Newton's Law of Universal Gravitation

Summary

In this video, Professor Dave explains Newton's law of universal gravitation, which describes the gravitational force that governs both planetary motion and objects falling on Earth. He uses thought experiments, like cannonball trajectories, to illustrate how objects in orbit are in constant free fall towards Earth. The video also explores how gravitational forces act between all objects with mass, though typically only noticeable with large bodies like planets. It touches on the concept of gravity as a field force, later expanded by Einstein's theory of relativity, and invites viewers to continue learning about modern physics.

Takeaways

• 🌍 Newton's law of universal gravitation describes how all objects with mass exert gravitational force on each other.
• 🌕 The same force that causes planets to orbit the Sun is what makes objects fall toward Earth.
• 💡 Newton understood that planets are in free fall toward the Sun, just as an apple falls toward Earth.
• 🚀 The thought experiment of a cannonball fired with immense speed illustrates how orbit works.
• 🛰️ Satellites and space stations orbit Earth because they are in a continuous state of free fall.
• 📏 The gravitational force between two objects depends on their masses and the distance between their centers.
• ⚖️ Every object with mass, from a car to a refrigerator, exerts gravity, though it is negligible compared to Earth's.
• 🧮 Newton developed an equation for gravitational force: F = G * (m1 * m2) / r^2.
• 🍎 The same gravitational force applies to all objects, regardless of mass, causing them to fall at the same rate if we ignore air resistance.
• 🔭 Later scientists, including Einstein, advanced Newton's ideas, explaining gravity as a field force, and general relativity expanded our understanding of space and gravity.

Q & A

What is Newton's law of universal gravitation?

Newton's law of universal gravitation states that every object in the universe that has mass exerts a gravitational force on every other object with mass. This force is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

How did Newton connect gravity with planetary motion?

Newton realized that the same force causing objects to fall towards Earth, gravity, also keeps the planets in their orbits around the Sun. He proposed that planets are in free fall towards the Sun, just like an apple falling towards Earth.

What is the significance of Newton's cannonball thought experiment?

Newton's cannonball thought experiment explains how objects can stay in orbit. If a cannonball is fired with enough speed, it would continuously fall towards Earth but never hit it, because it falls at the same rate as Earth's curvature. This is similar to how satellites and planets remain in orbit.

Why do all objects fall to Earth with the same acceleration?

All objects fall with the same acceleration due to gravity, regardless of their mass, because while heavier objects experience a stronger gravitational force, they also have more inertia, balancing out the acceleration. This acceleration is approximately 9.8 meters per second squared.

What is the equation for Newton's law of universal gravitation?

The equation for Newton's law of universal gravitation is F = G * (m1 * m2) / r², where F is the gravitational force, G is the gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²), m1 and m2 are the masses of the two objects, and r is the distance between their centers.

How did Newton calculate the gravitational force between two objects?

Newton's formula for gravitational force multiplies the masses of the two objects, then divides by the square of the distance between their centers. The result is multiplied by the gravitational constant (G) to determine the force.

Why do the Earth and Moon rotate around a common center of mass?

The Earth and Moon exert equal gravitational forces on each other, causing them to rotate around a common center of mass. However, because the Earth is much more massive, this center of mass is located inside the Earth, making it seem like only the Moon orbits the Earth.

Why doesn't the Earth's acceleration towards a falling object like an apple appear noticeable?

The Earth's mass is so large compared to an object like an apple that its acceleration due to their mutual gravitational attraction is negligible, making it imperceptible. Meanwhile, the apple's acceleration is noticeable due to its much smaller mass.

How did Newton's work on gravity revolutionize our understanding of celestial and terrestrial motion?

Newton's work unified the concepts of terrestrial motion (like falling objects) and celestial motion (planetary orbits) under the same gravitational force, providing a framework for understanding how gravity governs the movement of all objects in the universe.

How did Einstein's theory of relativity advance Newton's understanding of gravity?

Einstein's general theory of relativity described gravity not just as a force, but as the curvature of spacetime caused by mass. This more sophisticated understanding of gravity explained phenomena like how large objects like planets and stars influence the structure of space and time.

Outlines

00:00

🌍 Newton's Law of Universal Gravitation and Planetary Motion

This paragraph introduces Newton's law of universal gravitation, describing how Newton realized that the same force causing objects to fall on Earth also governs the motion of planets around the Sun. Newton proposed that planets are in free fall towards the Sun in a similar manner to how objects fall towards Earth. The paragraph explains Newton's thought experiment using a cannonball to illustrate how an object in orbit falls continuously but never reaches the ground. It also draws a parallel between satellites and space stations orbiting Earth in the same fashion, thanks to their high speed. This gravitational force applies to all objects with mass, though its effects are more noticeable when near large bodies like planets.

05:01

🍎 Understanding Gravitational Acceleration

This paragraph discusses the concept that all objects, regardless of their mass, fall towards Earth with the same acceleration, assuming no wind resistance. The idea might seem counterintuitive, but it is explained through Newton's second law, where the gravitational force acting on an object is proportional to its mass, but so is its inertia, leading to uniform acceleration. The paragraph concludes with a derivation showing how the acceleration due to gravity is a function of Earth's mass and radius, not the object's mass. It emphasizes the revolutionary nature of Newton's work in explaining both terrestrial and celestial motion, while also mentioning that Newton couldn't fully explain how objects exert gravitational force across distances, leaving this to later advancements by scientists like Einstein.

Keywords

💡Newton's Law of Universal Gravitation

This is a fundamental principle proposed by Isaac Newton that states every object in the universe attracts every other object with a force proportional to their masses and inversely proportional to the square of the distance between them. In the video, this law is introduced as the force governing both the motion of planets in orbit and the falling of objects toward Earth, such as the apple in the famous story.

💡Gravitational Force

Gravitational force is the attractive force between two objects that have mass. In the video, this force is described as the same one responsible for both the planets' orbits around the Sun and objects falling towards Earth. The explanation includes the cannonball thought experiment to show how gravitational force leads to orbiting motion.

💡Centripetal Force

Centripetal force is the inward force required for an object to follow a curved or circular path. In the video, Newton recognizes that the centripetal force acting on planets in orbit is caused by gravity. This force keeps the planets from moving in a straight line and instead pulls them in a curved orbit around the Sun.

💡Free Fall

Free fall occurs when an object moves under the influence of gravity alone, without resistance from other forces like air resistance. The video uses the example of both satellites orbiting the Earth and Newton's apple to describe how objects can be in a constant state of free fall but never hit the ground due to the curvature of the Earth or the Sun.

💡Orbit

An orbit is the curved path of an object around a star, planet, or moon due to gravity. The video explains that objects in space, such as satellites or planets, stay in orbit because they are in continuous free fall, with their motion balanced by the gravitational pull of the larger body they are orbiting around.

💡Gravitational Constant (G)

The gravitational constant, denoted by 'G', is a fundamental physical constant used in the calculation of gravitational force between two masses. Its value is 6.67 × 10⁻¹¹ N·m²/kg². The video explains that this constant was experimentally determined by Henry Cavendish and is essential for quantifying the gravitational force between objects.

💡Newton's Second Law of Motion

Newton's second law states that force equals mass times acceleration (F = ma). In the video, this law is applied to explain how the force of gravity affects different objects, showing that while the gravitational force on more massive objects is greater, their acceleration is the same as less massive objects because of their greater inertia.

💡Inertia

Inertia is the resistance of an object to changes in its state of motion. In the video, inertia is mentioned when discussing why more massive objects fall at the same rate as less massive ones despite experiencing greater gravitational force—because they also have more inertia, balancing the effect of the force.

💡Center of Mass

The center of mass is the point at which the mass of a system or object is concentrated. In the video, this concept is used to explain how the Earth and Moon both exert gravitational force on each other and rotate around a common center of mass, though the center lies within the Earth due to its greater mass.

💡Einstein's General Theory of Relativity

Einstein's theory of general relativity redefined the understanding of gravity, proposing that it is not a force but the result of the curvature of spacetime caused by massive objects. The video briefly mentions this theory as a more advanced framework that helps explain the structure of space and the formation of celestial bodies, advancing beyond Newton's classical theory of gravity.

Highlights

Newton's law of universal gravitation links the motion of planets to the same force that causes objects to fall towards Earth.

Newton realized that the planets are in a state of free fall toward the Sun, much like objects fall towards Earth.

The cannonball thought experiment demonstrates how an object fired with immense force would orbit the Earth rather than hitting the ground.

Satellites and space stations are in a continuous state of free fall around Earth, orbiting indefinitely due to their speed and altitude.

Newton's equation for gravitational force involves the constant of universal gravitation (G) and the masses of two objects, divided by the square of the distance between them.

Gravitational force is exerted by all objects with mass, though only massive objects like planets generate noticeable effects.

Newton's insight was so profound that he had to invent calculus to explain his law of gravitation mathematically.

Both Earth and the Moon exert equal gravitational forces on each other, but because Earth is far more massive, the Moon accelerates much more.

When an apple falls towards Earth, both the apple and Earth accelerate towards each other, but Earth's movement is negligible due to its massive size.

Objects of different masses, like an apple and a bowling ball, fall at the same rate towards Earth, as their acceleration due to gravity is the same.

Gravitational acceleration (9.8 m/s²) is derived from the gravitational constant, Earth’s mass, and the distance to Earth's center.

Newton's work unified terrestrial and celestial mechanics, showing that the same force governs both.

Newton could not explain how gravity acted across distances, a problem later addressed by defining gravity as a field force.

Einstein's general theory of relativity provided a more sophisticated understanding of gravity and its role in shaping space.

The study of gravity continues today, building on Newton and Einstein's work to understand the structure of the universe.

Transcripts

Professor Dave here, I want to tell you

about Newton's law of universal gravitation.

We learned about Newton's

laws of motion, but there's one more law

of his to discuss, and it's a big one.

It's Newton's law of universal

gravitation. In possibly one of the

greatest strokes of genius in the

history of mankind, Newton looked at the

motion of the planets in their nearly

circular orbits around the Sun and

understood that the centripetal force

causing this motion was precisely the

same force that causes objects to fall

down towards Earth, which we call the

gravitational force. In this way he

proposed that the planets are in a kind

of free fall towards the Sun just the

same way that the Apple that mythically

hit him on the head to produce this

insight was in free fall towards the

earth. The objects differ greatly in size

but the concept is the same. Newton

corroborated this notion with a thought

experiment. When a cannon fires a cannon ball,

the ball eventually hits the ground.

If another cannonball is fired with

greater force, it will go a little

further before hitting the ground. If a

cannonball could be fired with an

incredibly immense force, it could

produce a speed so great that the ball

would never hit the ground, since it

would fall at the same rate that Earth's

curvature is produced. It would thus

always be falling towards the earth but

never hitting it. Such an object would be

said to be in orbit around the Earth.

Of course no cannon can do this, but we have

finally achieved this feat with all of

our satellites and space stations, which

are brought up to orbit on rockets .These

are very far from Earth's surface and

they are moving with such great speed

that they, along with anyone on board, are

always falling towards the earth but

never hitting it, in a free fall just

like Newton's apple. This means they

orbit around the Earth indefinitely at a

fixed speed and radius. The same can be

said for all the planets around the Sun.

This gravitational force can describe

the motion of every object

in space and it is the case that every

object that contains mass will exert

gravity on every other massive object.

Of course to feel the effects of gravity we

must be near an enormous object, like a

planet, but it is completely accurate to

say that gravitational force is exerted

by your car, your refrigerator, even you

yourself. It is just that this force is

completely negligible compared to the

gravitational force exerted by the earth.

Newton developed an equation to quantify

the magnitude of the gravitational force

between two objects, and it looks like

this, where F is equal to the constant of

universal gravitation, G, times the mass

of the first object, times the mass of

the second object, divided by the

distance between them squared. This

constant, like any other constant, simply

exists so that a natural phenomenon like

gravity can be expressed in our own

arbitrary man-made units, and it is equal

to 6.67 times 10 to the negative 11

Newton meters squared over kilogram

squared. These are the units that will

cancel out the units on the masses and

radius so as to give a value for force

in newtons. This value is not known to

Newton at the time but was determined

experimentally about a hundred years

later by Henry Cavendish. When discussing

the radius between two objects we will

take the distance between their centers

rather than their surfaces, as Newton

showed that the gravitational force

exerted by an object depends only on its

mass and not on its volume, meaning that

when discussing gravity we can treat

everything as a point like mass. He had

to invent the calculus to do so, much to

the dismay of math students everywhere.

When examining a system like the earth

and the moon, we must understand that

both of these objects exert

gravitational force on the other, and

that these forces are equal in magnitude

meaning that both of these bodies rotate

around their combined center of mass, but

don't forget that F equals ma, so equal

forces will not produce equal

accelerations if the masses are

different. As it happens, the earth is

much much more massive than the moon so

the mutual gravitational force is able

to accelerate

the moon more than the earth, and the

center of mass for the system lies

within the earth itself, which is why we

simply observe the moon going around the

earth. The same can be said for Newton's

falling apple. The apple accelerates

towards the earth and the earth

accelerates towards the apple, but the

earth is more massive than the apple by

an inconceivable factor, so the

acceleration of the earth is not even

measurable whereas we can visually

confirm the acceleration of the apple.

Furthermore, we want to understand that

an apple will fall to the earth with the

same acceleration as a bowling ball or

any other massive object, if we disregard

wind resistance. Although

counterintuitive to some, we can

rationalize this if we understand that

while the force of gravity is able to

impart greater acceleration on a more

massive object, the more massive object

also has greater inertia or resistance

to being accelerated, so the end result

is that all objects accelerate towards

Earth in the same way, at 9.8 meters per

second squared. This fact is easy to

derive if we do some algebraic

manipulation. We know that a falling

object exhibits behavior according to

Newton's second law, F equals ma, where

the force that generates the falling is

equal to the mass of the object times

its acceleration, but this force is the

gravitational force, so we can also model

the falling behavior with G m1 m2 over r

squared, where m 1 is the mass of the

object and m2 is the mass of the earth.

If we set these equal to each other, the

mass of the object is found on both

sides and will cancel out, so we can see

that the acceleration due to gravity is

equal to the gravitational constant

times the mass of the Earth divided by

the radius squared. This means that the

mass of an object does not affect the

rate of free fall.

Newton's work on gravity

was revolutionary. It correlated an

incredible amount of data, from

terrestrial motion to celestial motion,

which is all any good theory can hope to

do. But he could not explain how objects

can exert the gravitational force on one

another from a distance. Later, scientists

solved this problem by labeling gravity

as a field force, stating that matter

generates gravitational fields in space.

This was a bit more satisfactory, but it

wasn't until Einstein's general theory

of relativity that we arrived at a more

sophisticated understanding of gravity,

which has helped us learn about the

structure of space itself, as well as how

planets and stars and galaxies form. We

are still trying to fully understand

gravity today, but the continuation of

this discussion will have to wait until

the modern physics course, so for now

let's check comprehension.

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