Newton's Universal Gravitation

Gravitation

The universal force of attraction that exists between all matter in the universe; every object with mass is attracted to every other object with mass

Gravitational Force

The attractive force that exists between any two objects that have mass

Gravity

The force of gravitational attraction exerted by the Earth (or any large celestial body) on objects near its surface; a specific case of the universal gravitational force

Newton's First Law of Motion

An object at rest remains at rest and an object in motion remains in motion at constant velocity unless acted upon by an unbalanced force

Newton's Second Law of Motion

The acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass; F = ma

Newton's Third Law of Motion

For every action, there is an equal and opposite reaction

Newton's Universal Law of Gravitation

Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers

Gravitational Force Formula

F = G(m₁m₂)/r²

Gravitational Constant (G)

6.67 × 10⁻¹¹ N·m²/kg²; shows why gravitational forces between everyday objects are incredibly small

Mass in Gravitational Force Formula

m₁ and m₂ represent the masses of the objects in kilograms (kg); directly proportional to gravitational force

Distance in Gravitational Force Formula

r represents the distance between the centers of the two objects in meters (m); inversely proportional to the square of the gravitational force

Direct Proportionality to Mass

If you double the mass of one object, gravitational force doubles; if you double both masses, gravitational force increases by a factor of 4 (2 × 2)

Triple Mass Effect

If only one of two objects has its mass tripled, the gravitational force triples (becomes 3F)

Inverse Square Law

Gravitational force is inversely proportional to the square of the distance; if distance doubles, force becomes 1/4 as strong; if distance triples, force becomes 1/9 as strong

Double Distance Effect

When distance doubles, gravitational force decreases by a factor of 4 (2²)

Triple Distance Effect

When distance triples, gravitational force decreases by a factor of 9 (3²)

Halve Distance Effect

When distance is reduced to half its original value, gravitational force increases by a factor of 4

Characteristics of Gravitational Force

Acts between ALL objects with mass; always attractive (never repulsive); acts at a distance without physical contact; depends on both masses and distance between them

Gravitation vs Gravity Scope

Gravitation is universal between ALL objects with mass anywhere; Gravity is specific to Earth's pull on nearby objects

Gravitation vs Gravity Formula

Gravitation: F = G(m₁m₂)/r²; Gravity: F = mg (or W = mg)

Why Gravitational Forces Between People Are Tiny

The gravitational constant G is very small (6.67 × 10⁻¹¹), so unless one object has enormous mass like Earth, gravitational forces are negligibly small

Center of Mass in Gravitational Calculations

Distance is measured from center to center because the center of mass is where all of an object's mass can be treated as concentrated for spherically symmetric objects

Mutual Forces in Gravitation

If object A attracts object B with force F, then object B attracts object A with the same force F (Newton's Third Law)

Mass vs Weight in Gravitational Context

Mass is the amount of matter in an object (constant everywhere); Weight is the gravitational force acting on an object (varies with location)

Weight Formula

Weight = mg, where g = 9.8 m/s² on Earth's surface

Gravitational Force Between Two Students Problem

Two students 1.50 m apart with masses 70.0 kg and 52.0 kg attract each other with force 1.08 × 10⁻⁷ N

Why Gravity is Weak Between Small Objects

Their masses are too small to produce any noticeable gravitational attraction despite gravity being universal

Three Identical Objects Net Force

Three identical 10.0 kg objects in a line, each 0.500 m apart; the center object experiences a net force of 0 N because forces from the two outer objects cancel

Unequal Masses Three Object System

Object A (10.0 kg), Object B (10.0 kg), Object C (15.0 kg) in a line, each 0.500 m apart; net gravitational force on B is approximately 1.33 × 10⁻⁹ N directed toward C

Rocket Mass Calculation

A rocket on Earth's surface experiences 4.98 × 10⁵ N gravitational force; mass of rocket is 5.07 × 10⁴ kg (approximately 50,700 kg)

Effect of Mass on Gravitational Force Analysis

Comparing objects with masses 60/70 kg versus 60/50 kg at same distance; larger masses produce larger gravitational force (7.01 × 10⁻⁸ N vs 5.01 × 10⁻⁸ N)

Effect of Distance on Gravitational Force Analysis

Same pair at r = 2.0 m produces F = 7.01 × 10⁻⁸ N; at r = 4.0 m produces F = 1.75 × 10⁻⁸ N; doubling distance reduces force to 1/4

Arrow Width in Gravitational Force Diagrams

Thicker arrows represent stronger gravitational forces; thinner arrows represent weaker gravitational forces

Concept of Small Mass Gravitational Force

When both objects have small masses, the gravitational force pulling them together is very small (shown by thin arrows)

Concept of Large Mass Gravitational Force

When objects have larger masses, the gravitational force pulling them together is larger (shown by thicker arrows)

Concept of Distance Effect on Gravitational Force

When objects are farther apart, the force between them weakens significantly; force at smaller distance is greater than force at larger distance

Momentum Formula

p = mv

Impulse Formula

J = F_net,avg × Δt

Impulse-Momentum Theorem

J = Δp

Change in Momentum Formula

Δp = F_net,avg × Δt

Stopping Force Formula

|F_net,avg| = |Δp| / Δt

Conservation of Momentum Two Objects

m₁v₁,i + m₂v₂,i = m₁v₁,f + m₂v₂,f

Conservation of Momentum General

p_before = p_after

Kinetic Energy Formula

KE = ½mv²

Gravitational Potential Energy Formula

PE = mgh

Mechanical Energy Formula

ME = KE + PE

Mechanical Energy Expanded Formula

ME = ½mv² + mgh

Conservation of Energy Ideal

ME_initial = ME_final

Conservation of Energy Expanded

KE_i + PE_i = KE_f + PE_f

Conservation of Energy Velocity Form

½mv_i² + mgh_i = ½mv_f² + mgh_f

Energy with Friction

ME_f = ME_i - E_dissipated

Astronauts Traveling to Moon

As they move away from Earth, Earth's gravity decreases by inverse square law; as they get closer to Moon, Moon's gravity increases; at 90% of the way, Moon's gravity becomes stronger than Earth's

Moon vs Person Gravitational Force Comparison

Moon's mass is about 7.35 × 10²² kg while a person's mass is 50-100 kg; enormous mass difference causes Moon-Earth force to be vastly larger than person-Earth force despite greater distance

Universal Nature of Gravity

Gravity acts on all objects with mass everywhere in the universe

Always Attractive Nature of Gravity

Gravitational forces always pull objects together; they never push objects apart