Newtonian Gravity: Crash Course Physics #8

Summary

Isaac Newton revolutionized physics with his laws of motion and universal gravitation. His concept of gravity, inspired by the fall of an apple, led to the realization that Earth's gravity affects the Moon's orbit. Newton's law of universal gravitation, F = GMm/r^2, describes the gravitational force between two masses, with G being a small constant. This law aligns with Kepler's laws of planetary motion, which Newton confirmed using his laws of motion and calculus. Today, Newton's principles continue to guide space exploration, including NASA's simulations of Martian gravity for spacesuit testing.

Takeaways

• 🌍 Isaac Newton transformed physics with his contributions, especially regarding motion and gravity.
• 🍎 Newton's famous discovery of gravity is linked to the story of an apple falling from a tree.
• 🔄 Newton connected the falling of objects on Earth to the orbital movements of celestial bodies like the Moon.
• 🌑 Newton realized that the Moon orbits Earth because of gravity, but it moves sideways fast enough to avoid crashing.
• 🔢 Newton formulated the law of universal gravitation, which states that gravitational force is proportional to the masses and inversely proportional to the square of their distance.
• 🧮 Newton introduced the gravitational constant, G, although its exact value was calculated later by Henry Cavendish.
• 🌞 Newton’s law of gravitation aligned with Kepler’s three laws of planetary motion, proving them with calculus.
• ⚖️ Kepler’s laws described planetary orbits as ellipses, with areas swept by planets being equal over time.
• 🪐 Newton explained small deviations in planetary orbits, caused by gravitational interactions between celestial bodies.
• 🚀 NASA still uses Newton’s calculations to simulate Martian gravity for testing spacesuits, confirming Newton’s lasting impact on science.

Q & A

What was the initial state of the scientific community's understanding of gravity before Newton?

Before Newton, the scientific community had no comprehensive concept of gravity. They knew that objects fell to the ground when dropped and observed that planets and moons orbited in a particular way, but they did not understand the connection between these phenomena.

How did Newton's apple story contribute to his understanding of gravity?

The apple story symbolizes Newton's realization that an invisible force, gravity, was pulling the apple towards the Earth. This led him to consider the possibility that the Earth was also being pulled by the apple, and that this force could extend to other celestial bodies like the Moon.

What was the significance of Newton's insight about the Moon's orbit in relation to gravity?

Newton's insight was significant because he understood that the Moon was being pulled towards the Earth by gravity, but it was moving sideways so quickly that it kept missing the Earth, which kept it in orbit. This was a key step in developing the concept of universal gravitation.

What factors did Newton identify as necessary to include in his equation for gravitational force?

Newton identified two main factors: the mass of the objects involved and the distance between them. He realized that the gravitational force must decrease with the square of the distance between the objects and increase with the product of their masses.

What is the law of universal gravitation as described by Newton?

Newton's law of universal gravitation states that every point mass attracts every other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them, which can be represented by the equation F = GMm/r^2.

Who was Henry Cavendish and what did he contribute to our understanding of gravity?

Henry Cavendish was a British scientist who, about a century after Newton, measured the gravitational constant (G) using precise experiments. He determined G to be approximately 6.67 * 10^-11 N*m^2/kg^2, which allowed for more accurate calculations of gravitational forces.

How did Newton's law of universal gravitation relate to Kepler's laws of planetary motion?

Newton's law of universal gravitation not only fit with Kepler's laws of planetary motion but also provided a physical explanation for them. Newton used his law of gravitation, combined with his laws of motion and calculus, to prove Kepler's laws, which described the orbits of planets as ellipses with the Sun at one focus.

What was Kepler's first law and how does it apply to planetary orbits?

Kepler's first law states that the orbit of a planet is an ellipse with the Sun at one of the two foci. This law applies to all elliptical orbits, not just those of planets, including Earth's moon orbiting the Earth.

Can you explain Kepler's second law and its implication for planetary orbits?

Kepler's second law, also known as the law of equal areas, states that a line joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that planets move faster when they are closer to the Sun and slower when they are farther away.

What is the significance of Kepler's third law in relation to the orbits of planets?

Kepler's third law relates the orbital period of a planet to the size of its orbit. It states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This law holds true for all planets orbiting the Sun, indicating a fundamental relationship in the solar system.

How does Newton's law of universal gravitation help in understanding the gravitational acceleration on Mars?

Using Newton's law of universal gravitation, we can calculate the gravitational acceleration on Mars by using the mass of Mars and its radius. The calculated acceleration is approximately 3.7 meters per second squared, which is about 38% of Earth's surface gravity, and is used to simulate Martian gravity for testing spacesuits.

Outlines

00:00

🍎 Newton's Discovery of Universal Gravitation

Isaac Newton revolutionized physics, particularly with his laws of motion and his concept of gravity. Initially, the scientific community had no unified theory of gravity, only observations of falling objects and celestial bodies' orbits. Newton's 'apple story' illustrates his insight that the force pulling an apple to the ground might be the same force keeping the Moon in orbit around Earth. He hypothesized that this force, gravity, could act at a distance and affect all objects. Newton's law of universal gravitation, F = GMm/r^2, was formulated to describe this force, where G is a constant representing the strength of the gravitational force, M and m are the masses of the two objects, and r is the distance between their centers. Although Newton didn't know the exact value of G, his law was instrumental in explaining the motions of celestial bodies and was later confirmed by Henry Cavendish's experiments.

05:02

📚 Newton's Gravitation and Kepler's Laws

Newton's law of universal gravitation not only aligned with Johannes Kepler's laws of planetary motion but also provided a physical explanation for them. Kepler's first law stated that planets orbit the Sun in elliptical paths with the Sun at one focus. The second law described how a planet sweeps out equal areas in equal times as it orbits, regardless of its distance from the Sun. Kepler's third law related the cube of the semi-major axis of a planet's orbit to the square of its orbital period, a constant ratio for all planets. Newton used his law of gravitation, combined with his laws of motion and calculus, to mathematically prove these laws and explain minor deviations in planetary orbits due to mutual gravitational influences. Newton's work also connected the gravitational force to his second law of motion, showing that gravitational acceleration is directly proportional to the mass of the larger object and inversely proportional to the square of the distance between the two objects, with G as the proportionality constant.

Keywords

💡Newton's Laws of Motion

Newton's Laws of Motion are three fundamental principles that describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. In the video, these laws are mentioned as a foundation for understanding how objects move, which is essential for comprehending Newton's work on gravity. The script does not delve into the specifics of each law but acknowledges their importance in the broader context of Newton's contributions to physics.

💡Gravity

Gravity is the force that attracts two bodies towards each other, proportional to their mass and inversely proportional to the square of the distance between their centers. The video focuses on Newton's understanding of gravity, which was a significant leap from the rudimentary knowledge of the time. It discusses how gravity not only causes objects to fall to the ground but also influences the orbits of celestial bodies like the Moon around the Earth.

💡Universal Gravitation

Newton's law of universal gravitation is a physical law that describes the gravitational attraction between objects with mass. The video explains how Newton conceptualized this law, which states that every point mass attracts every other point mass by a force acting along the line intersecting both points. The law is crucial for understanding the video's theme as it is the central concept that Newton used to explain the motion of celestial bodies and the falling of an apple.

💡Apple Anecdote

The 'apple anecdote' refers to the popular story that Newton was inspired to formulate his theory of gravity after watching an apple fall from a tree. While the video acknowledges that this story might be embellished, it serves to illustrate the concept of gravity's universality. The apple serves as a relatable example to explain how gravity affects objects both on Earth and in space.

💡Acceleration

In the context of the video, acceleration is the rate of change of an object's velocity with respect to time. It is a key component in Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The video uses the concept of acceleration to explain how gravity affects objects differently depending on their distance from the Earth's surface.

💡Mass

Mass is a fundamental property of matter that determines the strength of a body's interaction with the gravitational field. In the video, mass is discussed as a critical factor in Newton's law of universal gravitation, where the gravitational force between two objects is directly proportional to the product of their masses. The script uses the mass of the Earth and the Moon to explain the differing accelerations they experience due to gravity.

💡Distance

Distance is a measure of the space between two points. In the video, the concept of distance is integral to understanding how gravity weakens with the square of the distance between two objects. This principle is used to explain why the Moon, despite being much farther from the Earth than an apple in a tree, is still influenced by Earth's gravity but does not fall to the ground.

💡Johannes Kepler

Johannes Kepler was a German astronomer whose three laws of planetary motion provided a description of the orbits of planets around the Sun. The video mentions Kepler's laws as a precursor to Newton's work, showing how Newton's law of universal gravitation not only aligned with Kepler's observations but also provided a physical explanation for them.

💡Ellipse

An ellipse is a geometric shape defined as the set of all points for which the sum of the distances to two fixed points (foci) is constant. In the video, Kepler's first law is mentioned, stating that the orbits of planets are elliptical, with the Sun at one of the foci. This concept is essential for understanding the video's theme as it relates to the gravitational interactions between celestial bodies.

💡Henry Cavendish

Henry Cavendish was a British scientist who, through precise measurements, was the first to calculate the value of the gravitational constant (G) in Newton's law of universal gravitation. The video highlights Cavendish's contribution as it provides a numerical value to the constant that Newton had left as a placeholder in his equation, thus making the law practically applicable.

💡Gravitational Acceleration

Gravitational acceleration is the acceleration of an object caused by the force of gravity. The video uses this concept to explain how the gravitational force at Earth's surface, represented by 'small g', is calculated. It also extends this concept to calculate the gravitational acceleration on Mars, which is used by NASA to test spacesuits in reduced gravity environments.

Highlights

Isaac Newton's laws of motion and his understanding of gravity revolutionized physics.

Newton's concept of gravity connected the falling of an apple to the ground with the orbits of planets and moons.

Newton's law of universal gravitation is crucial for describing gravity's effects on a solar system scale.

The story of Newton's apple is a symbol of his realization about gravity, though its historical accuracy is debated.

Newton proposed that the force pulling the apple to Earth could also affect the Moon's orbit.

Newton's insight was that the Moon's sideways motion kept it from falling to Earth due to gravity.

Newton's law of universal gravitation suggests that gravity affects any two objects, including planets orbiting the Sun.

Newton's equation for gravitational force is based on an object's mass and acceleration.

The gravitational force decreases with the square of the distance between two objects.

Newton's law includes the mass of both objects involved in the gravitational interaction.

Newton introduced a constant, G, to represent the small gravitational force between objects.

Henry Cavendish later measured the value of G to be approximately 6.67 * 10^-11 N*m^2/kg^2.

Newton's law of universal gravitation aligns with Kepler's laws of planetary motion.

Newton's work allowed for the explanation of slight deviations from Kepler's predictions due to mutual gravitational pulls.

Newton's law of universal gravitation fits the expected form of a net force equation from his second law of motion.

NASA uses Newton's law to simulate Martian gravity for testing spacesuits on Earth.

The gravitational acceleration on Mars's surface is calculated to be about 3.7 meters per second squared.

Newton's mathematical framework continues to be essential for modern space exploration.

Transcripts

When people say that Isaac Newton completely transformed the field of physics, they really aren't kidding.

Now, we’ve already talked about his three laws of motion, which we use to describe how things move.

But another of Newton’s famous contributions to physics was his understanding of gravity.

When Newton was first starting out, scientists’ concept of gravity was pretty much nonexistent.

I mean, they knew that when you dropped something, it fell to the ground, and from careful observation,

they knew that planets and moons orbited in a particular way.

What they didn’t know was that those two concepts were connected.

Of course, just like with motion, we now know that there’s a lot more to gravity than

what Newton was able to observe.

Even so, when it comes to describing the effects of gravity on the scale of, say, our solar system,

Newton’s law of universal gravitation is incredibly useful.

And it all started with an apple.

… Probably.

[Theme Music]

Odds are, you’ve been told the story of Newton’s apple at some point.

The story goes that one day, he was sitting under an apple tree in his mother’s garden,

when an apple fell out of the tree.

That’s when Newton had his grand realization: Something was pulling that apple down to Earth.

And that led to another idea: What if the apple was pulling on Earth, too, but you just

couldn’t tell, because the effect of the apple’s force on Earth was less obvious?

A few years later, Newton was sitting in the same garden when he had another stroke of inspiration:

What if the same force that pulled the apple to the ground could affect things much farther

from Earth’s surface -- like the Moon?

It was kind of counterintuitive, because the Moon orbits Earth, instead of crashing straight

into the ground like an apple that falls off a tree.

But Newton realized that the Moon was still being pulled toward Earth -- it was just moving

sideways so quickly that it kept missing. That’s what was keeping it in orbit.

If gravity was keeping the Moon in orbit, what if it affected the behavior of any two objects --

like a planet orbiting the Sun?

That’s the official version of the story -- the one Newton himself used to tell.

Most historians think he was embellishing at least a little, but there probably is some truth to it.

Whether or not the thing with the apple actually happened, Newton thought his idea seemed promising.

The idea that gravity might affect everything, including the orbits of other planets and moons.

So he started looking for an equation that would accurately describe the way the gravitational force made objects behave --

whether it was an apple falling on the ground, or the Moon orbiting Earth.

Newton knew that however this gravitational force worked, it would probably behave like

any other net force on an object -- it would be equal to that object’s mass, times its acceleration.

The mass part was easy enough -- it would just be the mass of the apple or the Moon.

It was going to be a little harder to figure out the factors that were affecting the acceleration part of the equation.

The first thing Newton realized he’d have to take into account was distance.

When an object is close to the Earth’s surface, like an apple in a tree, gravity makes it

accelerate at about 10 meters per second squared.

But the Moon has an acceleration that’s only about a 3600th of that falling apple.

The Moon also happens to be about 60 times as far from the center of Earth as that apple would be --

and 60 squared is 3600.

So Newton figured that the gravitational force between two objects must get smaller the farther apart they are.

More specifically, it must depend on the distance between the two objects squared.

Then there was mass.

Not the mass of the apple or the Moon -- the mass of the other object involved in the

gravitational dance: in this case, Earth.

Newton realized that the greater the masses of the two objects pulling on each other,

the stronger the gravitational force would be between them.

Once he’d taken into account the distance between two objects, and their masses,

Newton had most of his equation for the way gravity behaved:

The gravitational force was proportional to the mass of the two objects multiplied together,

divided by the square of the distance between them.

But it had to be a lot smaller, or else you’d see a force pulling together most everyday objects.

Like, that Rubik’s cube is staying right where it is instead of being pulled towards me.

So the gravitational force between us must be very small.

So Newton added a constant to his equation -- a very small number that would make the

gravitational force just a tiny fraction of what you’d calculate otherwise.

He called it G.

And he called this full equation, F = GMm/r^2, the law of universal gravitation.

Newton had no idea what number big G would be, though. He just knew it would be a tiny

number, and put the letter G into his equation as a placeholder.

About a century later, Henry Cavendish, another British scientist, made careful measurements

with some of the most sensitive instruments of the time, and figured out that G was equal

to about 6.67 * 10^-11 N*m^2/kg^2.

So indeed, Newton was right about big G having to be quite small.

But even though he didn’t know the exact value of big G at the time,

Newton had enough to establish his law of universal gravitation.

He described gravity as a force between any 2 objects, and published his equation for calculating that force.

Then Newton took things a step further -- well, technically three steps further.

About 50 years earlier, an astronomer named Johannes Kepler had come up with three laws

that described the way orbits worked.

And those predictions almost perfectly matched the orbits that astronomers were seeing in the sky.

So, Newton knew that his law of universal gravitation had to fit with Kepler’s laws,

or he’d have to find some way to explain why Kepler was wrong.

Luckily for Newton, his law of gravitation not only fit with Kepler’s laws,

he was able to use it, in combination with his three laws of motion and calculus, to prove Kepler’s laws.

According to Kepler, the orbits of the planets were ellipses -- as opposed to circles --

with the Sun at one focus of the ellipse -- one of the two central points used to describe how the ellipse curves.

And that’s what’s known as Kepler’s first law, and it actually applies to any elliptical

orbit -- not just those of the planets.

Our moon’s orbit around Earth is also an ellipse, and Earth is at one focus of that ellipse.

Kepler’s second law was that if you draw a line from a planet to the sun, it’ll always

sweep out the same-sized area within a given amount of time.

When Earth is at its farthest point from the Sun, for example, over the course of one day

we’ll have covered an area that looks like a very long, very thin, kinda-lopsided pizza slice.

And when we’re at our closest point to the Sun, one day’s worth of the orbit will sweep

out an area that’s more like a short, fat pizza slice.

Kepler’s second law tells us that if we measure them both, those two pizza slices

will have the exact same area.

His third law is a little more technical, but it’s basically an observation about

what happens when you take the longest -- or semimajor -- radius of a planet’s orbit

and cube it, then divide that by the period of the planet’s orbit, squared.

According to Kepler, that ratio should be the same for every single planet --

and now we know that it is, almost exactly.

For every single planet that orbits our Sun, that ratio is either 3.34 or 3.35.

And! Newton was able to explain why the actual, observed orbits in the night sky sometimes

deviated very slightly from Kepler’s predictions -- for example, by having those slightly different ratios.

What Kepler didn’t know, and Newton figured out, was that the planets and moons were all

pulling on each other, and sometimes, that pull was strong enough to change their orbits just a little bit.

There’s one more thing we should point out about Newton’s law of universal gravitation,

which is that it fits what we expect the equation for a net force should look like, according to Newton.

From Newton’s second law of motion, we know that a net force is equal to mass times acceleration.

What the law of universal gravitation is saying, is that when the net force acting on an object

comes from gravity, the acceleration is equal to the mass of the bigger object -- like Earth --

divided by the distance between the two objects, times big G.

So, you know how we’ve been describing the gravitational acceleration at Earth’s surface as small g?

Well, small g is actually equal to big G, times Earth’s mass, divided by Earth’s radius, squared.

...math!

And we can use this equation for gravitational acceleration to help NASA out with a challenge

they’re grappling with right now.

We want to send humans to Mars. But we have to make sure that their spacesuits will work

properly in Martian gravity.

One way that NASA tests spacesuits is by flying astronauts on special planes --

sometimes called Vomit Comets.

They fly in arcs that let the spacesuit-testers experience reduced weight -- or none at all

-- for short periods of time.

To simulate Martian gravity, the flight plan will need to aim for the gravitational acceleration

you’d experience if you started hopping around on the surface of Mars.

So, what would that acceleration be?

Well, from Newton’s law of universal gravitation, we know that the acceleration of stuff at

Mars’s surface would be equal to big G, times the mass of Mars, divided by Mars’s radius squared.

We also happen to know Mars’s mass and radius already, which ... helps.

So, plugging in the numbers, we can calculate the gravitational acceleration at Mars’s surface:

it should be about 3.7 meters per second squared.

That’s the acceleration you’d experience on Mars, and what the Vomit Comet pilots try to attain when they fly --

about 38% of the acceleration that you experience when you jump off the ground here on Earth.

So, hundreds of years after Newton’s day, NASA is still using his math.

Yeah, I’d say he was a pretty big deal.

Today, you learned about how Newton came up with his law of universal gravitation.

We also talked about Kepler’s three laws, and calculated the gravitational acceleration on the surface of Mars.

Crash Course Physics is produced in association with PBS Digital Studios. You can head over

to their channel to check out amazing shows like Deep Look, The Good Stuff, and PBS Space Time.

This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio

with the help of these amazing people and our equally amazing graphics team is Thought Cafe.