Gravitational Force Exerted by a Sphere

Gravitational Force Exerted by a Sphere

To find the gravitational force exerted by a sphere of mass \(M\) on a particle of mass \(m\) outside of that sphere, we must first subdivide that sphere into many very skinny shells and find the gravitational force exerted by anyone of those shells on \(m\). We'll see, however, that finding the gravitational force exerted by such a shell is in of itself a somewhat tedious exercise. In the end, we'll see that the gravitational force exerted by a sphere of mass \(M\) on a particle of mass \(m\) outside of the sphere (where \(D\) is the center-to-center separation distance between the sphere and the particle) is completely identical to the gravitational force exerted by a particle of mass \(M\) on the other particle of mass \(m\) such that \(D\) is there separation distance.

Orbital Rings and Planet Building: Prelude to Colonizing the Solar System

Orbital Rings and Planet Building: Prelude to Colonizing the Solar System

An orbital ring connected to the Earth by space elevators would reduce the cost of going to space to an amount comparable to an airplane ticket. This would cause a boom in the space tourism industry and eventually millions and even billions of people and tons of cargo will be moving from the Earth’s surface to space annually, and vise versa. This would necessitate an expansion in our space-based infrastructure to include space-based solar panels, a lunar mass driver, the routine mining of asteroids, and especially enormous space habitats (for all those billions of people to live in) such as the Standford Torus, the Bernal Sphere, or the O’Neil Cylinder. Orbital rings also allow you to build artificial planets and Dyson spheres, which would allow us to completely colonize the solar system. They would also allow us to build a Birch planet, a single planet with a surface area which exceeds the total surface area of all the planets in the Milky Way galaxy.

Drake's Equation and Searching for Life in the Milky Way

Drake's Equation and Searching for Life in the Milky Way

In this lesson, we’ll discuss the prospect of life in the Milky Way galaxy beyond the Earth. We'll begin by discussing the speculations made in a paper written by Carl Sagan about the possibility of life in Jupiter's atmosphere. From there, we shall derive a formula which describes the habitable zone of a star. Using this formula and data obtained by the Kepler Space Telescope, we can estimate the total number of "Earth-like" planets in the Milky Way. From there, we discuss the fraction of those planets on which simple and intelligent life evolve; then we'll discuss the fraction of those planets on which advanced communicating civilizations evolve and what fraction of those civilizations are communicating right now.

The Diversity of Exoplanets in the Galaxy

The Diversity of Exoplanets in the Galaxy

In this lesson, we’ll attempt to give a brief catalog of the very different classes of planets in the universe. We'll discuss Pulsar planets, hot Jupiters, Super Earths, ice and water worlds, and many more. 

Proof of the Theorem: \(\lim_{ϴ→0}\frac{sinϴ}{ϴ}=1\)

Proof of the Theorem: \(\lim_{ϴ→0}\frac{sinϴ}{ϴ}=1\)

In this lesson, we’ll use the squeeze theorem and elementary trigonometry to prove that \(\lim_{x→0}\frac{sinx}{x}=1\).

Megastructures: Shkadov Thrusters

Megastructures: Shkadov Thrusters

In this video, we’ll discuss Shkadov thrusters: a method of moving stars, star systems, and even entire galaxies.

Proof of Green's Theorem

Proof of Green's Theorem

For a vector field \(\vec{F}(x,y)\) defined at each point \((x,y))\) within the region \(R\) and along the continuous, smooth, closed, piece-wise curve \(c\) such that \(R\) is the region enclosed by \(c\), we shall derive a formula (known as Green’s Theorem) which will allow us to calculate the line integral of \(\vec{F}(x,y)\) over the curve \(c\).

Gravitational Force Exerted by a Rod

Gravitational Force Exerted by a Rod

Using Newton's law of gravity and the concept of the definite integral, we can find the total gravitational force exerted by a rod on a particle a horizontal distance \(d\) away from the rod.

Gravitational Force Exerted by a Sphere

Gravitational Force Exerted by a Sphere

To find the gravitational force exerted by a sphere on a particle of mass \(M\) outside of that sphere, we must first subdivide that sphere into many very skinny shells and find the gravitational force exerted by anyone of those shells on \(m\). We'll see, however, that finding the gravitational force exerted by such a shell is in of itself a somewhat tedious exercise. In the end, we'll see that the gravitational force exerted by a sphere of mass \(M\) on a particle of mass \(m\) outside of the sphere (where \(D\) is the center-to-center separation distance between the sphere and particle) is completely identical to the gravitional force exerted by a particle of mass \(M\) on the mass \(m\) such that \(D\) is their separation distance.

Introduction to Double Integrals

Introduction to Double Integrals

In previous lessons, we learned that by taking the integral of some function \(f(x)\) we can find the area underneath that curve by summing the areas of infinitely many, infinitesimally skinny rectangles. In this lesson, we'll use the concept of a double integral to find the volume underneath any smooth and continuous surface \(f(x,y)\) by summing the volumes of infinitely many, infinitesimally skinny columns.

How to Produce Water and Oxygen on Mars

How to Produce Water and Oxygen on Mars

The lack of oxygen in Mars' atmosphere and running liquid water on its surface is very inconveniant for any humans living their since oxygen and liquid water are necessary for humans to survive. Fortunatelly, there is an abundance of frozen water on Mars' surface. In this lesson, we'll discuss various techniques which can be used to extract all of this water. Once the water is obtained, by performing electrolysis on the water we can distill all of the oxygen from that water we need.

Colonizing and Terraforming Venus

Colonizing and Terraforming Venus

The first serious proposal in scientific literature on terraforming other worlds in the universe was about terraforming Venus. The planetary scientist Carl Sagan imagined seeding the Venusian skies with photosynthetic microbes capable of converting Venus's \(C0_2\)-rich atmosphere into oxygen. Other proposals involve assembling a vast system of orbital mirrors capable of blocking the Sun's light and cooling Venus until this hot and hellish world became very frigid and rained \(C0_2\) from its atmosphere. The solleta would also be capable of simulating an Earth day/night cycle. To create oceans and an active hydrosphere on Venus, we could hurl scores of icy asteroids from the Kuiper belt to Venus and, upon impacting the Venusian atmosphere, would rapidly disintegrate releasing enormous quantities of water vapor into the atmosphere which subsequently condense to form the first seas on Venus. Or perhaps Saturn's moon Enceladus—containing a colossal subsurface ocean dwarfing that of the Earth's—could be sacrificed towards the end of creating the first seas on Venus. But even if humans never terraform this hellish world, they could still live their—partially at least—by deploying thousands of blimps into the Venusian skies capable of supporting a long-term, human presence of perhaps over a million people. Venusian sky cities. But eventually, after many millennia of terraforming Venus, a rich ecosystem of life—including us—could live on Venus's surface.

Optimization Problem

Optimization Problem

If \((x,y)\) represents any point on the circle, if \(P\) is a point fixed at the coordinate point \((4,0)\), and if \(d\) represents the distance between those two points then, by using only calculus, we can find the point \((x,y)\) on the circle associated with the minimum distance \(d\).

Maximizing the Area of a Rectangle

Maximizing the Area of a Rectangle

Given that the perimeter \(2x+2y\) of any arbitrary rectangle must be constant, we can use calculus to find that particular rectangle with the greatest area. The solution to this problem has practical applications. For example, suppose that someone had only 30 meters of fencing to enclose their backyard and they wanted to know what fencing layout would maximize the size and total area of their backyard. Using calculus, we can answer such questions.

Finding the Minima and Maxima of a Function

Finding the Minima and Maxima of a Function

Calculus—specifically, derivatives—can be used to find the values of \(x\) at which the function \(f(x)\) is at either a minimum value or a maximum value. For example, suppose that we let \(x\) denote the horizontal distance away from the beginning of a hiking trail near a mountain and we let \(f(x)\) denote the altitude of the mountainous terrain at each \(x\) value. \(f(x)\) reaches a minimum value when the function "flattens out"—that is, when \(f'(x)\) becomes equal to zero. These particular values of \(x\) are associated with the bottom and top of the mountain. The condition that \(f'(x)=0\) only tells us that \(f(x)\) is at either a minimum or a maximum. To determine whether or not \(f(x)\) is at a minimum or a maximum, we must use the concept of the second derivative. This will be the topic of discussion in this lesson.

Colonizing the Kuiper Belt and Oort Cloud

Colonizing the Kuiper Belt and Oort Cloud

The icy asteroids and comets in the Kuiper belt will one day (perhaps during the 22nd century) be hurled towards Mars’ atmosphere where they’ll disintegrate and release nitrogen into the atmosphere—a crucial step in the project of terraforming Mars. In the future, asteroids will perhaps be used as spaceships powered by nuclear reactors. The fuel for these nuclear reactors—deuterium and helium-3—could be harvested from the asteroids and the atmospheres of gas giants, respectively. As the famous physicist Freeman Dyson once noted, since the Oort Cloud contains all the ingredients necessary to support life, this realm of a trillion or more comets will likely be a way point for a long voyage to the Alpha Centuari star system.

Why Colonize the Universe?

Why Colonize the Universe?

In Carl Sagan's book Pale Blue Dot, he argued that humans evolved a love for exploration as an essential part of our survival as a species. It was this evolutionary trait which compelled our hunter-gather ancestors to leave their home—Africa—when times were getting rough and to meander across the planet. As planetary catastrophes become increasingly likely as time rolls by, Sagan argues that this same "survival strategy" will perhaps compel humanity to colonize the solar system, and beyond. Even the universe itself will one day become too dangerous for humans to live in and we’ll need to voyage to another universe to survive.

Calculating the Arc Length of a Curve

Calculating the Arc Length of a Curve

In this lesson, we'll use the concept of a definite integral to calculate the arc length of a curve.

The Kardeshev Scale

The Kardeshev Scale

The Kardeshev scale ranks how advanced a technological civilization is based on its power consumption. A Type I civilization is a civilization which has harnessed all of their planet's renewable energy sources and who can control the natural forces of their planet such as the weather and volcanoes; a Type II civilization has harnessed the total power output of their home-star and routinely move and dissemble stars; a Type III civilization is one which has spread across the entire galaxy and harnessed the total power output of their galaxy; a Type IV civilization is one with faster-than-light (FTL) speed spacecraft and that has harnessed the total power output of all the galaxies in their universe; Type V civilizations are like gods which have colonized other universes and can spontaneously create other universes at will.

Solving Problems using Line Integrals

Solving Problems using Line Integrals

In the previous lesson, we defined the concept of a line integral and derived a formula for calculating them. We learned that line integrals give the volume between a surface \(f(x,y)\) and a curve \(C\). In this lesson, we'll learn about some of the applications of line integral for finding the volumes of solids and calculating work. In particular, we'll use the concept of line integrals to calculate the volume of a cylinder, the work done by a proton on another proton moving in the presence of its electric field, and the work done by gravity on a swinging pendulum.