Derivate the forumla for the acceleration due to gravity

[zoobyshoe]

John,

Do you have any objection to the way "acceleration" is used in this example from a physics text:

"Suppose next that a body is thrown vertically upward with an initial velocity of 96 ft/sec. How far will it rise above the starting point, and how long will it be in motion? In order to solve this problem it is important to realize that as soon as it leaves the hand the moving body experiences a constant downward acceleration of 32 ft/sec², at every instant. This causes the upward velocity to decrease to zero at the point of maximun height and then produces an increasing downward velocity as the body falls back down to its starting point. It may help to understand this behavior if we anticipate somewhat our disscussion of dynamics in order to point out that accelerations are produced by forces. After the body leaves the hand the only force (air resistance neglected) acting on it is the pull of the earth. Furthermore, since this is a constant force (near the surface of the earth), the acceleration is therefore constant and is downward in the direction of the force."

Fundamentals of University Physics
W. Wallace McCormick 1969

 

[russ_watters]

RE: "Surely g is an accelleration due to gravity!"

This would imply that if an object is placed inside the Earth's gravitational field, it will accelerate at g. Not necessariy so, but the confused student will do this everytime.

The confused student just needs to have explained to him/her exactly what it means, not leaving out things like "...at sea level" and "...neglecting air resistance."

I really think that "gravitational field constant" will just create more confusion.

 

[robphy]

I really think that "gravitational field constant" will just create more confusion.

Given the unfortunate use of "acceleration due to gravity" used in current textbooks, I agree. However, as I argued above, I would advocate replacing the term in textbooks with "gravitational field" in analogy to the "electric field".

The confused student just needs to have explained to him/her exactly what it means, not leaving out things like "...at sea level" and "...neglecting air resistance."

Indeed, "knowing when a formula applies" is a stumbling block for many students.

Hence, if one wants to emphasize the notion of acceleration (as one does in kinematics), I suggested above the more descriptive term "freefall acceleration" for "g". In my opinion, "freefall acceleration [at sea level]" is better than "acceleration due to gravity [neglecting air resistance] [neglecting any other forces] [at sea level]".



Consider now the definition of weight, W=F_grav=mg. In the context of a book at rest on a table, "g" is probably better thought of as the "gravitational field" [at the book's location] (in analogy to the "electric field" E in F_elec=qE) rather than "the acceleration the book would experience if it were in freefall", an interpretation which must make use of Newton's Second Law.

 

[krab]

JohnDubYa: I think you make too big a deal of this. I don't hold to the notion that ambiguous terminology always impedes learning. Sure, "acceleration due to gravity" can be misleading, but even worse are words like "force" and "work". The latter 2 words existed in English before there were physics definitions, so students have to do a lot of thinking to e.g. get over the fact that pushing something that does not move results in no (physics) work done on that object. In my own experience, conflicts like these force a person to think. I am of the opinion that all real learning comes from thinking, not from remembering definitions. So e.g. when I learned g=acc. due to gravity, I pondered the fact that the reason I am not accelerating is because the floor knows exactly the force needed to cancel gravity. I learned a lot from this thought. Calling g by some generic name would also rob the students of making a direct connection between g and one of the (non-obvious) wonders of nature: that things freefall at the same rate. Contemplating this fact has taught me a lot.

 

[jcsd]

g is an accelartion and guess what causes it! gravity! Now I'm sure every student is aware that objects in gravitaional fields don't necessarily accelerate, but it's easy to remember that F = ma and that 'g' just slots into a to create F = mg. IMO describing it as a "universal constant" is X1000 more confusing.

 

[JohnDubYa]

Zooby, the description sounds okay to me.

JohnDubYa: I think you make too big a deal of this. I don't hold to the notion that ambiguous terminology always impedes learning."

It doesn't ALWAYS impede learning, but it sometimes does. I have seen the effects of calling g an acceleration, and they are very real.

RE: " Sure, "acceleration due to gravity" can be misleading, but even worse are words like "force" and "work".

Sure, but that is a battle that cannot be won because these definitions are so firmly entrenched throughout all of science. And I know of no better words to use as replacements.

RE: "In my own experience, conflicts like these force a person to think."

So you are advocating that we inject confusion into the curriculum to force students to sort it out on their own?

I think we should make physics as approachable as possible. This means we should use clear, precise language that minimizes confusion. Not everyone is an A student. Some students can work their way through the muck, but many others can't.

I don't disagree that calling g an acceleration and having students figure out the inconsistency can have a beneficial educational effect. But is the payoff worth it? Have you ever read a confusing instruction manual? When you finally figured it out, were you glad that the instruction manual was confusing? Sure, you may have learned something interesting while trying to sort out what to do, but the end result was just anger, wasted time, and a negative attitude towards the product. That is what I want to avoid.

Clarity and accuracy are the two most important features of quality writing.

 

[jcsd]

g is an accelration, it may not necessarily refer to the actual acceleration of an object, but nevertheless it is an acceleration.

 

[Nenad]

the equivalence principle states tha there is no difference between gravity and acceleration.

 

[JohnDubYa]

RE: "g is an accelration, it may not necessarily refer to the actual acceleration of an object, but nevertheless it is an acceleration."

g is a number. Acceleration is a physical event. What you mean to say is that objects, in certain situations, can accelerate at values numerically equal to g.

Sure, the distinction is subtle, but in my opinion important. Because defining g as an acceleration causes students problems.

RE: "the equivalence principle states tha there is no difference between gravity and acceleration."

Hmmm... I have a book sitting on my desk. Gravity acts on it, but it is not accelerating. Huh?

The units of acceleration are m/s^2. If gravity and acceleration are the same thing, do we assign m/s^2 as the units of gravity? If so, I have never seen gravity defined in such a manner. In fact, how exactly are you defining "gravity"?

A student in an introductory physics course is going to ask these questions. How would you answer them without leaving him scratching his head?

 

[zoobyshoe]

Zooby, the description sounds okay to me.

OK. So do you object to him saying that on the way up the ball undergoes a constant acceleration downward at 32ft/sec²? I would anticipate that you would since the ball is not even traveling downward, toward the earth, on it's way up.

 

[krab]

g is a number. Acceleration is a physical event. What you mean to say is that objects, in certain situations, can accelerate at values numerically equal to g.

Just a nit here, but an important one: g is not a number. It is a physical quantity. If you omit units, g can be any number.

 

[jcsd]

RE: "g is an accelration, it may not necessarily refer to the actual acceleration of an object, but nevertheless it is an acceleration."

g is a number. Acceleration is a physical event. What you mean to say is that objects, in certain situations, can accelerate at values numerically equal to g.

Sure, the distinction is subtle, but in my opinion important. Because defining g as an acceleration causes students problems.

As krab points out g has units of m/s^2, so it certianly is not just a number, it is also a vector and it is most defintely an acceleration. If it were not an acceleration we shouldn't be allowed to substiute it into F = ma. Even in non-relativistic physics acceleration can be 'transformed away' by using d'Alembert's principle, so whether an accelartion is an 'actual acceleration' (i.e. a co-ordinate acceleration) is subject to frame of reference. In fact when you transform an inertial frame to a non-inertial reference frame you find that there is a very real simlairty between the acceleration in the inertial reference frame and g.

 

[quarkman]

I am currently reviewing a freshman text and am in the middle of the section on gravity. Maybe I am complicating the subject but this is what the text states:

"The gravitational acceleration a computed with [a = GM/r^2] is not the same as the free-fall acceleration g that we would measure for the falling particle ... The two accelerations differ for three reasons: (1) Earth is not uniform, (2) it is no0t a perfect sphere, and (3) it rotates. Moreover, because g differs from a, the weight mg of the particle differs from the gravitational force on the particle as given by [F = GMm/r^2] for the same three reasons." Halliday, Resnick, Walker "Fundamentals of Physics Extended: Fifth Ed." 1997, pg. 326.

I originally found this confusing, but I feel it helps distinguish between the approximation g = 9.8 m/s/s and the actual force the mass of the Earth would exert on a particle at a distance r from its center of mass.

 

[LURCH]

Hmmm... I have a book sitting on my desk. Gravity acts on it, but it is not accelerating. Huh?

I think the basic principles of relativity would respond, "not accelerating relative to you". But according to the equivalence principle earlier cited, could not beset with equal accuracy that both you and the book are accelerating parallel to one another?

 

[VantagePoint72]

Yes, and also don't forget that Einstein defines freely falling reference frames as the benchmark for accelerated motion (spacetime itself and its curvature being the benchmark, freely falling reference frames being the way to recognize said curvature.) According to a reference point that is accelerating relative to you, an object at rest to you would be accelerating. Thus, according to the benchmark defined by Einstein (it's not a preferred frame by which to define motion, this would violate SR, it's only GR's embodiment of the equivalence principle), the desk your book is sitting on is accelerating upward as is the book, and much like when you feel pressed into your seat when your car accelerates, the book is pressed into the desk. It's the equivalence principle in action.

 

[JohnDubYa]

RE: "So do you object to him saying that on the way up the ball undergoes a constant acceleration downward at 32ft/sec2? I would anticipate that you would since the ball is not even traveling downward, toward the earth, on it's way up."

Huh? I don't see why you would think I would object. His explanation is perfectly reasonable.

RE: "I think the basic principles of relativity would respond, "not accelerating relative to you". But according to the equivalence principle earlier cited, could not beset with equal accuracy that both you and the book are accelerating parallel to one another?"

So what? How does that possibly answer the question. This appears to be a non-sequitur.

In retrospect (having seen all of your arguments), calling g the free-fall acceleration wouldn't bother me too much. (I still like "gravitational field" better.) Calling it the acceleration due to gravity, however, does. And according to quarkman's argument, it isn't even accurate.

I think the allusions to relativity are pointless in this debate.

 

[VantagePoint72]

Why the issue with calling it the acceleration due to gravity? It doesn't say that is how fast a given object will accelerate towards the earth, it just says that's the magnitude of gravity's contribution to its acceleration: approx. 9.81 m/s^2 towards the centre of mass of the earth. It's perfectly accurate. My previous references to relativity weren't pointless, relativity is the refinement of classical physics. While it's effects are hardly noticable at velocities we experience, it's explanations of the reason things happen they way do is always applicable. If you ignore it then you aren't looking at the full picture. The point is, an object in a gravitational field is accelerating. Always. That's what gravity IS. Calling g the acceleration due to gravity is absolutely accurate, it's the general, all encompassing way of saying it. Calling it "free-fall acceleration" is more specific. It excludes the situation of the book on the desk in one of your previous posts. The book is still accelerating! Acceleration due to gravity is a broader definition, and hence more applicable.

 

[JohnDubYa]

RE: "Why the issue with calling it the acceleration due to gravity? It doesn't say that is how fast a given object will accelerate towards the earth, it just says that's the magnitude of gravity's contribution to its acceleration: approx. 9.81 m/s^2 towards the centre of mass of the earth. It's perfectly accurate."

But very confusing to weaker students. They will think that a block sliding down a plane must accelerate at g. After all, gravity is making the block slide down the plane, and the acceleration due to gravity is g. So the block must accelerate at g.

Sure, WE know better. But they don't. And you can throw in as many caveats as you wish, the bad terminology will ultimately dominate.

This is one of the most common mistakes made in introductory physics in my experience, and is caused by a misleading description.

RE: "My previous references to relativity weren't pointless, relativity is the refinement of classical physics."

This argument centers around defining g in introductory physics. Definitions that depend on an understanding of special relativity are not going to help students.

RE: "The point is, an object in a gravitational field is accelerating. Always."

Sure, tell that to a student in an introductory physics class and watch the dropout rate.

The first thing that a student will ask is, "Are you saying that the forces acting on an object can never cancel?" What do you say in response?

RE: "That's what gravity IS. Calling g the acceleration due to gravity is absolutely accurate, it's the general, all encompassing way of saying it. Calling it "free-fall acceleration" is more specific. It excludes the situation of the book on the desk in one of your previous posts. The book is still accelerating!"

The student is going to ask, in which direction? What do you say in response?

 

[ArmoSkater87]

This argument about the terminology has dragged on this long probably because there hasnt been an actual students opinion...well I am here to solve that problem, I am a high school student that took physics this past year. Acceleration due to gravity is most certainly the easiest way to get across to anyone, what g actually is. Its very clear, and obviously self explanatory. Saying anything else would confuse the crap out of a student. It might not seem that way to some, but it will because not every student that wants to know physics is necessarily a very bright one. Anyways, I, as a student agree that with everyone that prefers "acceleration due to gravity" instead of "universal constant". In fact if i didnt know any physics, i would think that g is the same for all planets and object in the universe from that terminology (universal constant).

 

[VantagePoint72]

OK, JohnDubya, I think I understand your point of view now. You're questioning the simplicity of calling it "acceleration due to gravity" not the accuracy of it. Fair enough, I can see where there would be confusion among some students about it. The point is though, scientifically it's the most correct name. What would answer if the student asked me which way the book is accelerating? Upward, as the equivalence principle predicts and is confirmed by the freely falling observer. And reading that again, yes that would confuse the heck out of most students. Even so, the fact that the book even IS accelerating isn't introduced till GR. So, leaving some of the more complicated details until later in their studies, by calling g the "acceleration due to gravity" the students who do choose to go on to more complicated subjects won't have to completely switch around their thinking. I know, like ArmoSkater I'm a student too who happened to be so taken in with physics in my first course, that I've been teaching myself relativity, quantum theory, etc. Earlier, I had been a little confused about why everything accelerates at g in the Earth's gravity, even if we view them as being at rest, but I didn't exactly lose sleep over it. And it made understanding the equivalence principle that much easier.

 

[JohnDubYa]

RE: "This argument about the terminology has dragged on this long probably because there hasnt been an actual students opinion..."

Actually, some of the posters in this thread are probably students.

RE: "well I am here to solve that problem, I am a high school student that took physics this past year. Acceleration due to gravity is most certainly the easiest way to get across to anyone, what g actually is."

Well, to YOU. But don't assume that everyone thinks like you.

As I said earlier, assigning g as the acceleration of objects that are not in free fall is a tremendous problem in introductory physics. Obviously there are a lot of confused students out there. I am suggesting that calling g the acceleration due to gravity is the primary cause of the confusion.

RE: "Its very clear, and obviously self explanatory. Saying anything else would confuse the crap out of a student."

If so, then how did you learn physics related to the electric field? After all, if the term "gravitational field" confuses the crap out of you, how could you possibly learn electromagnetism with its reference to the electric field? The two are directly analagous.

RE: "It might not seem that way to some, but it will because not every student that wants to know physics is necessarily a very bright one. Anyways, I, as a student agree that with everyone that prefers "acceleration due to gravity" instead of "universal constant".

I don't recall the term "universal constant" being offered in this thread. Where did you read that?

 

[JohnDubYa]

RE: "What would answer if the student asked me which way the book is accelerating? Upward, as the equivalence principle predicts and is confirmed by the freely falling observer."

And a student is going to wonder what allows you to examine the physics of a event in a non-inertial reference frame. And why you have chosen that particular reference frame, when special relativity assumes that there is no preferred reference frame.

RE: "And reading that again, yes that would confuse the heck out of most students."

Most students? I am not aware of any introductory physics student who would have any hope of understanding the topic with such a definition.

RE: "Even so, the fact that the book even IS accelerating isn't introduced till GR. So, leavi"ng some of the more complicated details until later in their studies, by calling g the "acceleration due to gravity" the students who do choose to go on to more complicated subjects won't have to completely switch around their thinking."

Actually, calling g the gravitational field shouldn't either.

 

[robphy]

What would answer if the student asked me which way the book is accelerating? Upward, as the equivalence principle predicts and is confirmed by the freely falling observer. And reading that again, yes that would confuse the heck out of most students. Even so, the fact that the book even IS accelerating isn't introduced till GR. So, leaving some of the more complicated details until later in their studies, by calling g the "acceleration due to gravity" the students who do choose to go on to more complicated subjects won't have to completely switch around their thinking.

In GR, the book at rest on a table on the Earth's surface is accelerating upward (as you say).
Why is it said to be "accelerating" in GR? It is because the book is not moving inertially. Its worldline is not geodesic. [In GR, acceleration is an invariant measure of worldline curvature.] By contrast, a freefalling book has a worldline which is geodesic. Indeed, a freefalling accelerometer will read zero.

Now, what agent is applying the force on the book (keeping it off a geodesic)?
It is the table pushing upward... the book's acceleration is "due to the table".
(Near the Earth's surface, the magnitude of this upward acceleration is 9.8 m/s^2.) [Note that, in GR, "forces" always refer to "non-gravitational forces".]

So, it seems to me, that invoking GR suggests that "acceleration due to gravity" is not meaningful. ("Freefall acceleration" would also not be meaningful.)

I know, like ArmoSkater I'm a student too who happened to be so taken in with physics in my first course, that I've been teaching myself relativity, quantum theory, etc. Earlier, I had been a little confused about why everything accelerates at g in the Earth's gravity, even if we view them as being at rest, but I didn't exactly lose sleep over it. And it made understanding the equivalence principle that much easier.

 

[ArmoSkater87]

JohnDubY: First of all, "universal constant" was mentioned earlier in the thread. Second...u say not everyone might think like I do, then why should everyone think like YOU?? I didnt necessarily mean to say that everyone should think of "acceleration due to gravity" like i do, i was simply offering my opinion on it as a student. I understand that you are too, i really don't have a problem with "free fall acceleration", it's not a bad way to explain g, but its more specific than nessasary. In my opinion, i think the student should get the bigger picture first. "Acceleration due to gravity" just seems very clear, g is an acceleration...cause by gravity. Seems pretty simple to any student taking physics. To a student the whole deal with "accelaration due to gravity" being inaccurate doesn't come to mind at all, since the problems they deal with uses g as an acceleration in some gravitational field. I'm just giving my view, no one has to agree with it. :D

 

[JohnDubYa]

I sure don't remember anyone offering "universal constant" as an acceptable term for g. *I* certainly didn't. But I agree that such a term is a terrible definition.

RE: "Second...u say not everyone might think like I do, then why should everyone think like YOU??"

I offer my opinion based on teaching well over 1000 students over the past few years, not on my own cognition. After all, *I* am not confused by the definition. But too many of my students have been.

RE: "I didnt necessarily mean to say that everyone should think of "acceleration due to gravity" like i do, i was simply offering my opinion on it as a student."

Okay, fair enough.

RE: "I understand that you are too, i really don't have a problem with "free fall acceleration", it's not a bad way to explain g, but its more specific than nessasary."

I think its specificity is exactly why it would work. By being very specific, it is less likely to be used in cases where it doesn't apply.

RE: "In my opinion, i think the student should get the bigger picture first. "Acceleration due to gravity" just seems very clear, g is an acceleration...cause by gravity. Seems pretty simple to any student taking physics."

Any student? That is a pretty big statement. Believe me, the number of students that will go home and screw it up is quite considerable (and unnecessary).

RE: "To a student the whole deal with "accelaration due to gravity" being inaccurate doesn't come to mind at all, since the problems they deal with uses g as an acceleration in some gravitational field. I'm just giving my view, no one has to agree with it. :D"

Let me give you an example of where things go wrong.

You ask a student to find the time it takes for a block to slide down a plane. You give them the mass of the block, the geometry of the plane, and the kinetic coefficient of friction.

The student is expected to use Newton's second law to find the acceleration along the plane, and then use that acceleration to find the time of motion.

But many students will say to themselves, "Gravity is pushing this block down the plane. The acceleration due to gravity is 9.8 m/s/s. Therefore, I will use g in my distance formula..."

And you can talk about the proper use of g until the cows come home, but as long as it is called the "acceleration due to gravity" they will use it in such situations. The term "free-fall acceleration" is much better, because at least it hints that there is a problem with using it since the block is obviously not in free fall. (I like gravitational field, but let's ignore that definition for the purpose of this discussion.)

 

[ArmoSkater87]

Okay, what you say if fair enough. I had no idea you were a teacher I though you were a student. I guess some students can be misled by it. To me it would be clear that the friction decreases the acceleration, but i guess not every student can realize that. But I think "acceleration due to gravity" sounds cooler. :D

 

[VantagePoint72]

Well, JohnDubya, you're the teacher, if you think it will make the concepts clearer to your students then go for it. While my preference will still be to refer to it as the acceleration due to gravity, what I call it won't have the slightest bearing on the people you teach anyway.

 

[JohnDubYa]

Well, I think this has been a great thread. I understand your points. If you ever get to teach, do a little experimentation and see if the term acceleration due to gravity is confusing to students.

And "gravitational fields" is definitely cooler than "acceleration due to gravity." Students think "wow, fields... man! Like, force fields, and electric fields, and Fields of the Nephilim, and ..."

 

[jcsd]

John, upon review you didn't call g a universal consant, Doc Al did, tongue firmrly in cheek. But you did call it the graviational field constant which is what I object to strongly as it's not a constant (the value of g varies measurably even on the Earth's surface); if somone was talking about "the graviational field constant" G rather than g would spring into my mind (infact seraching for this term returns very few results, some of which refer to g others of which refer to G, though I am suprised to see that it does seem some school curriculums do use this term for g). Freefall acceleration is a much more acceptable term and it's it's one that is often preferred.

 

[Doc Al]

John, upon review you didn't call g a universal consant, Doc Al did, tongue firmrly in cheek.

Where did I do that? What I actually said was:

I'll probably start calling g the "gravitional field strength/constant at the Earth's surface".

Yes, my tongue was somewhat in cheek, since most students would give a blank stare at such a statement, at least at the time when "g" is first introduced. But on further reflection, considering that most students learn kinematics before dynamics before gravitation, I would probably say:
In covering the kinematics of projectile motion:

the freefall acceleration due to gravity = g (downwards)​

(As krab pointed out, this is a crucial fact that must be taught.)

In covering dynamics (Newton's second law):

the "weight" of body is w = mg (downward) (here pointing out that "g" is a measure of the strength of the Earth's gravitational field--more to come)​

In covering Newton's law of gravity:

here we can explicitly derive that "g" is the gravitational field strength (Force per unit mass) at the Earth's surface​

So I would have to start calling "g" the magnitude of the freefall acceleration due to gravity, and then later add that it is a measure of the gravitational field strength at the Earth's surface. Which is what I do already. Oh well.

Even though we've beat this topic to death, it still has been instructive. If nothing else, it has made me more aware of how easy it is to confuse students with sloppy terminology.