Chapter 2: Inertia and the Foundations of Classical Mechanics

Inertia and Historical Perspectives

Aristotle's Views on Motion

Aristotle proposed early ideas about motion, distinguishing between natural and violent motion.

• Natural motion: Objects move straight up or down on Earth; celestial motion is circular.

• Example: The Sun and Moon continually circle the Earth (geocentric model).

• Violent motion: Caused by external pushes or pulls (e.g., wind moving a ship).

Copernicus

Copernicus challenged the geocentric model, proposing that the Earth and other planets orbit the Sun.

• Heliocentric model: The Sun is at the center of the solar system.

Galileo's Contributions

Galileo made key discoveries about motion and inertia.

• Objects of different weights fall at the same rate in the absence of air resistance.

• A moving object needs no force to keep moving in the absence of friction.

• Objects speed up on downward slopes, slow down on upward slopes, and maintain speed on horizontal planes.

Equilibrium and Motion

Equilibrium of Moving Things

Equilibrium describes the state where all forces on an object are balanced.

• Static equilibrium: No change in motion (e.g., a hockey puck at rest on ice).

• Dynamic equilibrium: Constant speed in a straight line (e.g., a hockey puck sliding at constant speed on ice).

Motion is Relative

All motion is described relative to a chosen reference point.

• Example: Walking on a road is relative to the Earth, but the Earth itself moves relative to the Sun.

Speed and Acceleration

• Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

• Average speed: $\text{Average speed} = \frac{\text{Total distance covered}}{\text{Time interval}}$

• Acceleration: $\text{Acceleration} = \frac{\text{Change in velocity}}{\text{Time interval}}$

• Steeper inclines result in greater accelerations.

• In the absence of air resistance, all objects fall with the same acceleration.

Free Fall

Describes the motion of objects under the influence of gravity alone.

• Velocity acquired from rest: $\text{Velocity} = \text{Acceleration} \times \text{Time}$

• For free fall, $a = 10\ \text{m/s}^2$ (approximate value for Earth's gravity).

• Distance fallen: $\text{Distance} = \frac{1}{2} \times \text{Acceleration} \times \text{Time}^2$

Forces and Newton's Laws

Force Causes Acceleration

• Acceleration is proportional to net force.

• To increase acceleration, increase the net force.

Friction

• Friction depends on the materials in contact and how hard they are pressed together.

• Friction is less on smooth surfaces (e.g., ice) than on rough surfaces (e.g., wood).

Mass and Weight

• Weight: $\text{Weight} = mg$

• The weight of an object on the Moon is less than on Earth due to lower gravity.

• The mass of an object is the same everywhere.

Mass Resists Acceleration

• Acceleration is inversely proportional to mass: $\text{Acceleration} = \frac{1}{\text{mass}}$ (for a given force).

• More mass means less acceleration for the same force.

Newton's Second Law of Motion

• Formula: $\text{Acceleration} = \frac{\text{Net force}}{\text{Mass}}$

• If net force is doubled, acceleration is doubled; if mass is doubled, acceleration is halved.

Gravity and Free Fall

• The greater the mass, the greater the force of gravity.

• All objects accelerate at the same rate in free fall (ignoring air resistance).

• When air resistance is significant, acceleration decreases and objects may reach terminal velocity.

Forces and Interactions

Forces and Interactions

• Forces always act in pairs between two objects.

• Example: Tires push on the road, and the road pushes back on the tires.

Action and Reaction Forces (Newton's Third Law)

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

• Action and reaction forces act on different objects.

• Examples: Rocket pushes on gas, gas pushes on rocket; Earth pulls on ball, ball pulls on Earth.

Action and Reaction on Different Masses

• $a = \frac{F}{m}$

• The same force on a small mass produces a large acceleration; on a large mass, a small acceleration.

Energy, Work, and Machines

Energy

• Energy is the ability to do work.

• It is a conserved quantity and can be transferred or transformed.

Kinetic Energy (KE)

• Energy of motion.

• $KE = \frac{1}{2} \times \text{Mass} \times \text{Speed}^2$

• If speed is doubled, kinetic energy is quadrupled.

Potential Energy (PE)

• Stored energy due to position.

• $PE = \text{Mass} \times \text{Acceleration due to gravity} \times \text{Height}$

• Example: A stretched bow or rubber band stores energy that can do work.

Work

• $W = \text{Force} \times \text{Distance}$

• Unit: Joule (J)

• More work is done when lifting heavier objects or lifting to greater heights.

Work-Energy Theorem

• Work done on an object equals the change in its kinetic energy: $\text{Work} = \text{Change in KE}$

• Doubling speed requires four times the work.

Power

• Power is the rate of doing work.

• Unit: Watt (W); $1\ \text{Watt} = 1\ \text{Joule/second}$

• $1\ \text{kilowatt} = 1000\ \text{watts}$

Machines

• Lever: Rotates on a fulcrum; allows a small force over a large distance to move a load.

• Pulley: Changes the direction of the input force; operates like a lever with equal arms.

• Mechanical advantage: Machines multiply force or change its direction.

Gravity and Universal Gravitation

The Newtonian Synthesis

• Newton discovered that gravity is universal and acts on all objects, including planets.

The Universal Law of Gravity

• The force of gravity between two objects depends on their masses and the distance between them.

• $F = G \frac{m_1 m_2}{\text{Distance}^2}$

Effect of Mass on $F_g$	Effect of Distance on $F_g$
If both masses double, $F_g$ quadruples	If distance doubles, $F_g$ is quartered
If one mass triples, $F_g$ triples	If distance is halved, $F_g$ increases by four times

Gravitational Fields

• Earth is surrounded by a gravitational field, represented by field lines pointing toward Earth.

• Inside a planet, the field decreases to zero at the center; outside, it decreases to zero at infinity.

The Universal Gravitational Constant, G

• $F = G \frac{m_1 m_2}{r^2}$

• G is a constant that quantifies the strength of gravity in the universe.

Einstein's Theory of Gravitation

• Gravity is a warping of space-time by mass.

• The curvature of space-time affects the motion of other objects.

Weight and Weightlessness

• Weight is the force an object exerts against a supporting surface.

• Weightlessness occurs when there is no support force (e.g., free fall).

• Example: Standing on a scale in an accelerating elevator changes the reading due to changes in support force.