Spring balance analysis from a system point of view

[cianfa72]

TL;DR Summary

Analysis of a spring balance with an attached body from Newton mechanics point of view; definition of "system" and forces involved

Hi,
I was thinking about a simple scenario in the framework of Newton (classic) mechanics.

Take a spring scale/balance fixed at one end (wall) with a body attached to the other end's hook. From an analysis point of view we can consider as "system" the spring scale + the wall + the body. Then the body's weight is actually an "external" force -- at least in classic mechanics -- and therefore I believe the Newton 3rd law/principle does not actually applies to it (in other words this "external" force from the point of view of the "system" is not an interaction force: it is in fact only due to an external given fixed field).

So there are here many forces involved in the system:

• the weight of the body due to the gravity
• the internal (interaction) forces that fulfill the Newton 3rd law: namely the force that the hook applies to the body and the force that the body applies to the hook (same direction and opposite) + the force that that the spring fixed end applies to the wall and the force that the wall applies to the spring (same direction and opposite)

Now since the body is at rest (and the mass of the spring scale is negligible + the mass of the wall is infinite) the Newton first law requires that the (vector) sum of all forces applied to the "system" cancels out. Using 3rd law it follows that the force applied from the the body to the spring is equal to the weight of the body.

Does it make sense ? Thank you.

 

[Hill]

There should be another force that balances "the force that the spring fixed end applies to the wall".

 

[BvU]

Hi,

first you write

"external" force from the point of view of the "system"

and later

the (vector) sum of all forces applied to the "system" cancels out

which to me is contradictory ...

You also confuse me a little using 'wall' instead of 'ceiling'
and give that ceiling an infinite mass ...

$\$

 

[cianfa72]

There should be another force that balances "the force that the spring fixed end applies to the wall".

Yes it is the interaction force that the wall/ceiling applies to the spring fixed end.

 

[Hill]

Yes it is the interaction force that the wall/ceiling applies to the spring fixed end.

No, I am talking about another force applied to the wall by something that the wall is attached to.

 

[cianfa72]

No, I am talking about another force applied to the wall by something that the wall is attached to.

I believe we can analyze the problem taking the "system" as "the spring balance + the body + the wall/ceiling". There is no external field acting on the wall/ceiling and the interaction force the spring fixed end applies to it results in a zero velocity of the ceiling (the ceiling has infinite mass and stays at rest).

 

[Hill]

I believe we can analyze the problem taking the "system" as "the spring balance + the body + the wall/ceiling". There is no external field acting on the wall/ceiling and the interaction force the spring fixed end applies to it results in a zero velocity of the ceiling (the ceiling has infinite mass and stays at rest).

If there is no external force acting on the ceiling, then it is not at rest and the whole thing is free falling regardless of the mass.

 

[cianfa72]

If there is no external force acting on the ceiling, then it is not at rest and the whole thing is free falling regardless of the mass.

In the model above we're assuming that no external field acts on the ceiling (in other words the ceiling is not subject to the gravitational field). There is only one "interaction force" that acts on the ceiling due to the fixed end of the spring but the ceiling mass is infinite therefore it remains at rest.

 

[cianfa72]

which to me is contradictory ...

yes, the sum of external and "system internal forces" is not zero (it is equal to the weight of the body). However this external force applied to the center of mass (located inside the ceiling) does not change its state of rest since the total mass of the system (logically concentrated in the center of mass) is infinite.

 

[Dale]

the body's weight is actually an "external" force -- at least in classic mechanics -- and therefore I believe the Newton 3rd law/principle does not actually applies to it

Newton’s 3rd law does apply to external forces. You may not care about the force on the earth, but that doesn’t mean it isn’t there.

Using 3rd law it follows that the force applied from the the body to the spring is equal to the weight of the body.

Yes, although the system you chose makes it more difficult to determine that. Choosing just the body as the system there are far fewer forces of interest. There is just the spring force and the weight. So the equality comes directly from the 2nd law, given $\vec a=0$.

The choice of the system is somewhat of an art. Usually it is best to choose the system that makes the analysis easiest. Here, choosing the wall as part of the system seems to not be helpful. It has a large unknown mass, a large unknown weight, and a large unknown contact force on it.

 

[Hill]

In the model above we're assuming that no external field acts on the ceiling (in other words the ceiling is not subject to the gravitational field). There is only one "interaction force" that acts on the ceiling due to the fixed end of the spring but the ceiling mass is infinite therefore it remains at rest.

The two assumptions - of no other force and of being at rest - are contradictory. Making the wall mass "infinite" just makes its acceleration "infinitesimal", but not 0.

 

[cianfa72]

The two assumptions - of no other force and of being at rest - are contradictory. Making the wall mass "infinite" just makes its acceleration "infinitesimal", but not 0.

I'm just a student, but my teacher said a finite force divided for an infinite mass results in zero acceleration.

 

[cianfa72]

Newton’s 3rd law does apply to external forces. You may not care about the force on the earth, but that doesn’t mean it isn’t there

The point I was trying to make is that a given external fixed field determines the force acting on a body (e.g. the weight) but the body doesn't act in turn on the fixed assigned field.

Yes, although the system you chose makes it more difficult to determine that. Choosing just the body as the system there are far fewer forces of interest. There is just the spring force and the weight. So the equality comes directly from the 2nd law, given .

Yes, given $\vec a=0$ from the 2nd law it follows that the force the spring acts on the body is equal and opposite to the body's weight. Nevertheless to conclude that the force measured by the spring scale equals the body's weight we need the 3rd law.

 

[Dale]

The point I was trying to make is that a given external fixed field determines the force acting on a body (e.g. the weight) but the body doesn't act in turn on the fixed assigned field.

In Newtonian physics real forces don't act that way. The gravitational field is not separate from nor independent of the object generating the force. Any Newtonian force acts between two masses and follows the 3rd law. The field is just a convenience, and the Newtonian gravitational force does come in a 3rd law pair.

Nevertheless to conclude that the force measured by the spring scale equals the body's weight we need the 3rd law.

I agree.

In any case, I think that including the wall in the system is a bad choice. You can still use the 3rd law with the system boundaries being the body. The 3rd law applies to external forces.

 

[cianfa72]

You can still use the 3rd law with the system boundaries being the body. The 3rd law applies to external forces.

So your advice is to take the body itself as system (no spring or wall/ceiling). The forces involved are:

• the body weight (external)
• the external force due to the spring acting on the body

Since the body's acceleration is zero from the 1st law we get that the external force $F_1$ acting from the spring on the body is equal and opposite to the the body's weight $P$. Then we apply the 3rd law to the pair $F_1$ and $F_2$ (the force due to the body acting on the spring) getting $F_2 = - F_1 = P$ and we've done.

 

[hutchphd]

In your problem everything is static. Therefore each piece of the system has no acceleration nor deformation and there is zero net force on each piece (subsystem of the system).
It's turtles all the way down

 

[cianfa72]

In Newtonian physics real forces don't act that way. The gravitational field is not separate from nor independent of the object generating the force. Any Newtonian force acts between two masses and follows the 3rd law. The field is just a convenience, and the Newtonian gravitational force does come in a 3rd law pair.

Yes of course. Nevertheless we can use the field "abstraction" to avoid including all the pieces in the system and simplify the problem.

 

[Dale]

Yes of course. Nevertheless we can use the field "abstraction" to avoid including all the pieces in the system and simplify the problem.

You don't need to do that. You can just say "I am not interested in the forces on the earth".

In any case, if the interest is to simplify the problem then you really do not want to include the wall as part of the system. That adds very large unknown gravitational and contact forces, as well as an unknown mass. And your idea of getting rid of forces with your "field abstraction" makes that problem worse since you would get rid of the gravitational force but not the large contact force between the earth and the wall.

It is far better to apply Newton's laws consistently.

 

[Dale]

So your advice is to take the body itself as system (no spring or wall/ceiling). The forces involved are:

• the body weight (external)
• the external force due to the spring acting on the body

Since the body's acceleration is zero from the 1st law we get that the external force $F_1$ acting from the spring on the body is equal and opposite to the the body's weight $P$. Then we apply the 3rd law to the pair $F_1$ and $F_2$ (the force due to the body acting on the spring) getting $F_2 = - F_1 = P$ and we've done.

Yes. I usually try to have the force of interest as an external force or as the 3rd law pair of an external force.

 

[cianfa72]

Yes. I usually try to have the force of interest as an external force or as the 3rd law pair of an external force.

ok, so assuming the body as system the force applied from the body to the spring is the 3rd law pair of the system external force due to the spring acting on the body, right ?

 

[Dale]

ok, so assuming the body as system the force applied from the body to the spring is the 3rd law pair of the system external force due to the spring acting on the body, right ?

Yes.

 

[cianfa72]

Just one more point. If we take as our system a subset of all objects interacting each other, then some of the overall system internal 3rd pairs actually become external forces acting on the subsystem (i.e. on the system we have chosen to consider). Does the theorem of motion of center of mass still hold for this subsystem ?

 

[Dale]

some of the overall system internal 3rd pairs actually become external forces acting on the subsystem

Yes. That is deliberate.

Does the theorem of motion of center of mass still hold for this subsystem ?

Yes. Newton's 2nd Law applies to the subsystem.

 

[cianfa72]

In your problem everything is static. Therefore each piece of the system has no acceleration nor deformation and there is zero net force on each piece (subsystem of the system).
It's turtles all the way down

To take a more simple example we can avoid the gravitational interaction in the equations considering a system made up of "body + spring + wall of infinite mass" all laid down on an horizontal plane with negligible friction. Apply now an external force on the body until equilibrium is achieved.

I believe in this scenario on the wall acts a force due to the fixed end of the spring attached to it (in the same direction of the external force applied on the body). In other words not all subsystem's components have zero net force applied to them. Nevertheless assuming infinite mass for the wall gives zero acceleration even for it.

 

[hutchphd]

Infinities are to be shunned, and in this case their invocation is entirely unnecessary. Building infinite mass walls is to be avoided as a waste of material. Turtles are better and cheaper.