Gravity and Weight: Newton's Second Law

[nDever]

Hey,

About Newton's Second Law. To compute our weight on Earth using standard g, we use F=ma replacing F and a with W and g (or -g), respectively. So then from this, I gather that weight is a consequence of gravitational acceleration (and mass). So then, it is appropriate to say that our weight, which is a force, is caused by another force (gravity)?

 

[Doc Al]

So then, it is appropriate to say that our weight, which is a force, is caused by another force (gravity)?

Your weight is the gravitational force on you exerted by the earth. It happens to equal mg near the Earth's surface.

 

[nDever]

Your weight is the gravitational force on you exerted by the earth. It happens to equal mg near the Earth's surface.

But g is an acceleration. Acceleration is caused by force (according to the first law of motion). What force causes g?

 

[Doc Al]

What force causes g?

The gravitational force of the Earth on the object.

 

[nDever]

Your weight is the gravitational force on you exerted by the earth. It happens to equal mg near the Earth's surface.

But g is an acceleration. Acceleration is caused by force (according to the first law of motion). What force causes g?

The gravitational force of the Earth on the object.

So our weight causes our own acceleration..?

 

[Doc Al]

So our weight causes our own acceleration..?

Of course. What we call 'our weight' is the force of the Earth's gravity pulling us down. If that's the only force acting on us, we will have a downward acceleration equal to g.

 

[Naty1]

Far be it from me to disagree with Doc Al but I just don't like this:

So our weight causes our own acceleration..?

Of course.

I would much prefer to say our MASS causes our own acceleration. This distinction becomes significant in relativity...and it turns out other stuff like pressure
also affects gravitational acceleration.

This from Doc Al is much better in my opinion:

Your weight is the gravitational force on you exerted by the earth.

Check here for a good discussion of various aspects of WEIGHT:

http://en.wikipedia.org/wiki/Weight

 

[nDever]

So the acceleration due to Earth's gravity is constant because we are in the reference frame of the Earth, correct? Whenever an object accelerates towards Earth, Earth accelerates towards that object at a rate directly proportional to the other object's mass, correct? Is this why the mass of the other object cancels out?

 

[D H]

So the acceleration due to Earth's gravity is constant because we are in the reference frame of the Earth, correct?

Acceleration due to Earth's gravity is approximately constant. It varies with latitude, with altitude, and even from place to place at the same latitude/altitude. Gravitational acceleration at the poles is about half a percent more than that at the equator, gravitational acceleration atop a high mountain measurably smaller than it is at sea level, and those place to place variations are key to finding things like oil.

Whenever an object accelerates towards Earth, Earth accelerates towards that object at a rate directly proportional to the other object's mass, correct?

Yes, but this acceleration is immeasurably small.

Is this why the mass of the other object cancels out?

No. It cancels out because of the forms of Newton's law of gravitation, $F=GmM/r^2$, and Newton's second law, $F=ma$. Via transitivity (if a=b and a=c, then b=c) we must have $ma = GmM/r^2$, or $a=GM/r^2$.

 

[nDever]

Acceleration due to Earth's gravity is approximately constant. It varies with latitude, with altitude, and even from place to place at the same latitude/altitude. Gravitational acceleration at the poles is about half a percent more than that at the equator, gravitational acceleration atop a high mountain measurably smaller than it is at sea level, and those place to place variations are key to finding things like oil.


Yes, but this acceleration is immeasurably small.


No. It cancels out because of the forms of Newton's law of gravitation, $F=GmM/r^2$, and Newton's second law, $F=ma$. Via transitivity (if a=b and a=c, then b=c) we must have $ma = GmM/r^2$, or $a=GM/r^2$.

Is the acceleration approximately constant because the force of gravitational pull acts strongly on more massive objects and weaker on less massive objects? The mass of the attracted object has no bearing whatsoever on its acceleration?

 

[Doc Al]

Is the acceleration approximately constant because the force of gravitational pull acts strongly on more massive objects and weaker on less massive objects?

Yes. The gravitational force on an object is directly proportional to its mass. But the acceleration, given by Newton's 2nd law, is inversely proportional to its mass. Thus those two factors balance out, giving a constant acceleration regardless of mass.

 

[nDever]

Yes. The gravitational force on an object is directly proportional to its mass. But the acceleration, given by Newton's 2nd law, is inversely proportional to its mass. Thus those two factors balance out, giving a constant acceleration regardless of mass.

I believe I understand.

If I were to push a 1 kg supermarket cart down an aisle with a constant force of 1 N, it would accelerate at 1 m/s/s. If I were to put more stuff in the cart such that it has 2 kg of mass and then I push it with the same 1 N of force, then the acceleration halves and it accelerates at .5 m/s/s. These are the effects of applied forces. Double the mass, half the acceleration. Triple the mass, one third the acceleration.

Now Gravity is just a different force where the magnitude of the force is directly proportional to the product of the two masses (lets ignore the distance, I understand that). So then, the force of gravity increases and decreases with the mass. Its the nature of gravity. If the mass is small, the force is small. If the mass is large, the force is large. So then if a small and large mass is dropped from the same height, gravity pushes lightly on the less massive object and heavily on the more massive object, thus the acceleration is the same.

In a sense, gravity acts like this.

Now I have two supermarket carts placed side by side, one with a bunch of stuff in it, the other with very little in it, and I get two people to push them at the same time. The cart with more in it gets pushed harder and the other with less items gets pushed lighter. Let them go, and they move at the same rate with respect to one another.

Do I have the right idea?

 

[Doc Al]

Do I have the right idea?

I think you do.

For a more mathematical view, review D H's post #9 where he shows that the acceleration due to gravity does not depend on the mass of the object.

 

[nDever]

Ah, I see. It has everything to do with the mathematical definition of acceleration in terms of force.

So then any two objects that are dropped from the same height within the gravitational field of any more massive body will fall at the same time. After gravity takes it's course, the only thing that the acceleration depends on is the mass of the larger body and the distance between it and the others. As altitude increases, g decreases.

The reason that Earth's g is about 9.8 m/s/s is only because of the way that the Earth's mass and our distance from its center works out. Because the Earth is large and we are relatively small, the everyday heights that we achieve allow us to average g out to be 9.8 m/s/s.

If this be the case, then it can be said that it is impossible for any two objects dropped from two different heights to land on the same point at the same time (with negligible air resistance). Is this sound?