Spring in free fall

[Luigi Fortunati]

In my drawing
https://www.geogebra.org/m/vkz6htmu
the two springs A and B (with mass) have the same length at rest: 8.

Spring A is in the remote space and maintains its length 8.

Spring B is stationary in a well in the center of the Earth.

Is spring B in free fall even though it is stationary?

Spring B contracts with respect to its resting length, as it does in my
drawing?

[Richard Livingston]

Post by Luigi Fortunati
In my drawing
https://www.geogebra.org/m/vkz6htmu
the two springs A and B (with mass) have the same length at rest: 8.
Spring A is in the remote space and maintains its length 8.
Spring B is stationary in a well in the center of the Earth.
Is spring B in free fall even though it is stationary?
Spring B contracts with respect to its resting length, as it does in my
drawing?

Your diagram is correct, spring B will be compressed due to the gravity of
the earth.

To say that spring B is in free fall is ignoring the fact that it extends over
an extended volume of space, and thus parts experience gravitational
strains that other parts don't. Certainly the ends of the spring are not
"in free fall", although you might argue that the center of gravity is.

Rich L.

[Luigi Fortunati]

Post by Richard Livingston

Post by Luigi Fortunati
In my drawing
https://www.geogebra.org/m/vkz6htmu
the two springs A and B (with mass) have the same length at rest: 8.
Spring A is in the remote space and maintains its length 8.
Spring B is stationary in a well in the center of the Earth.
Is spring B in free fall even though it is stationary?
Spring B contracts with respect to its resting length, as it does in my
drawing?

Your diagram is correct, spring B will be compressed due to the gravity of
the earth.
To say that spring B is in free fall is ignoring the fact that it extends
over an extended volume of space, and thus parts experience gravitational
strains that other parts don't. Certainly the ends of the spring are not
"in free fall", although you might argue that the center of gravity is.
Rich L.

True.

But also Einstein's elevator extends over an extended volume of space
and, therefore, only its center is in free fall but not the top and
bottom ends, where gravity is different.

So, if we can't say my spring is in free fall, we can't say it for
Einstein's elevator either!

Luigi

[[Mod. note -- If Einstein's elevator were as big as the Earth, then
you'd be right. But the size of Einstein's elevator is much less than
the size of the Earth (= the spatial scale over which the gravitational
field varies).

In the presence of non-uniform gravitational fields, "free fall" is a
*local* phenomenon, i.e., it's rigorously defined (only) in terms of a
limit where the size of our experimental region (i.e., the "elevator")
goes to zero.
-- jt]]

[Luigi Fortunati]

Post by Luigi Fortunati

Post by Richard Livingston

Post by Luigi Fortunati
In my drawing
https://www.geogebra.org/m/vkz6htmu
the two springs A and B (with mass) have the same length at rest: 8.
Spring A is in the remote space and maintains its length 8.
Spring B is stationary in a well in the center of the Earth.
Is spring B in free fall even though it is stationary?
Spring B contracts with respect to its resting length, as it does in my
drawing?

Your diagram is correct, spring B will be compressed due to the gravity of
the earth.
To say that spring B is in free fall is ignoring the fact that it extends
over an extended volume of space, and thus parts experience gravitational
strains that other parts don't. Certainly the ends of the spring are not
"in free fall", although you might argue that the center of gravity is.
Rich L.

True.
But also Einstein's elevator extends over an extended volume of space
and, therefore, only its center is in free fall but not the top and
bottom ends, where gravity is different.
So, if we can't say my spring is in free fall, we can't say it for
Einstein's elevator either!
Luigi
[[Mod. note -- If Einstein's elevator were as big as the Earth, then
you'd be right. But the size of Einstein's elevator is much less than
the size of the Earth (= the spatial scale over which the gravitational
field varies).
In the presence of non-uniform gravitational fields, "free fall" is a
*local* phenomenon, i.e., it's rigorously defined (only) in terms of a
limit where the size of our experimental region (i.e., the "elevator")
goes to zero.
-- jt]]

In my drawing
https://www.geogebra.org/m/vkz6htmu
I added a slider to modify the length of the spring at will: you say it
is in "free fall" only when it is at zero and for no other value?

[[Mod. note --
No, I am saying that unless the spring is infinitesimally small, some
part(s) of the spring may be in free fall AND other part(s) may (in
practice, are very likely to) be not-in-free-fall. More generally,
we need to be precise as to *what* either is or isn't in free-fall:
simplying saying "the spring" isn't sufficiently precise. I expanded
on this reasoning in a separate posting in this thread, which has not
yet appeared.
-- jt]]

[Jonathan Thornburg [remove -color to reply]]

Post by Luigi Fortunati
In my drawing
https://www.geogebra.org/m/vkz6htmu
the two springs A and B (with mass) have the same length at rest: 8.
Spring A is in the remote space and maintains its length 8.
Spring B is stationary in a well in the center of the Earth.
Is spring B in free fall even though it is stationary?
Spring B contracts with respect to its resting length, as it does in my
drawing?

[[I'm discussing this in the context of Newtonian mechanics.]]

I'll start with the easy question: Yes, spring B contracts with
respect to its resting length, because it's in a spatially-variable
gravitational field which points down at the top of spring B, and
up at the bottom of spring B.

Now to the trickier question: what about "free fall"?

The concept of "being in free fall" is easy for a point mass.
But Luigi is asking about extended bodies (bodies with non-trivial
size and internal structure). Here there are different ways to think
about what "being in free fall" means. In particular, we need to
distinguish between two quite different questions:

Question #1: Is the object's center of mass in free-fall? This is
true if and only:
(a) the net non-gravitational force acting on the object (i.e., the
vector sum of all non-gravitational forces acting on the object)
is zero. Or, we could say;
(b) the net *external* non-gravitational force acting on the object
(i.e., the vector sum of all *external* non-gravitational forces
acting on the object, where "external" means "applied by something
that's not itself part of the object") is zero.

[Note that formulations (a) and (b) are actually exactly
equivalent, because the vector sum of all *internal*
non-gravitational forces acting on an object (i.e., forces
where one part of the object exerts a force on another part
of the object), must be zero by Newton's 2nd law.

In practice, formulation (b) is usually more convenient,
because it lets us ignore internal forces, e.g., in this
case it lets us ignore the forces one part of spring B
exerts on another part of spring B.]

The answer to Question #1 for Luigi's spring B is "yes": spring B's
center of mass is in free-fall, because there are no non-gravitational
forces acting on the spring.

Question #2: Is each part (or some specific part(s)) of the object
in free-fall? For some small part X of the object (small enough that
we can neglect it's internal structure, and assume that the gravitational
field is constant across it's diameter), X is in free-fall if and only if
the (vector) sum of any external non-gravitational forces acting on X is
zero.

If X is any part of spring B other than the part right at the center of
the Earth, then the answer to question #2 is "no": there is a non-zero
net force exerted on X by other parts of the spring.

SUMMARY:

The center-of-mass of Luigi's spring B *is* in free fall, but almost all
of the individual parts of spring B are *not* in free fall. A common
(albeit slightly imprecise) shorthand terminology for this is to say that
"spring B is in free-fall in a tidal gravitational field".

[Luigi Fortunati]

Post by Jonathan Thornburg [remove -color to reply]

Post by Luigi Fortunati
In my drawing
https://www.geogebra.org/m/vkz6htmu
,,,the two springs A and B (with mass) have the same length at rest: 8.
Spring A is in the remote space and maintains its length 8.
Spring B is stationary in a well in the center of the Earth.
Is spring B in free fall even though it is stationary?
Spring B contracts with respect to its resting length, as it does in my
drawing?

[[I'm discussing this in the context of Newtonian mechanics.]]
I'll start with the easy question: Yes, spring B contracts with
respect to its resting length, because it's in a spatially-variable
gravitational field which points down at the top of spring B, and
up at the bottom of spring B.
Now to the trickier question: what about "free fall"?
The concept of "being in free fall" is easy for a point mass.
But Luigi is asking about extended bodies (bodies with non-trivial
size and internal structure). Here there are different ways to think
about what "being in free fall" means. In particular, we need to
Question #1: Is the object's center of mass in free-fall? This is
(a) the net non-gravitational force acting on the object (i.e., the
vector sum of all non-gravitational forces acting on the object)
is zero. Or, we could say;
(b) the net *external* non-gravitational force acting on the object
(i.e., the vector sum of all *external* non-gravitational forces
acting on the object, where "external" means "applied by something
that's not itself part of the object") is zero.
[Note that formulations (a) and (b) are actually exactly
equivalent, because the vector sum of all *internal*
non-gravitational forces acting on an object (i.e., forces
where one part of the object exerts a force on another part
of the object), must be zero by Newton's 2nd law.
In practice, formulation (b) is usually more convenient,
because it lets us ignore internal forces, e.g., in this
case it lets us ignore the forces one part of spring B
exerts on another part of spring B.]
The answer to Question #1 for Luigi's spring B is "yes": spring B's
center of mass is in free-fall, because there are no non-gravitational
forces acting on the spring.
Question #2: Is each part (or some specific part(s)) of the object
in free-fall? For some small part X of the object (small enough that
we can neglect it's internal structure, and assume that the gravitational
field is constant across it's diameter), X is in free-fall if and only if
the (vector) sum of any external non-gravitational forces acting on X is
zero.
If X is any part of spring B other than the part right at the center of
the Earth, then the answer to question #2 is "no": there is a non-zero
net force exerted on X by other parts of the spring.
The center-of-mass of Luigi's spring B *is* in free fall, but almost all
of the individual parts of spring B are *not* in free fall. A common
(albeit slightly imprecise) shorthand terminology for this is to say that
"spring B is in free-fall in a tidal gravitational field".

I agree with everything you write but, if so, no extended body could ever be in free fall!

[Luigi Fortunati]

Post by Jonathan Thornburg [remove -color to reply]

Post by Luigi Fortunati
In my drawing
https://www.geogebra.org/m/vkz6htmu
the two springs A and B (with mass) have the same length at rest: 8.
Spring A is in the remote space and maintains its length 8.
Spring B is stationary in a well in the center of the Earth.
Is spring B in free fall even though it is stationary?
Spring B contracts with respect to its resting length, as it does in my
drawing?

.....
Question #2: Is each part (or some specific part(s)) of the object
in free-fall? For some small part X of the object (small enough that
we can neglect it's internal structure, and assume that the gravitational
field is constant across it's diameter), X is in free-fall if and only if
the (vector) sum of any external non-gravitational forces acting on X is
zero.
If X is any part of spring B other than the part right at the center of
the Earth, then the answer to question #2 is "no": there is a non-zero
net force exerted on X by other parts of the spring.
The center-of-mass of Luigi's spring B *is* in free fall, but almost all
of the individual parts of spring B are *not* in free fall.

Based on what you yourself write, an extended body (without external
constraints) in remote space is entirely in free fall in every single
particle (none excluded), while the same extended body that falls into a
well on the Earth, is not entirely in free fall because (almost) all its
own particles (although without external constraints) bind each other!

All of this confirms what I wrote: free fall near gravitational masses
is *different* from free fall in remote space.

[Phillip Helbig (undress to reply)]

Post by Luigi Fortunati
Based on what you yourself write, an extended body (without external
constraints) in remote space is entirely in free fall in every single
particle (none excluded), while the same extended body that falls into a
well on the Earth, is not entirely in free fall because (almost) all its
own particles (although without external constraints) bind each other!
All of this confirms what I wrote: free fall near gravitational masses
is *different* from free fall in remote space.

The difference is that tidal effects have to be taken into account near
gravitational masses. All such discussions explicitly or implicitly
assume that the tidal effects are "small enough". In other words, in
the limit of arbitrarily small tidal effects, the two are equivalent.