: Fundamental Forces

Force	Relative Strength	Range	Carrier Particle
Strong Nuclear	1	$10^{-15}$ m	Gluons
Electromagnetic	$1/137$	Infinite	Photon
Weak Nuclear	$10^{-6}$	$10^{-18}$ m	$W^+, W^-, Z^0$ bosons
Gravity	$6 \times 10^{-39}$	Infinite	Graviton (hypothetical)

Additional info: These notes provide a foundation for understanding the physical principles underlying movement and force, which are essential for advanced studies in physics, engineering, and related sciences.