# How can thing move?

#### david smith

Dear all

When I first learned about forces and motion I was taught F=ma in terms of apply a force to a mass and the acceleration of that mass = x. I was quite clear about how something moved ie apply a force to a mass - it moves or tends to move.

But then I came onto the D' Alembert variation of F-ma=0 ie everything in equilibrium.
Very useful it is too.

In terms of F= ma so ma = F and F-F = 0. So the I thought well how can things move then? I was told by many that the insertion of inertia into the equation so that F-ma = 0 is just a mathematical trick and useful for resolving forces and moments in a mechanical system.

However to me at least inertia is real and easily proved. You can't apply a force unless an equivalent force pushes back.

So I decided that motion is caused by the transfer of kinetic energy of momentum in the direction of the force applied. In terms of conservation of momentum, so that even though there is equal and opposite inertial force there is not equal and opposite momentum and the motion carries on in the direction of the force.

But I cant quite resolve that concept! Since the force applied to the mass does work on the mass as it accelerates therefore (in my mind) the inertial force must do equal and opposite (negative) work to the applied force. As Ek = 1/2mv^2 and work = fd theta and this becomes W = 1/2mv^2theta then Ek of applied force should equal negative Ek of inertial force.

So how do things move??

Can anyone explain? have I got my concepts mixed up?

Thanks Dave Smith

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#### Andy Resnick

A rock climber can ascend narrow chimneys using a standard method. The climber can even rest in place partially up the chimney. At rest, the climber is in equilibrium. Does that mean when at rest, there are no forces?

#### Mapes

Homework Helper
Gold Member
Hi Dave, welcome to PF. One serious problem is that your terms "kinetic energy of momentum" and "Ek of applied force" are very difficult to understand. Energy, momentum, and force are all very different, and you probably would not say something like "length of temperature" or "height of voltage."

To use your example, when A accelerates B, it does work on B and increases B's energy. The energy of A is decreased, so we could say that B does an equal amount of work on A, except opposite in sign. (I find the term "negative work" confusing.) So total energy remains constant. I'm not sure why you think you can set the two values (one positive, one negative) equal to each other ("Ek of applied force should equal negative Ek of inertial force.")

Edit: Oh, I think I see what you meant. Is it that the energy change in B is, say, 5 J; the energy change in A is -5 J; and 5 = -(-5)?

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#### david smith

A rock climber can ascend narrow chimneys using a standard method. The climber can even rest in place partially up the chimney. At rest, the climber is in equilibrium. Does that mean when at rest, there are no forces?
Andy

Yes there are forces at work and they are in equilibrium

Dave

#### david smith

Hi Dave, welcome to PF. One serious problem is that your terms "kinetic energy of momentum" and "Ek of applied force" are very difficult to understand. Energy, momentum, and force are all very different, and you probably would not say something like "length of temperature" or "height of voltage."

To use your example, when A accelerates B, it does work on B and increases B's energy. The energy of A is decreased, so we could say that B does an equal amount of work on A, except opposite in sign. (I find the term "negative work" confusing.) So total energy remains constant. I'm not sure why you think you can set the two values (one positive, one negative) equal to each other ("Ek of applied force should equal negative Ek of inertial force.")
Mapes

Its quite possible I am getting confused here but, I can't imagine a way length an temperature are linked however energy and momentum and force: Kinetic energy = 1/2 mv^2 and momentum = mv so kinetic energy requires momentum. The applied force does not exist on its own it requires a mass + an acceleration so acceleration = d/t^2 or a change in momentum so force = mass * change in momentum. Therefore a force = change of momentum = change in Kinetic energy. And so, in my mind, I can (perhaps loosely) say the energy of momentum and the Ek of force.

So how do you explain motion in terms of change from a state of rest or constant velocity.

To use your example, when A accelerates B, it does work on B and increases B's energy. The energy of A is decreased, so we could say that B does an equal amount of work on A, except opposite in sign.
I get that, so are you saying the forces are equal and it is the exchange of energy that causes motion?

I don't require a definition as I think I know all the definitions. Don't get me wrong I can and do work with all the equations of newtonian mechanics (even if the above doesn't sound like it). I just suspend my disbelief and do the maths.

Thanks for your help, Dave Smith

#### Andy Resnick

David,

Let me back up: there's a couple of places you say something like "the origin of an applied force is the acceleration of some object". Is that what you mean?

#### david smith

David,

Let me back up: there's a couple of places you say something like "the origin of an applied force is the acceleration of some object". Is that what you mean?
Andy

Yes both, if there is a force there must be an acceleration (or tendency to acelerate) and you can't apply a force unless you have another force to resist it ie 2 accelerations in opposite directions. EG you can't jump in the air if you don't have the ground to push off.

Cheers Dave

#### Andy Resnick

Ah. Now I understand your confusion.

The formula F = ma (or F = dp/dt) is, for various reasons, very subtle. First, it's important to realize that we never measure forces. There's no such thing as a 'force-o-meter'. This is not the case for acceleration (or velocity, or position)- we have rulers and stopwatches to measure these things.

There may be objections to this- surely a spring measures force? Or a strain gauge? Or a pressure gauge? None of those things really do, they rely on constitutive (material properties) relationships which relate a force (dynamic quantity) to a displacement (kinematic quantity).

F=ma conceals this important point- it sets equal a *kinematic* quantity (acceleration), with a *dynamic* quantity (force). We do not ultimately know what causes a force. We have lots of models, to be sure, in as much detail and complexity as you can stomach. But in the end, we still have no force-o-meter.

If you like, you may think of 'mass' as being a constant of proportionality between an applied force (whatever the origin of said force) and a resultant motion- but that has limits as well (massless particles). Consequently, it may be helpful to spend time temporarily expunging 'force' from your mental concepts and instead think in terms of momentum and energy.

#### russ_watters

Mentor
I didn't read the entire thread, but I'm not seeing in the first half of the OP anything to be misunderstood. f=ma means f=ma. Both forces are represented in the equation: the left side is the force applied to the object and the right side is the equal and opposite force applied back.

Re-arranging the equation to f-ma=0 doesn't change anything. 0 isn't the acceleration, it is the sum of the forces.

But then you repeat your question without actually connecting it to the math you just showed, so it isn't clear what you don't like about this.

Anyway, though, by throwing energy in there in the second half of the post, you are just confusing yourself. If you ever do sort out the math of it, you'll find it just reduces back to f=ma anyway. So just drop that entire thing and just get your arms around the idea that f=ma is a true and complete statement of how a force results in acceleration.

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#### russ_watters

Mentor
Hmm, maybe this will help. In order to apply a force to an object and not have it move, you need two force pairs on it, not one. If you have an object bolted to the ground and you push on it, you and the object exchange a force and the object and the bolt exchange a force. The forces sum to zero, with two positive and two negative.

If you have only one force pair, one of those forces manifests itself as ma in order to maintain the balance. If you start with F1=F2 and know that there is no corresponding force pair to stop the acceleration, then F2 must be a force due to inertia and acceleration. F2=ma.

Physicists often treat gravity as equivalent to acceleration, so the same principle applies to the unmoving climber on the side of the mountan. The force pair between the climber and mountain is F1=F2 and F2 still equals ma.

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#### Mapes

Homework Helper
Gold Member
I enjoyed reading Andy's answer. As supplement, Greenwood says this in Principles of Dynamics:

"The concept of force as a fundamental quantity in the study of mechanics has been criticized by various scientists and philosophers of science from shortly after Newton's enunciation of the laws of motion until the present time. Briefly, the idea of a force, and a field force in particular [Greenwood defines field forces as those involved with action at a distance], was considered to be an intellectual construction which has no real existence. It is merely another name for the product of mass and acceleration which occurs in the mathematics of solving a problem. Furthermore, the idea of force as a cause of motion should be discarded since the assumed cause and effect relationships cannot be proved." (boldface mine)

#### rcgldr

Homework Helper
Equal and opposite forces when an object is accelerated.
Reaction forces are the result of acceleration times inertia (inertia equals mass in the case of translational acceleration). For angular movement, you have reaction torque = angular acceleration times angular inertia. Reaction forces or torques don't stop acceleration, they are the response to acceleration. Without acceleration, reaction forces or torques don't exist.

When calculating accelerations, reaction forces are not included.

Say a pushing force is applied to a block on a frictionless surface, the block accelerates according to the equation, A = F/M. The block has inertia, and resists acceleration with an equal and opposite "reaction" force that exactly equals the force that is producing the acceleration (F = MA) but this determines the rate of acceleration, it doesn't cancel the acceleration.

There may be objections to this- surely a spring measures force?
Close enough for me. If the object involved has a known stress / strain curve, then the change in length of the object under tension or compression could measured be used to calculate force. Tension could be measured by the vibration rate, for example the strings on a piano or guitar, higher tension => higher pitch.

#### Cyrus

The D'almebert "force" is not a real force, you treat ma as if it were a force - but its not..

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#### Cyrus

Ah. Now I understand your confusion.

The formula F = ma (or F = dp/dt) is, for various reasons, very subtle. First, it's important to realize that we never measure forces. There's no such thing as a 'force-o-meter'. This is not the case for acceleration (or velocity, or position)- we have rulers and stopwatches to measure these things.

There may be objections to this- surely a spring measures force? Or a strain gauge? Or a pressure gauge? None of those things really do, they rely on constitutive (material properties) relationships which relate a force (dynamic quantity) to a displacement (kinematic quantity).

F=ma conceals this important point- it sets equal a *kinematic* quantity (acceleration), with a *dynamic* quantity (force). We do not ultimately know what causes a force. We have lots of models, to be sure, in as much detail and complexity as you can stomach. But in the end, we still have no force-o-meter.

If you like, you may think of 'mass' as being a constant of proportionality between an applied force (whatever the origin of said force) and a resultant motion- but that has limits as well (massless particles). Consequently, it may be helpful to spend time temporarily expunging 'force' from your mental concepts and instead think in terms of momentum and energy.
I enjoyed reading this post, nice.

#### Andy Resnick

<snip>

Close enough for me. If the object involved has a known stress / strain curve, then the change in length of the object under tension or compression could measured be used to calculate force. Tension could be measured by the vibration rate, for example the strings on a piano or guitar, higher tension => higher pitch.
Yes, but do you see what you did? You switched from talking about 'force' (Newton) to STRESS (Cauchy).

IMHO, Cauchy's first law is much more relevant that Newton's second law, for this very reason. We can indeed measure stress- pressure and strain gauges. This is not measuring a force! Applying a Newtonian force to a pressure gauge will give different results, depending on the geometry of the pressure gauge.

#### david smith

All, Russ, Jeff, Mapes, Andy, Cyrus, thakyou for you time and effort and your input.

Aha! this is more like it, these are the types of answers I've been looking for.

Let me explain, I work as a Podiatrist and deal with pathological problems of gait and posture in terms of biomechanics. Therefore I am dealing with kinetics and kinematics -forces and movement.

In my degree we were always taught that 'forces and moments in a mechanical system , whether stationary or in motion,are always in equilibrium' IE D'Alemberts version of Newtons law F=ma => F-ma=0.

(Russ
Re-arranging the equation to f-ma=0 doesn't change anything. 0 isn't the acceleration, it is the sum of the forces.
- I know zero is in terms of balance of forces and moments and not acceleration.)

So my quandry has always been if the forces and moments are equal on each side of the equation how can there be motion?

1A: Just accept it (My reply: Can't) Russ wrote-
So just drop that entire thing and just get your arms around the idea that f=ma is a true and complete statement of how a force results in acceleration.
Maybe I should Russ but isn't it a complete statment of how we explain acceleration in terms of force?

Russ
If you have only one force pair, one of those forces manifests itself as ma in order to maintain the balance. If you start with F1=F2 and know that there is no corresponding force pair to stop the acceleration, then F2 must be a force due to inertia and acceleration. F2=ma. Physicists often treat gravity as equivalent to acceleration, so the same principle applies to the unmoving climber on the side of the mountan. The force pair between the climber and mountain is F1=F2 and F2 still equals ma.
confused!! - so are you agreeing with me then, ie that forces are always in equilibrium and do not cause motion??:uhh:

2A: Inertia is an imaginary force, Cyrus wrote-
The D'almebert "force" is not a real force, you treat ma as if it were a force - but its not..
(My reply: seems real enough to me -EG try punching a wrecking ball hanging by its chain and see how the inertial force breaks your hand)

Jeff wrote
Reaction forces are the result of acceleration times inertia (inertia equals mass in the case of translational acceleration). For angular movement, you have reaction torque = angular acceleration times angular inertia. Reaction forces or torques don't stop acceleration, they are the response to acceleration. Without acceleration, reaction forces or torques don't exist.
So there is inertia and there are reaction FORCES

Jeff
When calculating accelerations, reaction forces are not included.
Oh! so there aren't any reaction forces, which is it? Is it valid (even if useful) to ignore a parameter just for convenience?

Jeff
Say a pushing force is applied to a block on a frictionless surface, the block accelerates according to the equation, A = F/M. The block has inertia, and resists acceleration with an equal and opposite "reaction" force that exactly equals the force that is producing the acceleration (F = MA) but this determines the rate of acceleration, it doesn't cancel the acceleration.
Exactly! so therefore if the forces can't cause the accleration, what does? (by your argument one ignores the inertial forces when calculating the acceleration, so then there is the inertia of the negative acceleration of the mass applying a force and the force of inertia of the positive acceleration of the mass resisting the applied force. Ignore the inertial forces, which is both sides of the equation ma = ma, and you get acceleration. How?

3A: Don't mix force theory with energetics theory. :grumpy:(My reply: Why?, isn't momentum and energy transfer directlly related to force? You can't transfer kinetic energy without applying a force and potential energy only becomes kinetic with the application of a force.)

Rus wrote
Anyway, though, by throwing energy in there in the second half of the post, you are just confusing yourself.
I'm definintely confused.!!!

Andy wrote
If you like, you may think of 'mass' as being a constant of proportionality between an applied force (whatever the origin of said force) and a resultant motion- but that has limits as well (massless particles). Consequently, it may be helpful to spend time temporarily expunging 'force' from your mental concepts and instead think in terms of momentum and energy.
Difficult, but you may be right.

Ah. Now I understand your confusion.

The formula F = ma (or F = dp/dt) is, for various reasons, very subtle. First, it's important to realize that we never measure forces. There's no such thing as a 'force-o-meter'. This is not the case for acceleration (or velocity, or position)- we have rulers and stopwatches to measure these things.
Good concept?

Andy wrote
There may be objections to this- surely a spring measures force? Or a strain gauge? Or a pressure gauge? None of those things really do, they rely on constitutive (material properties) relationships which relate a force (dynamic quantity) to a displacement (kinematic quantity).

F=ma conceals this important point- it sets equal a *kinematic* quantity (acceleration), with a *dynamic* quantity (force). We do not ultimately know what causes a force. We have lots of models, to be sure, in as much detail and complexity as you can stomach. But in the end, we still have no force-o-meter.
Not so difficult now, if I intuitively consider your premise above.

Mapes wrote
"The concept of force as a fundamental quantity in the study of mechanics has been criticized by various scientists and philosophers of science from shortly after Newton's enunciation of the laws of motion until the present time. Briefly, the idea of a force, and a field force in particular [Greenwood defines field forces as those involved with action at a distance], was considered to be an intellectual construction which has no real existence. It is merely another name for the product of mass and acceleration which occurs in the mathematics of solving a problem.
Can yougive some references please Mapes

By this premise and Andy's above my question becomes a mute point. "considered to be an intellectual construction which has no real existence". If force is a construct for, or a byproduct of, the calculation of other parameters of kinematics, then force is imaginary and cannot cause motion. So what does?

Mapes
the idea of force as a cause of motion should be discarded since the assumed cause and effect relationships cannot be proved.

This is all good for me

Russ
Hmm, maybe this will help. In order to apply a force to an object and not have it move, you need two force pairs on it, not one. If you have an object bolted to the ground and you push on it, you and the object exchange a force and the object and the bolt exchange a force. The forces sum to zero, with two positive and two negative.

If you have only one force pair, one of those forces manifests itself as ma in order to maintain the balance. If you start with F1=F2 and know that there is no corresponding force pair to stop the acceleration, then F2 must be a force due to inertia and acceleration. F2=ma.
Re reading this I see what you are staying (i think) So if I push against someone who is pushing back we each have Inertial forces (1 pair) and apllied forces (2 pair). Remove one of the applied force and we have motion. This means we have unbalanced forces and so one inertial force must accelerate to give us the balanced equation. Hmmm! Have I got that right. Not heard that concept before, I'll have to think about that one

Cheers all Dave

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#### regor60

Dear all

When I first learned about forces and motion I was taught F=ma in terms of apply a force to a mass and the acceleration of that mass = x. I was quite clear about how something moved ie apply a force to a mass - it moves or tends to move.

But then I came onto the D' Alembert variation of F-ma=0 ie everything in equilibrium.
Very useful it is too.

In terms of F= ma so ma = F and F-F = 0. So the I thought well how can things move then? I was told by many that the insertion of inertia into the equation so that F-ma = 0 is just a mathematical trick and useful for resolving forces and moments in a mechanical system.

However to me at least inertia is real and easily proved. You can't apply a force unless an equivalent force pushes back.

So I decided that motion is caused by the transfer of kinetic energy of momentum in the direction of the force applied. In terms of conservation of momentum, so that even though there is equal and opposite inertial force there is not equal and opposite momentum and the motion carries on in the direction of the force.

But I cant quite resolve that concept! Since the force applied to the mass does work on the mass as it accelerates therefore (in my mind) the inertial force must do equal and opposite (negative) work to the applied force. As Ek = 1/2mv^2 and work = fd theta and this becomes W = 1/2mv^2theta then Ek of applied force should equal negative Ek of inertial force.

So how do things move??

Can anyone explain? have I got my concepts mixed up?

Thanks Dave Smith

F is an external force on the object, ma is not an external force...

#### Cyrus

This is your problem, you are simply looking at F-ma=0 and saying, it equals zero, so it must be in equilibrium. But now you are assuming there is some OTHER m'a'=0.

in other words,

F-ma=0=m'a'

and concluding this a'=0, and therefore it does not move. You have imposed a m'a' term to satisfy a FALSE NOTION you made up in your head that it must stay at rest. There is no such, m'a' =0.

The fact is F-ma=0. This does NOT mean a=0! Very important you realize this. THIS a, is the acceleration, and it is NOT zero. This a' term I made up is just that, something YOU made up without realizing, its not equal to zero, and infact it does not even exist.

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#### russ_watters

Mentor
In my degree we were always taught that 'forces and moments in a mechanical system , whether stationary or in motion,are always in equilibrium' IE D'Alemberts version of Newtons law F=ma => F-ma=0.
Are you a biomedical engineer?? I'm a mechanical engineer and our concepts of this should be identical.
1A: Just accept it (My reply: Can't) Russ wrote- Maybe I should Russ but isn't it a complete statment of how we explain acceleration in terms of force?
Physicists may see this differently, but I see this as a needless complication of something that can mathematically be reduced to f=ma anyway.
confused!! - so are you agreeing with me then, ie that forces are always in equilibrium and do not cause motion??:uhh:
No, the forces are in equilibrium which means there must be motion. The equilibrium (the reaction force) is provided by ma. If you didn't have F2=ma in the equation and you applied a force of, say, 1N, you'd have 1-0=0 for F1-F2=0 -- an untrue statement.

[cyrus was basically saying the same thing]

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#### rcgldr

Homework Helper
> >When calculating accelerations, reaction forces are not included.

> Oh! so there aren't any reaction forces, which is it? Is it valid (even if useful)
> to ignore a parameter just for convenience?

Reaction forces are the result of acceleration, they are real, but they are not used to calculate acceleration. Acceleration = force (or torque) divided by inertia, and reaction forces aren't included in this acceleration equation.

"How can thing move?"

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