Understanding Gravity: The Relationship Between Mass and Acceleration

[barnaby]

This is something that's been troubling me for a while:

If I were to pull a 1kg mass with a force of 10N, it would accelerate at 10ms^-2. If I applied the same force to an object twice as massive, its acceleration would be half as large.

However, everything accelerates at ~9.81ms^-2 under the gravity of the Earth. This suggests to me that gravity pulls harder on more massive objects than it does on less massive ones - and I don't understand how that can be.

 

[neutrino]

everything accelerates at ~9.81ms^-2 under the gravity of the Earth.

Not quite. The value of (approximately) 9.8ms^-2 is at the surface of a perfectly spherical object with a mass of roughly 6e24Kg, and a radius of 6400Km. (Or a point particle with the same mass, with the test mass 6400Km away.) In other words, an idealised Earth. In many textbook problems, say ones involving projectiles, you might have been asked to assume that that acceleration due to gravity stays constant with height. That is because the value varies very little for
heights << radius of Earth. In reality, the value of acceleration changes, since the force of gravity is inversely proportional to the square of the distance between the two objects.

 

[robphy]

Newton's Universal Law of Gravitation:
the magnitude of the gravitational force on m due to M
$$F_{grav}= \frac{GMm}{r^2}$$
is proportional to the target mass m (and the source mass M).

 

[Janus]

Gravity pulls harder on more massive objects because the force of gravity between two objects is porportional to the product of the two masses, as shown in Newtons's equation:

$$F_g = \frac{GM_1 M_2}{d^2}$$

 

[barnaby]

I appreciate that that is true, but I can't really accept that as an explanation for *why* - as it just puts into an equation what I sort of already knew - I suppose the question I really want to ask is "what is gravity?"...

___

So $$\frac{GM_1 M_2}{d^2}$$ gives the force with which one object's 'gravity' will pull on another object? What is G in the equation? How do you define/calculate it?

___

I'm sorry if I'm annoying you - the only explanation my physics teacher provides is 'because God made it that way'...

 

[neutrino]

the only explanation my physics teacher provides is 'because God made it that way'...

What?

As to your question, G is a constant, which has a value of approximatley 6.67 x 10¹¹ Nm²Kg^-2 in the SI units.

This site might be helpful
http://hyperphysics.phy-astr.gsu.edu/hbase/grav.html#grav

 

[robphy]

I appreciate that that is true, but I can't really accept that as an explanation for *why* - as it just puts into an equation what I sort of already knew - I suppose the question I really want to ask is "what is gravity?"...

___

So $$\frac{GM_1 M_2}{d^2}$$ gives the force with which one object's 'gravity' will pull on another object? What is G in the equation? How do you define/calculate it?

___

I'm sorry if I'm annoying you - the only explanation my physics teacher provides is 'because God made it that way'...

We don't really know why... although experiment shows that it's a pretty good description of gravity.
A lot current research is interested in understanding why...
...but to get there we have to understand as much as we can about it...
including what its implications are and how it can be checked against experiment.

 

[barnaby]

What?

That's what happens when you're taught science by someone who believes in Intelligent Design... :uhh:

Thanks for the link.

 

[Parlyne]

Don't feel to bad about being troubled by this particular property of gravity. No less a figure than Einstein found this issue troublesome - so much so that it led rather directly to the formulation of general relativity. Conceptually, the argument is something like this - other than gravity, every time we see a force proportional to the mass of the object it's acting on, it turns out that the force is really just the effect of the object's inertia when viewed in a non-inertial frame of referece. The canonical example of this is the centrifugal force. When you're in a rotating frame of reference (a car making a turn, a rollercoaster on a loop-the-loop, etc.), you feel like you're being pushed outwards from the center of rotation, when, in fact, what's really happening is that you're being pulled away from the straight-line path that inertia would have you take in the absence of the force causing your circular motion.

Einstein took this idea quite seriously and posited that gravity itself is just such an effect. This would mean that the only truly inertial frames of reference are those of objects in free-fall. The big complication from this is that difference free-falling paths in a non-uniform gravitational field do not have a common frame of reference. In other words, there are no universal inertial frames, only locally defined ones!

 

[Kurdt]

This has troubled many physicists over many centuries. There is no reason for inertial mass (the mass found in F=ma) to be equivalent to gravitational mass (the mass of the object in Newton's universal law of gravitation). General relativity gives a nice explanation as to why this might be so.

As for your teacher, I'd seriously contemplate complaining to someone in power if those are the answers they're giving in class to student questions.

EDIT: Forgot to mention that if you're interested in reading a little more into the equivalence principle then the following living review is quite good. There are a few parts with some heavy maths in but mainly its just a lot of good explanations.

http://arxiv.org/PS_cache/gr-qc/pdf/0504/0504086v1.pdf

 

[rbj]

I appreciate that that is true, but I can't really accept that as an explanation for *why* - as it just puts into an equation what I sort of already knew - I suppose the question I really want to ask is "what is gravity?"...

___

So $$\frac{GM_1 M_2}{d^2}$$ gives the force with which one object's 'gravity' will pull on another object? What is G in the equation? How do you define/calculate it?

___

I'm sorry if I'm annoying you - the only explanation my physics teacher provides is 'because God made it that way'...

We don't really know why... although experiment shows that it's a pretty good description of gravity.
A lot current research is interested in understanding why...
...but to get there we have to understand as much as we can about it...
including what its implications are and how it can be checked against experiment.

we have some theoretical reason to believe that the force of gravity be proportional to both of the masses. and because an inverse-square relationship is natural in a 3-D space, there seems to me to be some theoretical preference to Newton's gravitational law.$$F_G = (4 \pi G)\frac{m_1 m_2}{4 \pi r^2}$$

i just think that $4 \pi G$ is a more "fundamental" constant of proportionality than just $G$.

 

[kudoushinichi88]

I'm sorry if I'm annoying you - the only explanation my physics teacher provides is 'because God made it that way'...

I believe in Intelligent Design, but I completely abhor such mentality; answering questions like that when there is well scientific and mathematical elucidation. It reflects indolence and capitulation without putting in any effort in understanding nature... how are we going to progress if we keep giving that answer?

Sorry for going off topic, but yeah, the Earth do pull larger objects with greater force.

As for what is gravity, to put it simply, it can be thought as a force (Newton), or curvature of space-time (Einstein). The Standard Model of Particles hypothesised the Graviton, the particle that mediates gravitational force between particles of matter. It can be thought that exchange of Gravitons between particles is what causes gravity.

As for what REALLY is gravity, no one really knows yet. Even if we discover a better explanation for gravity in the future, there will always be some doubt in the newly discovered theory... Like what Richard Feynman said, you need some doubt in science to progress... or something like that.

Correct me if I'm wrong.