This comment by NGD deserves to be at the top of the page, and I wish I had said exactly this in my video!!!! If Brahe hadn't collected the planetary data, and if Kepler hadn't studied it, then Newton wouldn't have known that F should be proportional to 1/r². Some fun history: Newton wasn't the first to propose that the planets felt a 1/r² force from the Sun. Newton's contribution was to explain where that 1/r² force came from and why a planet could be treated as though all of its mass was concentrated at the center. He also worked out the math for elliptical (as opposed to perfectly circular) orbits. Ismaël Bullialdus and Robert Hooke also did some work on that idea of a 1/r² force pulling the planets toward the center of their orbit.
@@gloystar that's a really great question. Physics teachers / textbooks will often show Newton's law of universal gravitation, then present Kepler's third law, and derive Kepler's third law from Newton's law of gravitation. Historically, that's somewhat backwards from what happened. This might be a more accurate way to put it: Kepler's third law was well-known, and it was also believed (prior to Newton's work) that planets must be feeling a force proportional to 1/r^2 keeping them in orbit. But no one knew where that force came from. Any attempts to propose an explanation would *have* to explain Kepler's third law. So the ability to derive Kepler's third law was a piece of evidence that supported Newton's law of universal gravitation. Given all that, I think it's still fair to say that Kepler's third law can be derived from Newton's law of universal gravitation, since gravity is more fundamental, and orbital motion arises due to it. But the evidence of planetary motion definitely led to the discovery of universal gravity (not the other way around).
@@danielm9463 That's exactly what I was thinking. I think Kepler's laws came from pure observations at first, and I've read somewhere that Newton already knew three things: 1) moon was 60r away, 2) Acceleration at surface of earth (g)=9.8, 3) Orbital period of the moon=27.3 days. Using this, he calculated g at moon's distance (as a centripetal acceleration) and compared that with g at earth's surface, and found out that it was 3600 times weaker, which is 1/(60)^2. This way, he knew that F is inversely proportional to r^2.
@@gloystar If that source is accessible to you, I'd be very grateful if you'd share it! That's a clever approach, and I hadn't read that this was part of Newton's calculation. What you said about Kepler's law coming from pure observation is exactly correct, historically speaking. (Kepler used the astronomical observations and data that Tycho Brahe collected.) What I've read is that, when Newton started working on this problem, much of the scientific community already believed that the planets felt a 1/r^2 force, and this came from Kepler's 3rd law. Connecting that to the moon would make a lot of sense. I know a huge part of what Newton did (which is rarely mentioned in high school contexts) was to show that this all works out for elliptical orbits. That required calculus, and if I'm remembering correctly, it also required Newton to prove that, in terms of gravitational effects, a spherical body can be treated as though all of its mass is located at a single point (the center of mass). I think Newton showed that calculation, too.
This is brilliant and despite my education and background in physics, this is the first time I’ve seen a clear argument as to how Newtons law of Gravity was derived. Thank you for putting this together. You’re one of the clearest teachers I’ve learned from.
I was searching for this explanation for a long time now and I can't believe how brilliant you're at explaining it ... I am in love with your way of explanation really 🥰💕 thanxxx
very informative. I always had queries why there was a multiplictive mathmatical operation on Gravitational Force and also Electrostaric Force. this has opened a lot of my mind...
Thank you Daniel for explaining this topic so simply and illustratively; now I am equipped with the understanding and intuition of gravity. I hope this video and you, won't be forgotten in the depths of youtube, and can still educate people in future.
Great derivation! I'm not sure the derivation for the inverse square relation using Kepler's third law is that clear (to me at least). I would go from F_g = 4*pi^2*m*r/t^2. And use Kepler's law to say t^2 is proportional to r^3. Then F_g is proportional to 4*pi^2*m/r^2. Then F_g is proportional to m/r^2 and hence the inverse square law follows
This is a great question. The approach you've described is totally logical, and it's one I considered when making the video. There were two reasons I didn't go that route. The first is that it gives us an expression for the force of gravity, F_g = (4π²m)(k/r²). This equation makes it seem that Newton's law of gravitation should have 4π²m somewhere in the numerator, and it would require a pretty nuanced explanation of why we don't actually expect that to appear. The second thing about writing F_g = (4π²m)(k/r²) is that it *appears* to suggest that F_g ∝ m. But of course, this equation F_g = (4π²m)(k/r²) isn't sufficient to infer that F_g ∝ m, and it's not the reason why Newton thought F_g ∝ m. Rather, the idea that F_g ∝ m and F_g ∝ M flowed from his understanding of his second and third laws of motion. I worried that isolating F_g and showing the equation F_g = (4π²m)(k/r²) would muddy the waters, and students might wonder: "Wait, why do we now have a second equation F_g = GmM/r² if we already know that F_g = (4π²m)(k/r²)? How do we reconcile the two?" I thought it would be better to show how, whatever we substitute in for F_g, it must provide the missing piece which completes Kepler's 3rd law. From the equation F_g = 4π²mr/T², the only way to complete Kepler's 3rd law is to substitute into F_g with k/r². I also like the approach of having students build intuition and learn to spot similarities in form between two different equations.
Great video! I would like to ask a question though. I've read that the masses that appear in this equation are gravitational masses and I don't understand why would someone distinguish between gravitational and inertial mass. Did Newton define these gravitational masses or did he just assume that W=m(inertial)g? If he did assume that W=m(inertial)g and he derived his law of gravitation using this assumption then I don't see why would someone think that the masses that appear in the universal law of gravitation would be different from the inertial masses in Newton's second law.
This is a great question. The short answer is that both of these equations (Fg = GmM/r^2 and Fnet = ma) can be used as a definition of mass itself. Since we have two different equations/definitions, the question naturally arises: do the two definitions always yield the same value of mass? In the first case (using the Fg equation), the equation implies a conceptual definition of mass along the lines of "the property of matter that creates gravitational attraction." In the second case (using the Fnet equation), the conceptual definition of mass is "the property of matter that resists acceleration when a net force is applied." Because these definitions are tied to equations, the implication is that we can (a) experimentally determine the gravitational mass using Fg = GmM/r^2, and (B) experimentally determine the inertial mass using Fnet = ma, then (c) compare to see if they match. Here's how that would work. We have to start by assuming we know the value of G through independent mechanisms. Imagine we have three objects A, B, and C. We place A and B in close proximity and measure their gravitational force F1 and separate distance r. Then we do the same for B and C, and for A and C. We set up a system of three equations: F1 = (G)(m_a)(m_b)/r^2 F2 = (G)(m_a)(m_c)/r^2 F3 = (G)(m_b)(m_c)/r^2 Since we know F1, F2, F3, G, and r, we can experimentally determine the 3 gravitational masses m_a, m_b, and m_c. Next, we can apply a net force F4 (which we measure) to each object. We measure each mass's resulting acceleration. We can again calculate the masses, from this second equation: m_a = F4 / a_a m_b = F4 / a_b m_c = F4 / a_c So the question is: would the values of m from the first experiment match the values of m from the second experiment? The true picture is a bit more complicated, because we don't have an independent way of measuring the universal gravitational constant G except using preexisting measurements of inertial mass m and M. This changes the question slightly from "is gravitational mass always the same as inertial mass" to "is the ratio of gravitational mass and inertial mass always the same?" This question quickly moves us into some of the core principles behind general relativity. Here's a nice Quora post by Viktor T. Toth that describes this in a bit more detail: www.quora.com/Is-gravitational-mass-and-inertial-mass-equivalent-complementary-different-or-totally-independent
Beautiful video thank you!! But I have a question: As far as I see, we used four fundamental laws during the derivation process: Newton's first, second, and third laws, and Kepler's third law. Newton's second law says F=ma and Kepler's third law says (T^2)/(r^3) is constant. Are these two laws empiric and observational, or were they derived from other laws?
I really love these questions, and I don't know if I've ever considered it that deeply, but it's worth doing. Kepler's 3rd law is definitely empirical and was discovered only through observation. It's also very much a description of observed patterns, which might be different from Newton's law of universal gravitation (or more broadly, Einstein's theory of general relativity)--both of which explain observation but are theoretically deeper, broader-reaching, and explain a diverse set of phenomena. I'm not sure if Newton's second law is empirical--it might actually be a mathematical definition of force. Force (similar to energy) isn't something we see or observe directly--I think of it as more of a mathematical abstraction / more removed from what we observe than, say, distance or speed. I really don't know how to characterize Newton's third law. I have some initial thoughts but would love to hear what you think.
Hi again, sorry for my late answer. I have been busy recently. I have thought about Newton's second law (F=m*a) and noticed that the definition of the terms in the formula can help us to understand it on a deeper level. I tried to understand the definition of these terms and tried to put them into my own words. Now, let's focus on what acceleration is and derive the second law from that. I think motion has two parts. The first is velocity, I say velocity is the vectorial change in an object's location per one unit of time. The second is acceleration, and I say acceleration is the vectorial change in an object's current behavior of changing its location per one unit of time. Now we notice that all this argument is about the change of things. Naturally, some phenomenons support change while some hinder it. Right here, I define force as any support and mass as any resistance to the change in an object's velocity. Consequently, acceleration is proportional to force and 1/mass, and we call this proportion Newton's second law. Let's look at Newton's first law to understand these definitions better: If no vectorial impact supports the change in velocity (net force = 0), velocity does not change (acceleration = 0) and the object maintains its current velocity. We can see this kind of relationship between support and hindrance to change in many places in nature: Voltage = Electric Current * Resistance etc. I would like to hear my possible mistakes, thanks for your interest.@@danielm9463
Really great question! It might be a slightly odd notation, because a proportionality usually indicates how two (and *only two*) variables are related to each other. The easiest way to see why the combined proportionality is valid is to work backwards. Let's start with a hypothetical equation y = c•x•z, where c is a constant and x and z are variables that can take on different values. As x and z change, so does y. We first want to understand how a change in z affects y. So we assume that x is fixed/constant, and then we write a proportionality between y and z by removing any constants from the equation--c and x. The equation y = cxz thus becomes y ∝ z. This tells us y is directly proportional to z. Next, we want to understand how a change in x impacts y. So we assume that z is fixed (unchanging), and then we rewrite the equation as a proportionality by taking out the constants (c and z). This gives: y ∝ x. But we've just uncovered something interesting. If we start with the two proportionalities (y ∝ z and y ∝ x), we could combine them to get y ∝ x•z. If we then added a constant c, we get back to the original equation: y = c•x•z. This procedure is valid in general: whenever we have two proportionalities, we can combine them together into a single proportionality. If we also add in a constant, then we can change the multiple proportionalities into a single equation.
I’ve read through the comments and I recognize you used Kepler’s law to derive Newtons, which is how, I gather, it occurred historically. The classes I’ve been in have typically used Newtons law to show Keplers law. Is there a chance you could do a video explaining the kind of data Kepler had gathered as well as the process he might’ve used to develop his laws? I don’t know if it would make a great video, I recognize that most of it is just parsing through collected data, but I feel like it would give a more complete understanding to how Newton or someone in his position rather, could have developed his laws.
These are great questions. So, here's a more nuanced understanding that I have: when Newton wrote the Principia, he actually did the derivation using data about the moon (its acceleration & distance from Earth). He presented this data in a "derivation" of the inverse-square component of his law of universal gravitation. After fleshing out the full law of universal gravitation, he showed that it explains Kepler's data (and he used calculus to do the calculations for elliptical orbits, which was a major achievement and was thought to be an impossible task). So the presentation in the Principia doesn't exactly match what I present in this video. However, the way Newton presents it in the Principia almost certainly isn't how he thought about it. Other physicists had already looked at Kepler's data and suggested that an inverse square force would explain the planets' motion if they were traveling in perfect circles, and Newton was aware of this work. (No one knew how to show this for elliptical orbits, and Newton's application of calculus was a very important part of the Principia.) Here's a link to a Quora answer that has a lot more detail and depth, which I recommend: qr.ae/pGziaX
Really excellant man u have explained why it is r square rather than simply saying its r square like that can u explain why G value is 6.67 why it cant be 7
Ultimately, we simply find ourselves in a universe where the universal gravitational constant is 6.67408 x 10^-11 N m^2 / kg^2. We don't have a reason why G has this value, but it's a good thing, because if G were much higher then the universe would only consist of black holes, and if G were much lower then stars and planets would have never formed. PBS Space Time has a video series on this "fine-tuned" nature of our universe. I believe this is the first video in a 3-part series: czcams.com/video/8wa1l7M5gU8/video.html
Good question! The v represents speed rather than velocity. The equation Fnet = mv^2/r only gives the *magnitude* of the net force. The *direction* of the net force is always toward the center of the circle, for uniform circular motion.
@@muslimwoolfy-winterequestr4344 Great question! There are two ways to answer it. If we follow the logic of the video, here's the answer: we need the force to be directly proportional to each mass and this isn't true when we have m + M. If we had F = G(M+m)/r^2, then F wouldn't be directly proportional to M or m because doubling M or doubling m won't necessarily double F. But with F = GmM/r^2, then F is directly proportional to both m and M. The other way to explain the answer is: the equation is written to describe what we see happening in nature and in the universe. The equation is written to follow our empirical observations/data, and multiplying mM fits the data (whereas m+M doesn't fit the data).
Great explanation. But I still don't understand how the force is proportional to the earth's mass. I mean, I see that the force should be proportional to the apple's mass. That follows from Kepler's 3rd law and F = ma, or rather F = mv^2/r. What I don't understand is how it follows from the third law that the force is proportional to the earth's mass, since the earth isn't moving at all. Meaning we can't apply F = Mv^2/r to it. Would anyone mind explaining that to me?
Great question! When we look at a falling object, we see that Fg = mg and Fg ∝ m. (This follows from Newton's 2nd law, Fnet = ma, not from Kepler's 3rd law and also not from Fnet = mv^2/r.) But this fact that Fg ∝ m is still true even when the object isn't falling. After all, gravity is always present, including when objects are stationary. It prevents us from floating up into the air. Furthermore, we're not heavier or lighter when falling--our weight is constant. So this property that Fg ∝ m is true in general, whether the object is moving or stationary. Once we establish the general principle that Fg ∝ m, we can apply it to the Earth itself. The Earth doesn't have to be falling for Fg ∝ m to be true--the Earth simply must feel a gravity force. And we know the Earth feels a gravity force because of Newton's Third Law. (If a ball is pulled down by the Earth with a gravity force Fg, then the Earth is pulled up by the ball with an equal force Fg.) So when we say that the Fg ∝ M, we're not relying on anything about the Earth's motion. We're simply relying on the fact that the Earth feels a gravity force (the reaction to the ball feeling a gravity force), and any gravity force on an object is proportional to the object's mass.
+Daniel M Thanks for the quick reply. Regarding my statement about Kepler's 3rd law, you can mathematically deduce that the gravitational force between the earth and the sun depends on the mass of the earth and the inverse of the square of the distance from it. Here's the link if you're interested: www.quora.com/How-can-Newtons-law-of-gravitation-be-derived-with-Keplers-law Scroll to the very bottom and you should find the derivation. But I wasn't able to understand how he concluded that the force depends on the sun's mass as well. I came on youtube hoping to find the answer to that question and yours was actually the only video I could find related to the topic. Your method is slightly different but here too I failed to understand why the force should depend on the other object's mass. I've read your explanation thoroughly but I'm afraid I still don't understand. I promise I'm not being obtuse on purpose. I guess what I'm having trouble with is the fact that Newton's 3rd law says that the force acting on a body should be independent of its mass. For example, if two balls collide, then both balls exert the same force on each other. It doesn't matter is one ball is tiny and the other massive. They will still exert the same force on each other. But by saying Fg ∝ M, we're saying that if the sun were more massive, the reaction force exerted by the earth on the sun would be greater. To me, this seems to be at odds with Newton's 3rd law. Hope you can clear this confusion.
Thanks for the link! In the post you mentioned, they derive the equation for universal gravity using both (a) Kepler's 3rd law and (b) Newton's 2nd law (a = F/m). The mass m is introduced via Newton's 2nd law rather than via Kepler's 3rd law. No problem at all--asking questions is always productive! In fact, Newton's 3rd law doesn't state that the force must be independent of mass. It's often true that the action and reaction forces don't depend on mass, but it's not **required** by Newton's 3rd law. Instead, the 3rd law simply states that, when two objects interact, they push on each other with equal forces in opposite directions. Newton's 3rd law states that any stipulations we make (like 'Fg ∝ M' and 'Fg ∝ m') must apply to both the action force and the reaction force. So let's imagine the Earth (of mass m) and the Sun (of mass M) pulling on each other. If the Earth's mass m suddenly grew a little, then the force of gravity Fg on the Earth would increase, and so would the force of gravity Fg on the Sun. The two forces would increase and still be equal. Likewise, if the Sun's mass M suddenly grew a little, then the force of gravity Fg on both the Earth and Sun would suddenly become larger, but it would still be equal.
+Daniel M Thanks for the reply. I understand the logic behind the argument but I tried doing it mathematically and wasn't able to. Which is weird because if it is logically sound, then proving it mathematically should be no big deal. And yet there just isn't a way to bring the sun's mass into the equation without directly stipulating that the force depends on it. This seems odd to me.
The equation we used is Fg ∝ m, where m is the mass of any object that feels a gravity force. The falling ball feels a gravity force, and thus Fg ∝ m. The Sun feels the same gravity force, and thus Fg ∝ M. According to Newton's third law, those two forces are the same. That was one of Newton's huge insights with the law of universal gravitation--extending the gravity force Fg to all objects.
The fact that the ratio of T^2/R^3 is the same for every planet is called Kepler's 3rd law. In Newton's time, it was known experimentally through data, by observing the planets. It wasn't derived--it provided evidence that supports Newton's law of universal gravitation.
The gravitational constant didn't come until much later, when Cavendish experimentally determined its value, and other scientists modified the proportionality of Newton's law of gravitation, and added the G to promote it to an equation. Prior to Cavendish, all we could know is the GM product of any given astronomical body. His experiment isolated G from M, so that he could "weigh" the Earth.
Daniel, when I pull an object, the object will fall on the surface of the earth. If i will pull an object on the string with huge force and the object is light the object will fall on the surface on a trajectory close to a vertical path. Make an experiment.
The G was added later thanks to the Cavendish Experiment. If you do a google or a youtube search, you'll get a ton of good hits on how it was conducted. It involved the hanging torsional pendulum that I showed a clip of.
Really great question! It might be a slightly odd notation, because a proportionality usually indicates how two (and *only two*) variables are related to each other. The easiest way to see why the combined proportionality is valid is to work backwards. Let's start with a hypothetical equation y = c•x•z, where c is a constant and x and z are variables that can take on different values. As x and z change, so does y. We first want to understand how a change in z affects y. So we assume that x is fixed/constant, and then we write a proportionality between y and z by removing any constants from the equation--c and x. The equation y = cxz thus becomes y ∝ z. This tells us y is directly proportional to z. Next, we want to understand how a change in x impacts y. So we assume that z is fixed (unchanging), and then we rewrite the equation as a proportionality by taking out the constants (c and z). This gives: y ∝ x. But we've just uncovered something interesting. If we start with the two proportionalities (y ∝ z and y ∝ x), we could combine them to get y ∝ x•z. If we then added a constant c, we get back to the original equation: y = c•x•z. This procedure is valid in general: whenever we have two proportionalities, we can combine them together into a single proportionality. If we also add in a constant, then we can change the multiple proportionalities into a single equation.
Ultimately, the law can't be proven. Even this video doesn't offer a proof! The big-picture goal of physics is to come up with laws, equations, and concepts that describe our universe and make predictions. This law is best thought of as descriptive of the gravitational phenomena we see. Logically, we're using induction rather than deduction.
Okay.. did you just use newton's law of gravitation to change Eg to k/r^2?? IF AND ONLY IF that is the case, then the numerator should be the product of masses, right?
Hm... I don't think so. The video never uses the equation for Eg (gravitational potential energy). The goal of this video is to derive Newton's law of gravitation. Here's where k/r^2 comes in: Newton knew that Fg = mv^2/r from circular motion, and he knew that T^2/r^3 = k. If we put those two things together, we can deduce that the Fg must be equal to k/r^2. Newton wasn't the first person to propose this. Bullialdus did so, and Hooke claimed to have originated the same idea that the gravity force must be proportional to the inverse square of distance from the Sun.
but uhh i tried what you told: putting mv^2/r and T^2/r^3 together and im getting a really weird answer lol.. can you explain this in your next vid? Or if you can explain me in the comment section it will be great!! plzzzzzzzzzzzzzz???????
Great question! Kepler's law was empirical--not theoretical. That means the only reason to square T and cube r is because that's what the data shows. Kepler recognized the pattern before anyone could explain it. When they looked at the planetary data, they found that squaring T and cubing r caused the data to follow a linear relationship. Newton was seeking a law that could explain why T was squared and r was cubed.
Kepler derived his third law empirically, by examining data that the great astronomer Tycho Brahe collected. A quote from Kepler: ". . . after I had by unceasing toil through a long period of time, _using the observations of Brahe_, discovered the true distances of the orbits, at last, at last, the true relation . . . overcame by storm the shadows of my mind, with such fullness of agreement between my seventeen years' labor on the observations of Brahe and this present study of mine that I at first believed that I was dreaming . . . ." (from Harmonies of the World) [emphasis added] You can read more here: books.google.com/books?id=czaGZzR0XOUC&pg=PA40&hl=en#v=onepage&q=kepler's%20third&f=false
I have the same doubt. Seeking for a law thats implicitly defined without relying on experimental observations. I came upon it when I encountered Columbs law of force between charges.
@debendra, ultimately, all physical laws must rely on experimental observation. The goals of physics are to describe the universe in the simplest mathematical terms.
@Daniel M, the problem is that either you assume 3 Kepler Laws to derive 1 Newton Law, as opposed to 1 newton Law to derive 3 kepler's laws. The latter is much morre powerful and favorable. Mathematics and Physics tend to assume less and conclude more. In fact, when Kepler went to Newton, where Kepler went almost insane trying to figure out the orbits, Newton derived from his gravitational law that the orbits are elliptical (Kepler's 1st law), that equal areas are swept out in equal times (Kepler's 2nd law), and that the square of period is proportional to the cube of the semimajor axis of the ellipse, or also the radius of orbit (Kepler's 3rd law). Therefore, Newton's law is more powerful then Kepler's laws, which rely on a lot of complex, hard to believe assumptions, whereas Newton's rely on a simple idea of gravitational field being proportional to the inverse square of radius. Which is better, to make a special single, easy tool that can be used in a diverse amount of examples, or many convoluted assumptive tools that are all just to prove 1 statement. If you think about what's more natural, it's obvious.
This is quite late, but better late than never. I have a question. You kept on mentioning that we "almost have Kepler's 3rd law", but why is it that we HAVE to have Kepler's 3rd law in the equation. Furthermore, why does Fg have to be the thing with r^2 in the denominator? What if we multiplied both sides of the equation by r^-2? I get that ((T^2)/(r^3)) is special because it is constant, but why does that allow it to earn its place into the equation. This might be a stupid question. Sorry, I'm in high school.
Definitely not a stupid question! This is a tricky thing to explain, but essentially it comes down to this: if you know you that T^2/r^3 is a constant, then you can demonstrate, as a mathematical consequence, that the circular motion must be caused by an acceleration (and force) proportional to 1/r^2. Before Newton started working on the problem, physicists already knew from Kepler's work that T^2/r^3 was a constant, and people had even suggested the planets are held in orbit by a force proportional to 1/r^2. But until Newton came along, no one knew what caused that 1/r^2 acceleration, or what the rest of the form was, or what other phenomenon it was connected to. Newton filled in all of those pieces. From centripetal acceleration, we know: a = v^2/r We can substitute in v = distance/time = 2πr/T: a = (2πr/T)^2/r = (4)(π^2)(r/T^2) a / 4π^2 = (r/T^2) But we know from Kepler that: k = r^3/T^2 k = r^2 * (r/T^2) Substituting in for r/T^2 gives: k = r^2 * (a / 4π^2) k = r^2 * a / 4π^2 4kπ^2/r^2 = a And this suggests an acceleration proportional to 1/r^2.
@@danielm9463 Ahhhh, That makes sense. I was confused on how k popped up in the equation, but using parts of the v^2/r equation and plugging that into k = r^2((r)/(T^2)) makes a ton of sense. Also interesting on how searching for a known constant in your equation tells you about the nature of, in this case, acceleration around a planet. Very cool! Thank you so much!
@@TahirAhmad-io6uw you put it better than I could have! That's exactly the idea I was trying to get at with this video. I agree--it is a really cool way for the physics to have developed. Good luck in your studies!!
@@danielm9463Thank you for the kind words! I was reading up on how Newton went about discovering his Universal Law of Gravitation, and this video helped me comprehend much of what I was reading. I wish you well!
You are correct! This is an approximation that we make at the high school level because the ellipse shape isn't studied. The more precise formulation of Kepler's third law refers to the semi-major axis.
Exactly...that is why we are also taught Einstein's field equations.....more important is the fact that even light (made up of photons with 0 mass) also gets curved near such planets...so according to Newtons law of gravitation, it is acted upon by 0 force...so?
You're exactly right! Einstein later showed that Newton's law of gravitation is incomplete (rather than wrong). This doesn't change the fact that, when dealing with most classical scenarios, the difference between Einstein's theory and Newton's law is negligible. For example, Newton's law of gravitation was good enough to get astronauts to the moon. So the statement "even light ... also gets curved" is true to an extent. At certain scales and distances, the bending will be negligible. The way people have encouraged me to think of it is like this: Newton's law has its place, and general relativity has its place.
The fact that the planets felt a 1/r² force was known empirically. The planetary data (which Brahe largely collected, I think) revealed a 1/r² force, not a 1/r force.
It comes from the surface area of the sphere of influence, which is proportional to r^2. The gravitational field is inversely proportional to the sphere's surface area. Had we lived in Flatland, it would be 1/r. In 4D hyperspace, it would be 1/r^3. The divergence theorem shows why this must be. Gravity is a non-divergent force field in free space, and only divergent (or rather convergent, with negative divergence) in a region of space with mass. Once external to that mass, gravitational field lines (more accurately: gravitational flux) are no longer being created, and just get redistributed to a smaller concentration. The smaller the concentration of field lines, the smaller the force per unit mass at that location in space.
“The origin of momentum creation is founded on the application of energy. Momentum serves as the initial cause of pushing and pulling. The energy, emanating from the internal core of a planet, is what gives rise to momentum, commonly referred to as gravity, and not due to the planet’s size, mass, or space-time curvature. For example, when energy is applied, it generates the necessary momentum for the actions of pushing and pulling. Without the application of energy, momentum cannot be established. Consequently, force cannot be exerted for pushing and pulling, as it lacks a physical existence. This, in turn, clarifies why force is not gravity.” ~Guadalupe Guerra
I love the poeticism of this, but do you intend it to be physics? If we use accepted physics definitions, then most of these statements are inaccurate.† What do you think, do you view this exposition (which again, I find beautiful) as attempting to do the same thing that physics attempts to do? The goal of physics is to define concepts mathematically in very precise ways and then use those mathematical definitions in equations that describe how the physical world acts, interacts, evolves, etc. So, for example, the statement "force ... lacks a physical existence" is a little silly because, in physics, force is only intended to be a precise mathematical concept that can be used in equations to describe what we observe in the physical world. Force doesn't lack a physical existence any more than momentum, energy, or any other physics concept. They're all mathematical constructs that help us describe the universe through equations. †Note: it's correct that generating momentum requires an input of energy--though technically you could also have an input of mass per E=mc^2. This idea that energy is needed to generate momentum is a consequence of conservation of energy. Momentum is defined as the product of mass and velocity, and thus nonzero momentum (mv > 0) also implies nonzero kinetic energy (0.5*mv^2 > 0). So to generate e.g. 10 joules of kinetic energy requires that the 10 J come from *somewhere* (e.g., from another object's kinetic energy, from potential energy, from chemical energy, etc.). However, Emmy Noether showed that even this principle of energy conservation is not true in general and only holds in the presence of time symmetry. Here's a concrete example: space is expanding. As it does, it generates momentum without any input of energy. Far-off galaxies are speeding up as they move away from us, and their increasing momentum comes seemingly from thin air.
@@danielm9463 The importance of change being constant is evident in scientific discoveries. Based on my own research, I propose that force should be defined as a description, rather than simply a push and pull. This perspective aligns with the dynamic nature of our understanding of the universe. ~ Guadalupe Guerra
@@amberheard2869 Yes! In fact, the notation I used might be slightly odd, because a proportionality usually indicates how two (and *only two*) variables are related. The easiest way to see why the combined proportionality is valid is to work backwards. Let's start with a hypothetical equation y = c•x•z, where c is a constant and x and z are variables that can take on different values. As x and z change, so does y. We first want to understand how a change in z affects y. So we assume that x is fixed/constant, and then we write a proportionality between y and z by removing any constants from the equation--c and x. The equation y = cxz thus becomes y ∝ z. This tells us y is directly proportional to z. Next, we want to understand how a change in x impacts y. So we assume that z is fixed (unchanging), and then we rewrite the equation as a proportionality by taking out the constants (c and z). This gives: y ∝ x. But we've just uncovered something interesting. If we start with the two proportionalities (y ∝ z and y ∝ x), we could combine them to get y ∝ x•z. If we then added a constant c, we get back to the original equation: y = c•x•z. This procedure is valid in general: whenever we have two proportionalities, we can combine them together into a single proportionality. If we also add in a constant, then we can change the multiple proportionalities into a single equation.
Newton's Universal Law of Gravitation was formulated before Kepler's Third Law. Newton published his law of gravitation in his work "Philosophiæ Naturalis Principia Mathematica" in 1687, while Johannes Kepler formulated his third law of planetary motion earlier in the 17th century. So, Newton's law predates Kepler's third law. So my question is, how does newton even knew the inverse square law before kepler's third law of planetary motion?
Kepler's 3rd law was published between 1609 and 1619 (i.e., some time in the early 17th century), which was well before Newton published the Principia in 1687. Kepler's work was very well known at the time when Newton was formulating his universal theory of gravitation, and before Newton published in 1687, it had even been posited that Kepler's 3rd law implied an inverse square force pulling on the planets--but it was Newton who suggested that this inverse square force was gravity, i.e., the same phenomenon that causes objects to fall toward the surface of the earth. And perhaps more importantly, Newton derived Kepler's 3rd law for elliptical orbit, which others thought was impossible and (if I recollect correctly) required him to invent some calculus for the full proof.
Assuming circular orbits is an oversimplification, to keep it practical to explain it at a high school level. The planetary orbits are approximately circular, and even by assuming they are circular orbits, the approximation is consistent with empirical data. The truth is, they are elliptical, and circular orbits are a special case of elliptical orbits. But the math is a lot more complicated to show why this law also works for elliptical orbits. Newton invented Calculus to show why they are elliptical orbits, and why Kepler's other laws are also true.
This is faulty; you are using Kepler's 3rd law (which is observational in nature) to get the law of gravity. But you need to do the opposite: derive the law of gravity fundamentally and then prove Kepler's 3rd law with it.
@Ilavenya, it's impossible to derive the law of gravity "fundamentally"--but perhaps I just don't understand what you mean yet. Newton's law of universal gravitation (and all physics, ultimately) is *observational* in nature. A true derivation for the law of universal gravitation doesn't exist--the law must be based on evidence. To that end, the primary evidence Newton had at his disposal were Kepler's laws. This video aims to present an overview of that evidence and the accompanying thinking, all of which led Newton to write down his law of universal gravitation.
I agree. I think that even if Kepler's law wasn't widely known, Newton would have been able to come up with universal gravitation. The inverse square law arises from geometry in many fields of physics like light intensity. Kepler's law was confirmation that gravity acted like a glow expanding out from the masses.
I'm not sure it's fair to say that it's *all* based on assumptions. At its core, science is empirical. The assumptions we make are judged based on how well they match/predict experimental data. To that end, the assumptions in this video are really good! They got humans to the Moon. On the other hand, the assumptions of Newtonian gravity aren't good enough for GPS systems, and they're not good enough to model the orbit of Mercury. For those things, Einstein's theory of general relativity is needed. There's a gradual progression in education, and this video is for high-schoolers, who have to make these assumptions in order to simplify the math (they can't yet solve field equations and don't know tensor calculus).
I didn't like this video because I think the author spent far too much time explaining how to get the formulas to work and no time at all explaining the why and how. It's titled "deriving newton's law of universal gravitation" and yet no deriving was done. To clarify, this feels like common core. "Here is a formula, here are the things to plug in." How did we get to this formula?
Thanks Collin for the comment! I'd love to try and convince you otherwise. In this video, I've provided the sort of mental thought processes that Newton himself may have used to develop his equation F = GmM/r². The purpose of a derivation is to explain, e.g., why it's mM on the right side and not m²M² or m+M. Those are the sorts of questions that we answer when we study physics at a deeper level--we're making mathematical arguments about the physical world. Dealing with this many equations absolutely can feel like 'plug and chug,' but you'll see more and more of these mathematical arguments the higher you go in your physics studies. You're correct that I'm not explaining where Newton's laws of motion come from, and I'm not deriving Kepler's 3rd law or the centripetal force equation--that's background knowledge that viewers must bring to the video. But I do explain why and how the formulas apply to the specific scenario of planetary motion. The example of a falling apple is argument about why we expect F ∝ m in Newton's final equation. The discussion of Newton's 3rd law is an argument about why we expect F ∝ M in the final equation. The discussion of Kepler's third law and circular motion is exactly the argument that Newton used for why we expect F ∝ 1/r² in the final equation. I'm attempting to derive the equation by making arguments about the constituent proportionalities, using only information that Newton would've had access to.
jimmy alderson Einstein gravity visualization of bending of space and time and Newton gravity visualization of attraction of two bodies towards their centre both are right at their own place 😢😢😢😢😢😢😢😢😢😢😢
Haha, okay, so Newton's law of universal gravitation was accurate enough to get humans to the moon. While general relativity did supersede Newton's law, general relativity (when applied to most classical contexts) and Newton's law produce indistinguishable results.
So technically if Kepler had not yet studied planetary motion Newton would have been lacking the inverse square part of his equation
This comment by NGD deserves to be at the top of the page, and I wish I had said exactly this in my video!!!! If Brahe hadn't collected the planetary data, and if Kepler hadn't studied it, then Newton wouldn't have known that F should be proportional to 1/r².
Some fun history: Newton wasn't the first to propose that the planets felt a 1/r² force from the Sun. Newton's contribution was to explain where that 1/r² force came from and why a planet could be treated as though all of its mass was concentrated at the center. He also worked out the math for elliptical (as opposed to perfectly circular) orbits. Ismaël Bullialdus and Robert Hooke also did some work on that idea of a 1/r² force pulling the planets toward the center of their orbit.
@@danielm9463 Probably yeah but then does this mean that Newton's mathematical proof of Kepler's third law is a circular argument?
@@gloystar that's a really great question. Physics teachers / textbooks will often show Newton's law of universal gravitation, then present Kepler's third law, and derive Kepler's third law from Newton's law of gravitation. Historically, that's somewhat backwards from what happened.
This might be a more accurate way to put it: Kepler's third law was well-known, and it was also believed (prior to Newton's work) that planets must be feeling a force proportional to 1/r^2 keeping them in orbit. But no one knew where that force came from. Any attempts to propose an explanation would *have* to explain Kepler's third law. So the ability to derive Kepler's third law was a piece of evidence that supported Newton's law of universal gravitation.
Given all that, I think it's still fair to say that Kepler's third law can be derived from Newton's law of universal gravitation, since gravity is more fundamental, and orbital motion arises due to it. But the evidence of planetary motion definitely led to the discovery of universal gravity (not the other way around).
@@danielm9463 That's exactly what I was thinking. I think Kepler's laws came from pure observations at first, and I've read somewhere that Newton already knew three things: 1) moon was 60r away, 2) Acceleration at surface of earth (g)=9.8, 3) Orbital period of the moon=27.3 days. Using this, he calculated g at moon's distance (as a centripetal acceleration) and compared that with g at earth's surface, and found out that it was 3600 times weaker, which is 1/(60)^2. This way, he knew that F is inversely proportional to r^2.
@@gloystar If that source is accessible to you, I'd be very grateful if you'd share it! That's a clever approach, and I hadn't read that this was part of Newton's calculation. What you said about Kepler's law coming from pure observation is exactly correct, historically speaking. (Kepler used the astronomical observations and data that Tycho Brahe collected.) What I've read is that, when Newton started working on this problem, much of the scientific community already believed that the planets felt a 1/r^2 force, and this came from Kepler's 3rd law. Connecting that to the moon would make a lot of sense. I know a huge part of what Newton did (which is rarely mentioned in high school contexts) was to show that this all works out for elliptical orbits. That required calculus, and if I'm remembering correctly, it also required Newton to prove that, in terms of gravitational effects, a spherical body can be treated as though all of its mass is located at a single point (the center of mass). I think Newton showed that calculation, too.
This is brilliant and despite my education and background in physics, this is the first time I’ve seen a clear argument as to how Newtons law of Gravity was derived. Thank you for putting this together. You’re one of the clearest teachers I’ve learned from.
i think newton saw this video before us
NOO
I was searching for this explanation for a long time now and I can't believe how brilliant you're at explaining it ... I am in love with your way of explanation really 🥰💕 thanxxx
very informative. I always had queries why there was a multiplictive mathmatical operation on Gravitational Force and also Electrostaric Force. this has opened a lot of my mind...
Thank you Daniel for explaining this topic so simply and illustratively; now I am equipped with the understanding and intuition of gravity. I hope this video and you, won't be forgotten in the depths of youtube, and can still educate people in future.
def educating me rn
Very intuitive video. Thanks!
You cleared all my confusion man..... Awesome..
I like how you explained it, you really considered your audience❤
You are an actual genius ! I really appreciate your video❤
Great lecture, Daniel. It deserves more views.
Thanks, Antonio!
Thank you, these things are not taught in school here in Tanzania...very clear description
Greetings! They are not taught here in Brooklyn either.
Beautifully put, bravo!
Great derivation! I'm not sure the derivation for the inverse square relation using Kepler's third law is that clear (to me at least). I would go from F_g = 4*pi^2*m*r/t^2. And use Kepler's law to say t^2 is proportional to r^3. Then F_g is proportional to 4*pi^2*m/r^2. Then F_g is proportional to m/r^2 and hence the inverse square law follows
This is a great question. The approach you've described is totally logical, and it's one I considered when making the video. There were two reasons I didn't go that route. The first is that it gives us an expression for the force of gravity, F_g = (4π²m)(k/r²). This equation makes it seem that Newton's law of gravitation should have 4π²m somewhere in the numerator, and it would require a pretty nuanced explanation of why we don't actually expect that to appear. The second thing about writing F_g = (4π²m)(k/r²) is that it *appears* to suggest that F_g ∝ m. But of course, this equation F_g = (4π²m)(k/r²) isn't sufficient to infer that F_g ∝ m, and it's not the reason why Newton thought F_g ∝ m. Rather, the idea that F_g ∝ m and F_g ∝ M flowed from his understanding of his second and third laws of motion. I worried that isolating F_g and showing the equation F_g = (4π²m)(k/r²) would muddy the waters, and students might wonder: "Wait, why do we now have a second equation F_g = GmM/r² if we already know that F_g = (4π²m)(k/r²)? How do we reconcile the two?" I thought it would be better to show how, whatever we substitute in for F_g, it must provide the missing piece which completes Kepler's 3rd law. From the equation F_g = 4π²mr/T², the only way to complete Kepler's 3rd law is to substitute into F_g with k/r². I also like the approach of having students build intuition and learn to spot similarities in form between two different equations.
hmm good point, thanks!
good video brother
Amazing
Why do we measured gravity with keplers law? Please tell
Excellent explanation from the ground up.
Thanks!
This guy needs million subscribe
You won my subscribe
Great video! I would like to ask a question though. I've read that the masses that appear in this equation are gravitational masses and I don't understand why would someone distinguish between gravitational and inertial mass. Did Newton define these gravitational masses or did he just assume that W=m(inertial)g? If he did assume that W=m(inertial)g and he derived his law of gravitation using this assumption then I don't see why would someone think that the masses that appear in the universal law of gravitation would be different from the inertial masses in Newton's second law.
This is a great question. The short answer is that both of these equations (Fg = GmM/r^2 and Fnet = ma) can be used as a definition of mass itself. Since we have two different equations/definitions, the question naturally arises: do the two definitions always yield the same value of mass?
In the first case (using the Fg equation), the equation implies a conceptual definition of mass along the lines of "the property of matter that creates gravitational attraction." In the second case (using the Fnet equation), the conceptual definition of mass is "the property of matter that resists acceleration when a net force is applied."
Because these definitions are tied to equations, the implication is that we can (a) experimentally determine the gravitational mass using Fg = GmM/r^2, and (B) experimentally determine the inertial mass using Fnet = ma, then (c) compare to see if they match.
Here's how that would work. We have to start by assuming we know the value of G through independent mechanisms. Imagine we have three objects A, B, and C. We place A and B in close proximity and measure their gravitational force F1 and separate distance r. Then we do the same for B and C, and for A and C. We set up a system of three equations:
F1 = (G)(m_a)(m_b)/r^2
F2 = (G)(m_a)(m_c)/r^2
F3 = (G)(m_b)(m_c)/r^2
Since we know F1, F2, F3, G, and r, we can experimentally determine the 3 gravitational masses m_a, m_b, and m_c.
Next, we can apply a net force F4 (which we measure) to each object. We measure each mass's resulting acceleration. We can again calculate the masses, from this second equation:
m_a = F4 / a_a
m_b = F4 / a_b
m_c = F4 / a_c
So the question is: would the values of m from the first experiment match the values of m from the second experiment?
The true picture is a bit more complicated, because we don't have an independent way of measuring the universal gravitational constant G except using preexisting measurements of inertial mass m and M. This changes the question slightly from "is gravitational mass always the same as inertial mass" to "is the ratio of gravitational mass and inertial mass always the same?" This question quickly moves us into some of the core principles behind general relativity. Here's a nice Quora post by Viktor T. Toth that describes this in a bit more detail:
www.quora.com/Is-gravitational-mass-and-inertial-mass-equivalent-complementary-different-or-totally-independent
@@danielm9463 Wow! Great explanation! Thank you very much.
Best example on CZcams
this is so magical.
Beautiful video thank you!! But I have a question:
As far as I see, we used four fundamental laws during the derivation process: Newton's first, second, and third laws, and Kepler's third law. Newton's second law says F=ma and Kepler's third law says (T^2)/(r^3) is constant. Are these two laws empiric and observational, or were they derived from other laws?
I really love these questions, and I don't know if I've ever considered it that deeply, but it's worth doing. Kepler's 3rd law is definitely empirical and was discovered only through observation. It's also very much a description of observed patterns, which might be different from Newton's law of universal gravitation (or more broadly, Einstein's theory of general relativity)--both of which explain observation but are theoretically deeper, broader-reaching, and explain a diverse set of phenomena. I'm not sure if Newton's second law is empirical--it might actually be a mathematical definition of force. Force (similar to energy) isn't something we see or observe directly--I think of it as more of a mathematical abstraction / more removed from what we observe than, say, distance or speed. I really don't know how to characterize Newton's third law. I have some initial thoughts but would love to hear what you think.
Hi again, sorry for my late answer. I have been busy recently.
I have thought about Newton's second law (F=m*a) and noticed that the definition of the terms in the formula can help us to understand it on a deeper level. I tried to understand the definition of these terms and tried to put them into my own words. Now, let's focus on what acceleration is and derive the second law from that.
I think motion has two parts. The first is velocity, I say velocity is the vectorial change in an object's location per one unit of time. The second is acceleration, and I say acceleration is the vectorial change in an object's current behavior of changing its location per one unit of time. Now we notice that all this argument is about the change of things. Naturally, some phenomenons support change while some hinder it. Right here, I define force as any support and mass as any resistance to the change in an object's velocity. Consequently, acceleration is proportional to force and 1/mass, and we call this proportion Newton's second law.
Let's look at Newton's first law to understand these definitions better: If no vectorial impact supports the change in velocity (net force = 0), velocity does not change (acceleration = 0) and the object maintains its current velocity.
We can see this kind of relationship between support and hindrance to change in many places in nature: Voltage = Electric Current * Resistance etc.
I would like to hear my possible mistakes, thanks for your interest.@@danielm9463
How can we assume that the product of all three proportions can be multiplied and still remain proportional to the force of gravity?
Really great question! It might be a slightly odd notation, because a proportionality usually indicates how two (and *only two*) variables are related to each other. The easiest way to see why the combined proportionality is valid is to work backwards. Let's start with a hypothetical equation y = c•x•z, where c is a constant and x and z are variables that can take on different values. As x and z change, so does y. We first want to understand how a change in z affects y. So we assume that x is fixed/constant, and then we write a proportionality between y and z by removing any constants from the equation--c and x. The equation y = cxz thus becomes y ∝ z. This tells us y is directly proportional to z. Next, we want to understand how a change in x impacts y. So we assume that z is fixed (unchanging), and then we rewrite the equation as a proportionality by taking out the constants (c and z). This gives: y ∝ x. But we've just uncovered something interesting. If we start with the two proportionalities (y ∝ z and y ∝ x), we could combine them to get y ∝ x•z. If we then added a constant c, we get back to the original equation: y = c•x•z. This procedure is valid in general: whenever we have two proportionalities, we can combine them together into a single proportionality. If we also add in a constant, then we can change the multiple proportionalities into a single equation.
Thank you :)
you're welcome!
I really have a question here, so if F is proportional to both M and m, how do you prove that now F is proportional to the product of two?
I’ve read through the comments and I recognize you used Kepler’s law to derive Newtons, which is how, I gather, it occurred historically. The classes I’ve been in have typically used Newtons law to show Keplers law. Is there a chance you could do a video explaining the kind of data Kepler had gathered as well as the process he might’ve used to develop his laws?
I don’t know if it would make a great video, I recognize that most of it is just parsing through collected data, but I feel like it would give a more complete understanding to how Newton or someone in his position rather, could have developed his laws.
These are great questions. So, here's a more nuanced understanding that I have: when Newton wrote the Principia, he actually did the derivation using data about the moon (its acceleration & distance from Earth). He presented this data in a "derivation" of the inverse-square component of his law of universal gravitation. After fleshing out the full law of universal gravitation, he showed that it explains Kepler's data (and he used calculus to do the calculations for elliptical orbits, which was a major achievement and was thought to be an impossible task). So the presentation in the Principia doesn't exactly match what I present in this video. However, the way Newton presents it in the Principia almost certainly isn't how he thought about it. Other physicists had already looked at Kepler's data and suggested that an inverse square force would explain the planets' motion if they were traveling in perfect circles, and Newton was aware of this work. (No one knew how to show this for elliptical orbits, and Newton's application of calculus was a very important part of the Principia.)
Here's a link to a Quora answer that has a lot more detail and depth, which I recommend:
qr.ae/pGziaX
@@danielm9463 Thanks for responding to this so quickly and including more info. Must have missed the notification a few days ago. Thanks again!
@@danielm9463 Great Reply. Thanks!
Where is gravitational constant? And F is not the same units as m1m2/r2 units.
Really excellant man u have explained why it is r square rather than simply saying its r square like that can u explain why G value is 6.67 why it cant be 7
Ultimately, we simply find ourselves in a universe where the universal gravitational constant is 6.67408 x 10^-11 N m^2 / kg^2. We don't have a reason why G has this value, but it's a good thing, because if G were much higher then the universe would only consist of black holes, and if G were much lower then stars and planets would have never formed. PBS Space Time has a video series on this "fine-tuned" nature of our universe. I believe this is the first video in a 3-part series: czcams.com/video/8wa1l7M5gU8/video.html
@@danielm9463 thanks man it was really helpfully
At around 8:04, you took the velocity to be the distance over time that is circumference/Time. But isn't velocity Displacement over time?
Good question! The v represents speed rather than velocity. The equation Fnet = mv^2/r only gives the *magnitude* of the net force. The *direction* of the net force is always toward the center of the circle, for uniform circular motion.
Excellent explanation bruh
very well explained !
Glad it was helpful!
@@danielm9463 may i ask i is it that in the formula its M times M why not M+M ?
@@muslimwoolfy-winterequestr4344 Great question! There are two ways to answer it. If we follow the logic of the video, here's the answer: we need the force to be directly proportional to each mass and this isn't true when we have m + M. If we had F = G(M+m)/r^2, then F wouldn't be directly proportional to M or m because doubling M or doubling m won't necessarily double F. But with F = GmM/r^2, then F is directly proportional to both m and M.
The other way to explain the answer is: the equation is written to describe what we see happening in nature and in the universe. The equation is written to follow our empirical observations/data, and multiplying mM fits the data (whereas m+M doesn't fit the data).
Keplers law relies on F = Gm1m2/r² does it not?
Nope, Kepler's Law was discovered first and was used to develop Newton's law of universal gravitation.
Great explanation. But I still don't understand how the force is proportional to the earth's mass. I mean, I see that the force should be proportional to the apple's mass. That follows from Kepler's 3rd law and F = ma, or rather F = mv^2/r. What I don't understand is how it follows from the third law that the force is proportional to the earth's mass, since the earth isn't moving at all. Meaning we can't apply F = Mv^2/r to it. Would anyone mind explaining that to me?
Great question! When we look at a falling object, we see that Fg = mg and Fg ∝ m. (This follows from Newton's 2nd law, Fnet = ma, not from Kepler's 3rd law and also not from Fnet = mv^2/r.)
But this fact that Fg ∝ m is still true even when the object isn't falling. After all, gravity is always present, including when objects are stationary. It prevents us from floating up into the air. Furthermore, we're not heavier or lighter when falling--our weight is constant. So this property that Fg ∝ m is true in general, whether the object is moving or stationary.
Once we establish the general principle that Fg ∝ m, we can apply it to the Earth itself. The Earth doesn't have to be falling for Fg ∝ m to be true--the Earth simply must feel a gravity force. And we know the Earth feels a gravity force because of Newton's Third Law. (If a ball is pulled down by the Earth with a gravity force Fg, then the Earth is pulled up by the ball with an equal force Fg.) So when we say that the Fg ∝ M, we're not relying on anything about the Earth's motion. We're simply relying on the fact that the Earth feels a gravity force (the reaction to the ball feeling a gravity force), and any gravity force on an object is proportional to the object's mass.
+Daniel M Thanks for the quick reply. Regarding my statement about Kepler's 3rd law, you can mathematically deduce that the gravitational force between the earth and the sun depends on the mass of the earth and the inverse of the square of the distance from it. Here's the link if you're interested:
www.quora.com/How-can-Newtons-law-of-gravitation-be-derived-with-Keplers-law
Scroll to the very bottom and you should find the derivation.
But I wasn't able to understand how he concluded that the force depends on the sun's mass as well. I came on youtube hoping to find the answer to that question and yours was actually the only video I could find related to the topic. Your method is slightly different but here too I failed to understand why the force should depend on the other object's mass.
I've read your explanation thoroughly but I'm afraid I still don't understand. I promise I'm not being obtuse on purpose. I guess what I'm having trouble with is the fact that Newton's 3rd law says that the force acting on a body should be independent of its mass. For example, if two balls collide, then both balls exert the same force on each other. It doesn't matter is one ball is tiny and the other massive. They will still exert the same force on each other. But by saying Fg ∝ M, we're saying that if the sun were more massive, the reaction force exerted by the earth on the sun would be greater. To me, this seems to be at odds with Newton's 3rd law. Hope you can clear this confusion.
Thanks for the link! In the post you mentioned, they derive the equation for universal gravity using both (a) Kepler's 3rd law and (b) Newton's 2nd law (a = F/m). The mass m is introduced via Newton's 2nd law rather than via Kepler's 3rd law.
No problem at all--asking questions is always productive! In fact, Newton's 3rd law doesn't state that the force must be independent of mass. It's often true that the action and reaction forces don't depend on mass, but it's not **required** by Newton's 3rd law. Instead, the 3rd law simply states that, when two objects interact, they push on each other with equal forces in opposite directions.
Newton's 3rd law states that any stipulations we make (like 'Fg ∝ M' and 'Fg ∝ m') must apply to both the action force and the reaction force. So let's imagine the Earth (of mass m) and the Sun (of mass M) pulling on each other. If the Earth's mass m suddenly grew a little, then the force of gravity Fg on the Earth would increase, and so would the force of gravity Fg on the Sun. The two forces would increase and still be equal. Likewise, if the Sun's mass M suddenly grew a little, then the force of gravity Fg on both the Earth and Sun would suddenly become larger, but it would still be equal.
+Daniel M Thanks for the reply. I understand the logic behind the argument but I tried doing it mathematically and wasn't able to. Which is weird because if it is logically sound, then proving it mathematically should be no big deal. And yet there just isn't a way to bring the sun's mass into the equation without directly stipulating that the force depends on it. This seems odd to me.
The equation we used is Fg ∝ m, where m is the mass of any object that feels a gravity force. The falling ball feels a gravity force, and thus Fg ∝ m. The Sun feels the same gravity force, and thus Fg ∝ M. According to Newton's third law, those two forces are the same. That was one of Newton's huge insights with the law of universal gravitation--extending the gravity force Fg to all objects.
what about the gravitational constant how did newton come up with this constant....
The fact that the ratio of T^2/R^3 is the same for every planet is called Kepler's 3rd law. In Newton's time, it was known experimentally through data, by observing the planets. It wasn't derived--it provided evidence that supports Newton's law of universal gravitation.
The gravitational constant didn't come until much later, when Cavendish experimentally determined its value, and other scientists modified the proportionality of Newton's law of gravitation, and added the G to promote it to an equation.
Prior to Cavendish, all we could know is the GM product of any given astronomical body. His experiment isolated G from M, so that he could "weigh" the Earth.
Daniel, when I pull an object, the object will fall on the surface of the earth. If i will pull an object on the string with huge force and the object is light the object will fall on the surface on a trajectory close to a vertical path. Make an experiment.
God bless u bro:)
how was the G added later?
oh and I got another question as to why the "Tsquared over r" was multiplied to both sides.
The G was added later thanks to the Cavendish Experiment. If you do a google or a youtube search, you'll get a ton of good hits on how it was conducted. It involved the hanging torsional pendulum that I showed a clip of.
it is amazing
yes it is!
Can someone explain the jump from multiple proportionality to the combined one? 10:01.
Really great question! It might be a slightly odd notation, because a proportionality usually indicates how two (and *only two*) variables are related to each other. The easiest way to see why the combined proportionality is valid is to work backwards. Let's start with a hypothetical equation y = c•x•z, where c is a constant and x and z are variables that can take on different values. As x and z change, so does y. We first want to understand how a change in z affects y. So we assume that x is fixed/constant, and then we write a proportionality between y and z by removing any constants from the equation--c and x. The equation y = cxz thus becomes y ∝ z. This tells us y is directly proportional to z. Next, we want to understand how a change in x impacts y. So we assume that z is fixed (unchanging), and then we rewrite the equation as a proportionality by taking out the constants (c and z). This gives: y ∝ x. But we've just uncovered something interesting. If we start with the two proportionalities (y ∝ z and y ∝ x), we could combine them to get y ∝ x•z. If we then added a constant c, we get back to the original equation: y = c•x•z. This procedure is valid in general: whenever we have two proportionalities, we can combine them together into a single proportionality. If we also add in a constant, then we can change the multiple proportionalities into a single equation.
Daniel M I can’t thank you enough for this valuable explanation other than subscribing...cheers...
Cool but can you prove it without using kepler's third law?
Ultimately, the law can't be proven. Even this video doesn't offer a proof! The big-picture goal of physics is to come up with laws, equations, and concepts that describe our universe and make predictions. This law is best thought of as descriptive of the gravitational phenomena we see. Logically, we're using induction rather than deduction.
Okay.. did you just use newton's law of gravitation to change Eg to k/r^2??
IF AND ONLY IF that is the case, then the numerator should be the product of masses, right?
Hm... I don't think so. The video never uses the equation for Eg (gravitational potential energy). The goal of this video is to derive Newton's law of gravitation. Here's where k/r^2 comes in: Newton knew that Fg = mv^2/r from circular motion, and he knew that T^2/r^3 = k. If we put those two things together, we can deduce that the Fg must be equal to k/r^2. Newton wasn't the first person to propose this. Bullialdus did so, and Hooke claimed to have originated the same idea that the gravity force must be proportional to the inverse square of distance from the Sun.
ohh sorry.. i typed Eg instead of Fg.. my bad.. But i must say im convinced..
but uhh i tried what you told: putting mv^2/r and T^2/r^3 together and im getting a really weird answer lol.. can you explain this in your next vid? Or if you can explain me in the comment section it will be great!! plzzzzzzzzzzzzzz???????
Why we square t and cube r
Great question! Kepler's law was empirical--not theoretical. That means the only reason to square T and cube r is because that's what the data shows. Kepler recognized the pattern before anyone could explain it. When they looked at the planetary data, they found that squaring T and cubing r caused the data to follow a linear relationship. Newton was seeking a law that could explain why T was squared and r was cubed.
So it was more of an observation thanks alot
You've got it right!
Because of Kepler's law and to balance the equation
very nice but I have a doubt. How kepler's 3rd law was invented?
Thanks :)
Kepler derived his third law empirically, by examining data that the great astronomer Tycho Brahe collected. A quote from Kepler:
". . . after I had by unceasing toil through a long period of time, _using the observations of Brahe_, discovered the true distances of the orbits, at last, at last, the true relation . . . overcame by storm the shadows of my mind, with such fullness of agreement between my seventeen years' labor on the observations of Brahe and this present study of mine that I at first believed that I was dreaming . . . ." (from Harmonies of the World) [emphasis added]
You can read more here:
books.google.com/books?id=czaGZzR0XOUC&pg=PA40&hl=en#v=onepage&q=kepler's%20third&f=false
I have the same doubt. Seeking for a law thats implicitly defined without relying on experimental observations. I came upon it when I encountered Columbs law of force between charges.
@debendra, ultimately, all physical laws must rely on experimental observation. The goals of physics are to describe the universe in the simplest mathematical terms.
@Daniel M, the problem is that either you assume 3 Kepler Laws to derive 1 Newton Law, as opposed to 1 newton Law to derive 3 kepler's laws. The latter is much morre powerful and favorable. Mathematics and Physics tend to assume less and conclude more. In fact, when Kepler went to Newton, where Kepler went almost insane trying to figure out the orbits, Newton derived from his gravitational law that the orbits are elliptical (Kepler's 1st law), that equal areas are swept out in equal times (Kepler's 2nd law), and that the square of period is proportional to the cube of the semimajor axis of the ellipse, or also the radius of orbit (Kepler's 3rd law). Therefore, Newton's law is more powerful then Kepler's laws, which rely on a lot of complex, hard to believe assumptions, whereas Newton's rely on a simple idea of gravitational field being proportional to the inverse square of radius. Which is better, to make a special single, easy tool that can be used in a diverse amount of examples, or many convoluted assumptive tools that are all just to prove 1 statement. If you think about what's more natural, it's obvious.
Why you write tsguared and r cubed
This is quite late, but better late than never. I have a question. You kept on mentioning that we "almost have Kepler's 3rd law", but why is it that we HAVE to have Kepler's 3rd law in the equation. Furthermore, why does Fg have to be the thing with r^2 in the denominator? What if we multiplied both sides of the equation by r^-2? I get that ((T^2)/(r^3)) is special because it is constant, but why does that allow it to earn its place into the equation. This might be a stupid question. Sorry, I'm in high school.
Definitely not a stupid question! This is a tricky thing to explain, but essentially it comes down to this: if you know you that T^2/r^3 is a constant, then you can demonstrate, as a mathematical consequence, that the circular motion must be caused by an acceleration (and force) proportional to 1/r^2. Before Newton started working on the problem, physicists already knew from Kepler's work that T^2/r^3 was a constant, and people had even suggested the planets are held in orbit by a force proportional to 1/r^2. But until Newton came along, no one knew what caused that 1/r^2 acceleration, or what the rest of the form was, or what other phenomenon it was connected to. Newton filled in all of those pieces.
From centripetal acceleration, we know:
a = v^2/r
We can substitute in v = distance/time = 2πr/T:
a = (2πr/T)^2/r = (4)(π^2)(r/T^2)
a / 4π^2 = (r/T^2)
But we know from Kepler that:
k = r^3/T^2
k = r^2 * (r/T^2)
Substituting in for r/T^2 gives:
k = r^2 * (a / 4π^2)
k = r^2 * a / 4π^2
4kπ^2/r^2 = a
And this suggests an acceleration proportional to 1/r^2.
@@danielm9463 Ahhhh, That makes sense. I was confused on how k popped up in the equation, but using parts of the v^2/r equation and plugging that into k = r^2((r)/(T^2)) makes a ton of sense. Also interesting on how searching for a known constant in your equation tells you about the nature of, in this case, acceleration around a planet. Very cool! Thank you so much!
@@TahirAhmad-io6uw you put it better than I could have! That's exactly the idea I was trying to get at with this video. I agree--it is a really cool way for the physics to have developed. Good luck in your studies!!
@@danielm9463Thank you for the kind words! I was reading up on how Newton went about discovering his Universal Law of Gravitation, and this video helped me comprehend much of what I was reading. I wish you well!
But orbits are not circular rather elliptical then using (2*pi*r / T ) is wrong?
You are correct! This is an approximation that we make at the high school level because the ellipse shape isn't studied. The more precise formulation of Kepler's third law refers to the semi-major axis.
Exactly...that is why we are also taught Einstein's field equations.....more important is the fact that even light (made up of photons with 0 mass) also gets curved near such planets...so according to Newtons law of gravitation, it is acted upon by 0 force...so?
You're exactly right! Einstein later showed that Newton's law of gravitation is incomplete (rather than wrong). This doesn't change the fact that, when dealing with most classical scenarios, the difference between Einstein's theory and Newton's law is negligible. For example, Newton's law of gravitation was good enough to get astronauts to the moon. So the statement "even light ... also gets curved" is true to an extent. At certain scales and distances, the bending will be negligible. The way people have encouraged me to think of it is like this: Newton's law has its place, and general relativity has its place.
Why r^2?? Why not r?
The fact that the planets felt a 1/r² force was known empirically. The planetary data (which Brahe largely collected, I think) revealed a 1/r² force, not a 1/r force.
It comes from the surface area of the sphere of influence, which is proportional to r^2. The gravitational field is inversely proportional to the sphere's surface area. Had we lived in Flatland, it would be 1/r. In 4D hyperspace, it would be 1/r^3.
The divergence theorem shows why this must be. Gravity is a non-divergent force field in free space, and only divergent (or rather convergent, with negative divergence) in a region of space with mass. Once external to that mass, gravitational field lines (more accurately: gravitational flux) are no longer being created, and just get redistributed to a smaller concentration. The smaller the concentration of field lines, the smaller the force per unit mass at that location in space.
“The origin of momentum creation is founded on the application of energy. Momentum serves as the initial cause of pushing and pulling. The energy, emanating from the internal core of a planet, is what gives rise to momentum, commonly referred to as gravity, and not due to the planet’s size, mass, or space-time curvature. For example, when energy is applied, it generates the necessary momentum for the actions of pushing and pulling. Without the application of energy, momentum cannot be established. Consequently, force cannot be exerted for pushing and pulling, as it lacks a physical existence. This, in turn, clarifies why force is not gravity.” ~Guadalupe Guerra
I love the poeticism of this, but do you intend it to be physics? If we use accepted physics definitions, then most of these statements are inaccurate.† What do you think, do you view this exposition (which again, I find beautiful) as attempting to do the same thing that physics attempts to do? The goal of physics is to define concepts mathematically in very precise ways and then use those mathematical definitions in equations that describe how the physical world acts, interacts, evolves, etc. So, for example, the statement "force ... lacks a physical existence" is a little silly because, in physics, force is only intended to be a precise mathematical concept that can be used in equations to describe what we observe in the physical world. Force doesn't lack a physical existence any more than momentum, energy, or any other physics concept. They're all mathematical constructs that help us describe the universe through equations.
†Note: it's correct that generating momentum requires an input of energy--though technically you could also have an input of mass per E=mc^2. This idea that energy is needed to generate momentum is a consequence of conservation of energy. Momentum is defined as the product of mass and velocity, and thus nonzero momentum (mv > 0) also implies nonzero kinetic energy (0.5*mv^2 > 0). So to generate e.g. 10 joules of kinetic energy requires that the 10 J come from *somewhere* (e.g., from another object's kinetic energy, from potential energy, from chemical energy, etc.). However, Emmy Noether showed that even this principle of energy conservation is not true in general and only holds in the presence of time symmetry. Here's a concrete example: space is expanding. As it does, it generates momentum without any input of energy. Far-off galaxies are speeding up as they move away from us, and their increasing momentum comes seemingly from thin air.
@@danielm9463 The importance of change being constant is evident in scientific discoveries. Based on my own research, I propose that force should be defined as a description, rather than simply a push and pull. This perspective aligns with the dynamic nature of our understanding of the universe. ~ Guadalupe Guerra
why is it mM?
Because, according to Newtons' 3rd law, the gravitational force would have to be proportional to *both* masses.
@@danielm9463 I know it is proportional to both mass but I am kinda curious about multiplication of both mass.
@@amberheard2869 Yes! In fact, the notation I used might be slightly odd, because a proportionality usually indicates how two (and *only two*) variables are related. The easiest way to see why the combined proportionality is valid is to work backwards. Let's start with a hypothetical equation y = c•x•z, where c is a constant and x and z are variables that can take on different values. As x and z change, so does y. We first want to understand how a change in z affects y. So we assume that x is fixed/constant, and then we write a proportionality between y and z by removing any constants from the equation--c and x. The equation y = cxz thus becomes y ∝ z. This tells us y is directly proportional to z. Next, we want to understand how a change in x impacts y. So we assume that z is fixed (unchanging), and then we rewrite the equation as a proportionality by taking out the constants (c and z). This gives: y ∝ x. But we've just uncovered something interesting. If we start with the two proportionalities (y ∝ z and y ∝ x), we could combine them to get y ∝ x•z. If we then added a constant c, we get back to the original equation: y = c•x•z. This procedure is valid in general: whenever we have two proportionalities, we can combine them together into a single proportionality. If we also add in a constant, then we can change the multiple proportionalities into a single equation.
Newton's Universal Law of Gravitation was formulated before Kepler's Third Law. Newton published his law of gravitation in his work "Philosophiæ Naturalis Principia Mathematica" in 1687, while Johannes Kepler formulated his third law of planetary motion earlier in the 17th century. So, Newton's law predates Kepler's third law. So my question is, how does newton even knew the inverse square law before kepler's third law of planetary motion?
Kepler's 3rd law was published between 1609 and 1619 (i.e., some time in the early 17th century), which was well before Newton published the Principia in 1687. Kepler's work was very well known at the time when Newton was formulating his universal theory of gravitation, and before Newton published in 1687, it had even been posited that Kepler's 3rd law implied an inverse square force pulling on the planets--but it was Newton who suggested that this inverse square force was gravity, i.e., the same phenomenon that causes objects to fall toward the surface of the earth. And perhaps more importantly, Newton derived Kepler's 3rd law for elliptical orbit, which others thought was impossible and (if I recollect correctly) required him to invent some calculus for the full proof.
Here's a good article with some of the history: engines.egr.uh.edu/episode/3015
But he assumed that all planets or objects follow a circular path around the sun ...which is not true.
they are basically circles
Assuming circular orbits is an oversimplification, to keep it practical to explain it at a high school level. The planetary orbits are approximately circular, and even by assuming they are circular orbits, the approximation is consistent with empirical data.
The truth is, they are elliptical, and circular orbits are a special case of elliptical orbits. But the math is a lot more complicated to show why this law also works for elliptical orbits. Newton invented Calculus to show why they are elliptical orbits, and why Kepler's other laws are also true.
This is faulty; you are using Kepler's 3rd law (which is observational in nature) to get the law of gravity. But you need to do the opposite: derive the law of gravity fundamentally and then prove Kepler's 3rd law with it.
@Ilavenya, it's impossible to derive the law of gravity "fundamentally"--but perhaps I just don't understand what you mean yet. Newton's law of universal gravitation (and all physics, ultimately) is *observational* in nature. A true derivation for the law of universal gravitation doesn't exist--the law must be based on evidence. To that end, the primary evidence Newton had at his disposal were Kepler's laws. This video aims to present an overview of that evidence and the accompanying thinking, all of which led Newton to write down his law of universal gravitation.
I agree. I think that even if Kepler's law wasn't widely known, Newton would have been able to come up with universal gravitation. The inverse square law arises from geometry in many fields of physics like light intensity. Kepler's law was confirmation that gravity acted like a glow expanding out from the masses.
The more I study this stuff the more I realize it’s all theoretical and based on assumptions about how things work.
I'm not sure it's fair to say that it's *all* based on assumptions. At its core, science is empirical. The assumptions we make are judged based on how well they match/predict experimental data. To that end, the assumptions in this video are really good! They got humans to the Moon. On the other hand, the assumptions of Newtonian gravity aren't good enough for GPS systems, and they're not good enough to model the orbit of Mercury. For those things, Einstein's theory of general relativity is needed. There's a gradual progression in education, and this video is for high-schoolers, who have to make these assumptions in order to simplify the math (they can't yet solve field equations and don't know tensor calculus).
I didn't like this video because I think the author spent far too much time explaining how to get the formulas to work and no time at all explaining the why and how. It's titled "deriving newton's law of universal gravitation" and yet no deriving was done.
To clarify, this feels like common core. "Here is a formula, here are the things to plug in." How did we get to this formula?
Thanks Collin for the comment! I'd love to try and convince you otherwise. In this video, I've provided the sort of mental thought processes that Newton himself may have used to develop his equation F = GmM/r². The purpose of a derivation is to explain, e.g., why it's mM on the right side and not m²M² or m+M. Those are the sorts of questions that we answer when we study physics at a deeper level--we're making mathematical arguments about the physical world. Dealing with this many equations absolutely can feel like 'plug and chug,' but you'll see more and more of these mathematical arguments the higher you go in your physics studies. You're correct that I'm not explaining where Newton's laws of motion come from, and I'm not deriving Kepler's 3rd law or the centripetal force equation--that's background knowledge that viewers must bring to the video. But I do explain why and how the formulas apply to the specific scenario of planetary motion. The example of a falling apple is argument about why we expect F ∝ m in Newton's final equation. The discussion of Newton's 3rd law is an argument about why we expect F ∝ M in the final equation. The discussion of Kepler's third law and circular motion is exactly the argument that Newton used for why we expect F ∝ 1/r² in the final equation. I'm attempting to derive the equation by making arguments about the constituent proportionalities, using only information that Newton would've had access to.
But lol@Newton cos he was wrong
jimmy alderson Einstein gravity visualization of bending of space and time and Newton gravity visualization of attraction of two bodies towards their centre both are right at their own place 😢😢😢😢😢😢😢😢😢😢😢
Way to lol at your forebears...
Haha, okay, so Newton's law of universal gravitation was accurate enough to get humans to the moon. While general relativity did supersede Newton's law, general relativity (when applied to most classical contexts) and Newton's law produce indistinguishable results.