Stellite motion problem.
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Original post by Mehrdad jafari
You are right, but I don't understand why you would use v2=aR instead of V2=GM/R, the equation (v2=aR) would assume that a is constant at any given distance
It doesn't assume anything, as the radius increases and velocity does too, then the acceleration must decrease. You can change any value as you please to satisfy the equality. Every value is malleable in the equation. It might be better to think of it as a=v2/r, even though they're equivalent.
Original post by Callum Scott
It doesn't assume anything, as the radius increases and velocity does too, then the acceleration must decrease. You can change any value as you please to satisfy the equality. Every value is malleable in the equation. It might be better to think of it as a=v2/r, even though they're equivalent.
In the case of a satellite orbiting a planet, using that equation would mean that a and R are independent, that changing one doesn't change the other. As I said, you could use it for objects in circular motion without any force of attraction between the object and the centre of the orbit in which case you could choose any number as you wish. But in the case of a satellite, the values should be within reason otherwise if, for example, the centripetal acceleration is too small and that R is too small too then the satellite will fall towards the planet. Of course, you might think you could increase the speed of the satellite for it not to fall but increasing the speed would result in an increase in the centripetal acceleration of the satellite, so it's not up to us. The centripetal acceleration of a satellite orbiting a planet is always equal to the gravitational field strength at that point. With that we could decide at what distance the satellite will orbit at a particular linear speed.
What is your level of studying, if you don't mind me asking?
(edited 8 years ago)
First of all, I'm going into A2 next week, but that doesn't change anything, I think you may be misunderstanding what I'm saying. Are you delving into the fact that, in reality; the satellite would have a non-zero mass and would orbit the common centre of mass etc?
If so, I'm operating under the assumption that the mass of the satellite is so small compared to the body that this effect is negligible. I'm in no way claiming that this equation is suitable for binary systems of objects with roughly equal masses or the like -- I haven't thought to spend my time trying to see if it's still valid, or what the better equation is, etc.
Judging from the fact that the thread author was discussing v2=ar, I assumed his level of education was similar to mine and so I operated under the standard assumptions that are made for students at A2 level.
If so, I'm operating under the assumption that the mass of the satellite is so small compared to the body that this effect is negligible. I'm in no way claiming that this equation is suitable for binary systems of objects with roughly equal masses or the like -- I haven't thought to spend my time trying to see if it's still valid, or what the better equation is, etc.
Judging from the fact that the thread author was discussing v2=ar, I assumed his level of education was similar to mine and so I operated under the standard assumptions that are made for students at A2 level.
Original post by Callum Scott
First of all, I'm going into A2 next week, but that doesn't change anything, I think you may be misunderstanding what I'm saying. Are you delving into the fact that, in reality; the satellite would have a non-zero mass and would orbit the common centre of mass etc?
If so, I'm operating under the assumption that the mass of the satellite is so small compared to the body that this effect is negligible. I'm in no way claiming that this equation is suitable for binary systems of objects with roughly equal masses or the like -- I haven't thought to spend my time trying to see if it's still valid, or what the better equation is, etc.
Judging from the fact that the thread author was discussing v2=ar, I assumed his level of education was similar to mine and so I operated under the standard assumptions that are made for students at A2 level.
If so, I'm operating under the assumption that the mass of the satellite is so small compared to the body that this effect is negligible. I'm in no way claiming that this equation is suitable for binary systems of objects with roughly equal masses or the like -- I haven't thought to spend my time trying to see if it's still valid, or what the better equation is, etc.
Judging from the fact that the thread author was discussing v2=ar, I assumed his level of education was similar to mine and so I operated under the standard assumptions that are made for students at A2 level.
For objects in circular morion as a result of being in a gravitational field, the centripetal force/acceleration is independent of the mass of the objects. This is because the gravitational mass is equal to inertial mass (you can search about it online). And so I'm not assuming that the mass of the satellite is negligible because then it wouldn't have weight and so couldn't stay in orbit around a planet. Though I do want to say that you are in a better position of learning. In science, we shouldn't be told why something is true, but why something isn't.
Original post by Mehrdad jafari
For objects in circular morion as a result of being in a gravitational field, the centripetal force/acceleration is independent of the mass of the objects. This is because the gravitational mass is equal to inertial mass (you can search about it online). And so I'm not assuming that the mass of the satellite is negligible because then it wouldn't have weight and so couldn't stay in orbit around a planet. Though I do want to say that you are in a better position of learning. In science, we shouldn't be told why something is true, but why something isn't.
I guess I'm not operating on the basis of newton's laws anyway. I [and I'm assuming the A level syllabus] operate under the assumption that an orbiting point circles a fixed point with a constant angular velocity. The relation between the magnitude of the acceleration, the tangential velocity or speed [not sure if they're synonymous] and the distance the body is from the centre point is, unequivocally, v2=ar. I just assumed that a = GM/r2, but I forgot that the v2 equation existed within a different mathematical model to the newtonian model and shouldn't have merged them together to begin with I suppose.
Is it not true to say that the limit as the mass of the circularly orbiting body reaches 0, in the newtonian model, would churn out v2=ar, however?
(similarly to how the limit as c approaches ∞ with the lorentz transformations [gamma or whatever] churns out the galilean classical mechanics)
(Also since the centre of mass of the system would edge closer to the centre of the larger body and thus more closely represent the 'A level' model)
and btw, what stage of study are you at, you seem pretty wise?
(edited 8 years ago)
Original post by Callum Scott
I guess I'm not operating on the basis of newton's laws anyway. I [and I'm assuming the A level syllabus] operate under the assumption that an orbiting point circles a fixed point with a constant angular velocity. The relation between the magnitude of the acceleration, the tangential velocity or speed [not sure if they're synonymous] and the distance the body is from the centre point is, unequivocally, v2=ar. I just assumed that a = GM/r2, but I forgot that the v2 equation existed within a different mathematical model to the newtonian model and shouldn't have merged them together to begin with I suppose.
Is it not true to say that the limit as the mass of the circularly orbiting body reaches 0, in the newtonian model, would churn out v2=ar, however?
(similarly to how the limit as c approaches ∞ with the lorentz transformations [gamma or whatever] churns out the galilean classical mechanics)
(Also since the centre of mass of the system would edge closer to the centre of the larger body and thus more closely represent the 'A level' model)
and btw, what stage of study are you at, you seem pretty wise?
Is it not true to say that the limit as the mass of the circularly orbiting body reaches 0, in the newtonian model, would churn out v2=ar, however?
(similarly to how the limit as c approaches ∞ with the lorentz transformations [gamma or whatever] churns out the galilean classical mechanics)
(Also since the centre of mass of the system would edge closer to the centre of the larger body and thus more closely represent the 'A level' model)
and btw, what stage of study are you at, you seem pretty wise?
Yeah, at A levels only circular motion with constant angular velocity is taught. Our case is the same. Tangential velocity and linear speed would mean he same thing. The equation GM/r2 only gives the centripetal acceleration (or the field strength) of a body orbiting a planet at a particular radius, but equating GM/r2=v2/r results in the equation for the speed as v2=GM/r.
Since the mass of a body orbiting a planet is independent of the tangential velocity and the acceleration of the body, the only influential mass in that equation is the mass of the planet causing the gravitational field. Although it would be reasonable to assume that as M reaches zero the equation for normal circular motion (v2=ar) without any newton's law of gravity at work would be derived, mathematically it wouldn't work like the derivation of Galilean transformations as GM/r would be zero. What do you exactly mean by the part in bold?
I've just finished my A levels. Hmm, thanks for your meaningful words but I'm not wise. It would certainly take experience to be wise
Original post by Mehrdad jafari
Yeah, at A levels only circular motion with constant angular velocity is taught. Our case is the same. Tangential velocity and linear speed would mean he same thing. The equation GM/r2 only gives the centripetal acceleration (or the field strength) of a body orbiting a planet at a particular radius, but equating GM/r2=v2/r results in the equation for the speed as v2=GM/r.
Since the mass of a body orbiting a planet is independent of the tangential velocity and the acceleration of the body, the only influential mass in that equation is the mass of the planet causing the gravitational field. Although it would be reasonable to assume that as M reaches zero the equation for normal circular motion (v2=ar) without any newton's law of gravity at work would be derived, mathematically it wouldn't work like the derivation of Galilean transformations as GM/r would be zero. What do you exactly mean by the part in bold?
I've just finished my A levels. Hmm, thanks for your meaningful words but I'm not wise. It would certainly take experience to be wise
Since the mass of a body orbiting a planet is independent of the tangential velocity and the acceleration of the body, the only influential mass in that equation is the mass of the planet causing the gravitational field. Although it would be reasonable to assume that as M reaches zero the equation for normal circular motion (v2=ar) without any newton's law of gravity at work would be derived, mathematically it wouldn't work like the derivation of Galilean transformations as GM/r would be zero. What do you exactly mean by the part in bold?
I've just finished my A levels. Hmm, thanks for your meaningful words but I'm not wise. It would certainly take experience to be wise
I'm not sure if the location of the centre of mass changes anything tbqh (I don't know enough yet), since the v2=ar equation simulates the orbiting body rotating around a fixed point, I assumed because in newton's mechanics, both objects orbit a common centre of mass, as the mass of the orbiting body reaches zero, the common centre of mass approaches the centre of the more massive body, and so the equation would approach v2=ar, because the newtonian model is then approaching the 'A level' model. with the limit as m approaches 0 equating newton's mechanics to the v2 model. But I'm unsure how newton's mechanics mathematically plays into the situation anyway tbh again. It doesn't matter anyway, I'll find out soon enough. I don't want to ask qualitative questions and get random facts until I can understand the maths behind it anyway.
And cool, I hope you did well and got into a decent uni for you :P, are you planning on studying physics? or even planning on going Uni, for that matter
Original post by Callum Scott
I'm not sure if the location of the centre of mass changes anything tbqh (I don't know enough yet), since the v2=ar equation simulates the orbiting body rotating around a fixed point, I assumed because in newton's mechanics, both objects orbit a common centre of mass, as the mass of the orbiting body reaches zero, the common centre of mass approaches the centre of the more massive body, and so the equation would approach v2=ar, because the newtonian model is then approaching the 'A level' model. with the limit as m approaches 0 equating newton's mechanics to the v2 model. But I'm unsure how newton's mechanics mathematically plays into the situation anyway tbh again. It doesn't matter anyway, I'll find out soon enough. I don't want to ask qualitative questions and get random facts until I can understand the maths behind it anyway.
And cool, I hope you did well and got into a decent uni for you :P, are you planning on studying physics? or even planning on going Uni, for that matter
And cool, I hope you did well and got into a decent uni for you :P, are you planning on studying physics? or even planning on going Uni, for that matter
I don't think that's quite correct as it would violate the idea of a mass orbiting a planet if the mass of the object is to be considered zero. Cool. You will learn more about Newton's laws of motion but it's his law of gravity which plays a significant part in orbiting bodies around a planet.
I didn't do well in my exams but I'm not thinking of going to uni anyway.
Best of luck!
That's unfortunate, but thanks for the wishes :P
Anyway, I've thought about what I've said and concluded it makes perfect sense
, trust me, my opinion isn't biased at all, lol!
To explain it, you need to understand limits; have you studied maths at A/AS level and understand how calculus uses limits by any chance?
For example:
How if a =0, a/a is undefined since you can't divide by zero
BUT, the limit has a approaches 0 of a/a = 1 because a is still treated as a number, albeit an infinitesimaly small one
Anyway, I've thought about what I've said and concluded it makes perfect sense
, trust me, my opinion isn't biased at all, lol!To explain it, you need to understand limits; have you studied maths at A/AS level and understand how calculus uses limits by any chance?
For example:
How if a =0, a/a is undefined since you can't divide by zero
BUT, the limit has a approaches 0 of a/a = 1 because a is still treated as a number, albeit an infinitesimaly small one
(edited 8 years ago)
Original post by Alen.m
so have you find another way to solve the problem? Your explanation would be much appreciated
I can't think of an explanation that doesn't require calculus to be quite honest, I think it's necessary to include calculus to talk about constant changing positions, velocities and accelerations.
What I was saying was that I know a way to get the idealised model from my previous explanation (where an object of no mass orbits a fixed point) and to link it to the Newtonian model -- if we think of the mass of the orbiting body as the limit approaching 0, instead of just being equal to 0.
I've found another explanation that doesn't require nearly as much calculus, only a slight use of limits, if you'd like to hear that one?
there would be number of explanation that could link to the problem that we trying to solve but none of them have given me a certain way to sort problem and I'm not 100% sure about my way of dealing with the problem either but to me this was the only way that could actually lead to something even though there might be problems with it
Original post by Alen.m
yes please
what level are you studying at? AS or A level or what? Have you learnt the chain rule for calculus?
I don't think that this proof is as rigorous, but it seems to work.
You need to have at least AS maths knowledge for the s = r*delta theta and some knowledge of the chain rule...
If you don't understand why the limit as theta approaches 0 of sin(delta theta/2)/(delta theta /2) = 1, I could explain that via a proof too if you understand everything else
If you don't understand any part in particular, I'll be happy to explain it
Im not sure what all of this got do with what level im studying in but for your information i do study A2 which is why i come up with the A2 question .
Apart from all of this, science is a region of making mistakes and learn from other experiences if you are trying to show off your skills, good luck with that mate. Im not here to waste my time on none sense like this . Im here to learn and use other people knowledge so i guess we are not in the same page .
Apart from all of this, science is a region of making mistakes and learn from other experiences if you are trying to show off your skills, good luck with that mate. Im not here to waste my time on none sense like this . Im here to learn and use other people knowledge so i guess we are not in the same page .
Original post by Alen.m
Im not sure what all of this got do with what level im studying in but for your information i do study A2 which is why i come up with the A2 question .
Apart from all of this, science is a region of making mistakes and learn from other experiences if you are trying to show off your skills, good luck with that mate. Im not here to waste my time on none sense like this . Im here to learn and use other people knowledge so i guess we are not in the same page .
Apart from all of this, science is a region of making mistakes and learn from other experiences if you are trying to show off your skills, good luck with that mate. Im not here to waste my time on none sense like this . Im here to learn and use other people knowledge so i guess we are not in the same page .
wowowow, calm down.. your level of understanding has everything to do with it. If you don't understand calculus then you can't really understand how to derive the equation v2=ar.
What is it you're actually trying to do?
Are you trying to make standard assumptions about an orbiting body around a fixed point and mathematically derive v2=ar [which is what I assumed you asked for, or are you trying to somehow 'prove' that it is true, or are you doing something else?
If we take the centre of mass of the Earth as a fixed point that a body orbits around (a body with negligible mass) with constant angular velocity and radius from the Earth's centre, I have shown 2 ways in which v2=ar can be derived from this situation. Im sorry to say that it can't really be made any simpler!
Why should i upload a A2 physic question when my underestanding of it is below that?you're funny
There might be areas that needs to be improved but that does not alow you to underestimate anyone underestanding and compare it with yours(if there's any). Im not trying to do anything as you can see from my replies i said im not sure about my way and never said it's absolute right unlike yours i do make a mistake
There might be areas that needs to be improved but that does not alow you to underestimate anyone underestanding and compare it with yours(if there's any). Im not trying to do anything as you can see from my replies i said im not sure about my way and never said it's absolute right unlike yours i do make a mistake
Original post by Alen.m
Why should i upload a A2 physic question when my underestanding of it is below that?you're funny
There might be areas that needs to be improved but that does not alow you to underestimate anyone underestanding and compare it with yours(if there's any). Im not trying to do anything as you can see from my replies i said im not sure about my way and never said it's absolute right unlike yours i do make a mistake
There might be areas that needs to be improved but that does not alow you to underestimate anyone underestanding and compare it with yours(if there's any). Im not trying to do anything as you can see from my replies i said im not sure about my way and never said it's absolute right unlike yours i do make a mistake
So you've done A2 physics and understand the chain rule and calculus etc?
If so, the fact that you've finished A2 and are asking A2 questions confuses me. I assumed that you were an AS student, starting A2 within the next week or so, and lacked the knowledge of calculus -- as any normal person would -- hence why I 'underestimated you'.
If you're an AS student, starting A2 soon, then what are you on about 'underestimating'?
Original post by Callum Scott
wowowow, calm down.. your level of understanding has everything to do with it. If you don't understand calculus then you can't really understand how to derive the equation v2=ar.
What is it you're actually trying to do?
Are you trying to make standard assumptions about an orbiting body around a fixed point and mathematically derive v2=ar [which is what I assumed you asked for, or are you trying to somehow 'prove' that it is true, or are you doing something else?
If we take the centre of mass of the Earth as a fixed point that a body orbits around (a body with negligible mass) with constant angular velocity and radius from the Earth's centre, I have shown 2 ways in which v2=ar can be derived from this situation. Im sorry to say that it can't really be made any simpler!
What is it you're actually trying to do?
Are you trying to make standard assumptions about an orbiting body around a fixed point and mathematically derive v2=ar [which is what I assumed you asked for, or are you trying to somehow 'prove' that it is true, or are you doing something else?
If we take the centre of mass of the Earth as a fixed point that a body orbits around (a body with negligible mass) with constant angular velocity and radius from the Earth's centre, I have shown 2 ways in which v2=ar can be derived from this situation. Im sorry to say that it can't really be made any simpler!
Unfortunately that's not true. This equation was first derived by Newton around 500 years ago when the trigonometric functions we not yet formulated. Also, Newton himself didn't know about chain rule when he formulated that equation. There is a simple method to prove that equation that assumes no knowledge of trigonometric functions and calculus. I will post it in a while.
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