Inertial Forces and Newton's Laws of Motion

Kevin Kirby · Feb 26, 2012

Members do not see these Ads. Sign Up.
I have taken the liberty of moving the very interesting discussion on inertial forces and Newton's Law of Motion to this new thread, from the thread Joint moments in forefoot vs rearfoot strike running.

Eric Fuller said:

I don't understand your first sentence. When you describe forces you need to name the object that applies the force and the object that the force is applied to.

Forces are not always equal and opposite. Look at newton's second law. Net force = mass x acceleration. To have a net force all forces are not going to be equal and opposite. We can use this to solve our question by working in the opposite direction that we have been. A person is standing. Their body accelerates upward. From Newton's second law we know that there must have been an upward net force. Let's say this net force acting on the body is from the ground. Even though there is an acceleration, there is still an equal and opposite force acting at the foot ground interface. To name these forces: there is force from the foot applied to the ground that is equal and opposite to the force from the ground acting on the foot. Only one of those equal and opposite forces are applied to the body. The other is not. However, if we look at the forces applied to the body, the force from the ground acting on the body is greater than than the force of gravity acting on the body. This is why the body accelerates upward.

The ankle planar flexors can move the body in isolation. However, if you want to remain standing, without any other external forces, or taking a step, you have to lean forward so that your center of mass is over your forefoot. Even, if you don't put your center of mass over your forefoot, you will still move upwards, but you will tend to also move backwards.

Another example, lie on a nice smooth hardwood floor while wearing a fleece jacket. Lie so that your feet are against the wall. Plantar flex your ankle and you will move without activating any other muscle besides the triceps surae.
Click to expand...

David Smith said:

I will,

"In a mechanical system, any mechanical system, whether static or in motion, there is always equilibrium of forces and moments" Quote (or close enough as remembered) Prof S Nicol Strathclyde Uni Dept of Bioengineering.

Proof:

Static equilibrium

=> (force (a) * lever arm (a)) - (force (b) * lever arm (b)) = zero

Dynamic equilibrium (with no acceleration i.e. constant linear velocity or angular velocity)NB no acceleration = no force.

=> (force (a) * lever arm (a)) - (force (b) * lever arm (b)) = zero

Dynamic equlibrium (with acceleration linear or angular) we are considering angular displacements so:

=> [(force {a} * lever arm {a}) +( inertal force {a1}) * lever arm (a1))] - [(force (b) * lever arm (b) + (inertial force (b1) * lever arm {b1})] = zero

so now we know that; force * lever arm = moment (M)
and Inertial force * Lever arm = inertial moment (I)

so - Re write like (Ma + Ia) - (Mb + Ib) = 0

Now obviously acceleration can only be in one direction at a given time and inertial force is in the opposite direction to the acceleration. Therefore if Ma is clockwise then moment due to Ia is anticlockwise and moment Ib is zero. So if we give clockwise a minus sign and anticlockwise positive then:

(You might note here that Inertial force can be/has been called fictitious since it does not have a direct equal and opposite force e.g. Ib is always zero relative to Ia.)

So now; (-Ma+Ia)+(Mb) = 0

So now rearrange the equation to -Ma+Mb = Ia which is the more familiar scenario of one moment being larger than the opposing moment (external V's Internal) that will result in acceleration of the limb of interest.

So remembering F=MA then to find the acceleration of the limb of interest A=F/M
Click to expand...

Kevin Kirby said:

Eric:

Can't agree with you on that one. Newton's Third Law of Motion states: For every action, there is an equal and opposite reaction.

http://www.physicsclassroom.com/clas...laws/u2l4a.cfm
Click to expand...

Dave Smith said:

From the equations in my last post you can see that there is a force that is not equal and opposite since it is the result of and reaction too acceleration ( as Eric said earlier)

So as in Eric's example when the man jumps then there is an inertial force plus gravity*mass acting opposite to the ground reaction force produced by muscular effort therefore the sum of forces is equal.

You can consider that the force applied to the body and the force applied to the earth are equal and opposite but they are acting on separate bodies and so each body has the same inertial force but the earth has no significant relative acceleration due to its enormously disproportionate mass.

You can look t the problem is several different way but I have always found it useful to use the proposition that all moments and forces sum to zero and then formulate that equation. Then when you want to find the value of interest you just rearrange the equation to find it.
Click to expand...

Kevin Kirby said:

Dave:

Since when is an inertial force not a force? Newton stated that for every reaction force there must be an equal and opposite reaction force. These forces may be either static or dynamic and may include inertial forces as your equations demonstrate.

Unless you can demonstrate to me that an inertial force is somehow not a force, then I would say that Newton's Third Law of Motion still applies and for every force that is applied to an object, there must be an equal and opposite force being applied by another object.

Here is a nice summary:

Newton's Third Law

Newton's third law: All forces in the universe occur in equal but oppositely directed pairs. There are no isolated forces; for every external force that acts on an object there is a force of equal magnitude but opposite direction which acts back on the object which exerted that external force. In the case of internal forces, a force on one part of a system will be countered by a reaction force on another part of the system so that an isolated system cannot by any means exert a net force on the system as a whole. A system cannot "bootstrap" itself into motion with purely internal forces - to achieve a net force and an acceleration, it must interact with an object external to itself.

Without specifying the nature or origin of the forces on the two masses, Newton's 3rd law states that if they arise from the two masses themselves, they must be equal in magnitude but opposite in direction so that no net force arises from purely internal forces.

Newton's third law is one of the fundamental symmetry principles of the universe. Since we have no examples of it being violated in nature, it is a useful tool for analyzing situations which are somewhat counter-intuitive. For example, when a small truck collides head-on with a large truck, your intuition might tell you that the force on the small truck is larger. Not so!

http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html
Click to expand...

Simon Spooner said:

Another example in which a force in addition to the force from the ankle plantarflexors and ground reaction forces on the plantar foot is applied, in this case from the ground on the entire dorsal surface of the body. And I thought my see-saw analogy was a stinker

So, back to RCSP and the ability of the ankle plantarflexors to lift the right heel without any assistance from other muscles and how that initial acceleration is achieved when the internal ankle plantarflexion moment is equal and opposite to the external ankle dorsiflexion moment because Newton's third law applies... you got to overcome body weight, which in my case is lots of Newtons, how many Newtons can the ankle plantarflexors generate? And how is that initial acceleration achieved?
Click to expand...

Dave Smith said:

Kevin, Yes inertia is a force but I was trying to make the point that you can have either perspective. The inertial force is a reaction to acceleration not another force.

So if I squeeze a ball between both hands it will have two opposing and equal forces acting on it, there is no separate force from inside the ball and the ball is motionless.

If I throw the ball (horizontal and parallel to the ground) then, all the time ball and hand are in contact, there is a single force from my hand acting on the ball but now because of acceleration there is a force from within the ball equal and opposite to the force from my hand. Of course the inertial force is real but it only appears as a reaction acceleration and also fulfils Newtons 3rd law.
Click to expand...

Simon Spooner said:

Newton might disagree, Dave. Hooke and Cauchy explained why, maybe Pascal too if the ball is hollow.

Assuming the ball and hand are moving forward during the throw, just prior to the release, isn't there more than a single force acting on the ball, that from the resistance from the air and gravity for example?
Click to expand...

Eric Fuller said:

Yes, the couplets are equal and opposite. But if all forces are equal and opposite then you could never have acceleration because there would never be a net force. It seems that people are having a problem with understanding what I'm saying because they think that all forces must be equal and opposite.

In Dave's example of throwing a ball, the hand applies a force to the ball and the ball applies a force to the hand. These are an equal and opposite couplet. However, these two forces are applied to different objects. If you examine just the ball, there is only a force from the hand (and gravity and wind resistance), but the force from the hand is the one that is making it accelerate horizontally. When looking at the ball in isolation, there is no equal and opposite force and this makes sense in terms of Newton's second law.
Click to expand...

Eric Fuller said:

Ok, so we will eliminate all other sources. A body in weightless space with the foot on the side of the space ship. The ankle plantar flexors contract, the foot pushes off of the space ship and the body accelerates. No other forces are needed.

Once ankle plantar flexion has started, the internal plantar flexion moment, or more specifically the plantar flexion moment about the ankle applied by the leg to the foot is not equal and opposite to the external dorsiflexion moment applied by the ground about the ankle joint axis to the foot. It cannot be because of Newton's second law for angular motion. The internal and external moments are not a couplet. One of the couplets is the moment applied by the leg to the foot and the other is the moment applied by foot to the leg. Contraction of the ankle plantar flexion muscles will simultaneously create a moment that will tend to rotate the distal aspect of the foot plantarly and a moment that will tend to rotate the top of the tibia posteriorly. Those are the equal and opposite moments.
Click to expand...

Kevin Kriby said:

I think we are having a discussion mostly of semantics now and also think that we must be very careful of our terminology or we will all get confused.

Newton's Third Law of Motion: For every action force, there will be an equal and opposite reaction force. Agreed?

Inertial force is still a force, even though it involves an acceleration of a mass and not dependent necessarily on gravity. Agreed?

If you throw a ball with your hand, the force from the ball pressing on the hand must always be equal and opposite to the force from the hand pressing on the ball. Agreed?

Part of the force from the hand acting on the ball being thrown is coming from the mass of the ball being acted on by gravity with the other part of the force acting on the ball from the hand coming from the inertial force of the ball resisting being accelerated and the wind resistance? Agreed?

Are we all in agreement now?
Click to expand...

Dave Smith said:

Yes this is also true but wind resistance in this case in insignificant (unless the ball is very light and large, something like a beachball) gravity could have an effect but that is why I added "(horizontal and parallel to the ground)" since then there is no influence by gravity on the horizontal motion

This is quite an important concept to understand, I remember at University and for some time afterwards I was always confused about how it is if all forces are equal that there can be motion. I knew rationally that there is acceleration from an applied force but could not reconcile what I observe with what the theory says, this is a common intuitive stumbling block apparently.

The summation of all forces are always equal to zero (Newton's 2nd)and yet only unequal forces, i.e. a net force that is not zero, cause acceleration (Newtons 3rd)??

On the face of it it doesn't make sense and yet it does when we understand the concept of a 'ficticious' yet real force that is only the reaction to, or result of, acceleration.

Also we can understand the concept of equal and opposite forces if we only apply it to the two bodies acting on one another (Newtons 1st a body will continue on it's course at a constant velocity unless acted upon by another body) So if two bodies act on each other the result is acceleration of both bodies in the direction of the applied and reaction forces but only one unbalanced force acts on each body (ignoring inertial force)

Sometimes its like trying to imagine the infinite universe you know the theory but the concept makes your head spin.
Click to expand...

Mike Weber said:

Dave I sure the above makes sense but it not sinking in.

Can you if you have time have another go for the kids in the back row?

Thanks in advance
Click to expand...

Kevin Kirby said:

Another difficult task is to try and teach these concepts in a clear manner to students so they don't become confused and then, like the discussion we are having currently, it is also difficult to use terminology correctly so that we all don't confuse each other.

Regarding the concept of accelerations and inertial forces, Albert Einstein, when he was working on his General Theory of Relativity, did a thought experiment to help him better understand the concept of acceleration and gravity. Einstein imagined a person in an elevator car in a weightless environment, in space, and then having a cable tied to the top of the elevator car pulling the elevator with a constant magnitude of acceleration upward (see illustration below). He realized, that to the person inside the elevator in space, that the constant acceleration of the elevator car in a weightless environment would feel exactly equivalent to that person as would the effects of gravitational acceleration.

In much the same way, we must realize that, if any object with a mass is accelerated, then reaction force must be applied to that object in order to accelerate it. This concept is embodied in Newton's Second Law of Motion, Force (F) = mass (m) x acceleration (a) or F=ma.

However, we must also realize that no object may apply a force to another object without equal and opposite forces being applied simultaneously between these two objects. This concept is embodied in Newton's Third Law of Motion: for every action there is an equal and opposite reaction.

Therefore, truly, there are never "unbalanced" forces of one object on another object or a situation where two objects have different action and reaction forces. However, there are plenty of situations where a force (F) acting on an object of mass (m) causes an acceleration (a) of that object (F=ma) so that part of the action-reaction forces generated are due to the resistance of that object being accelerated. This portion of the action-reaction forces which are due to accelerating a mass are what are commonly known as inertial forces. As long as inertial forces are present, then the special circumstances of equilibrium can not be used to predict the forces acting between objects.

It is for these reasons that I chose, over two decades ago, to use the special mechanical situation of rotational equilibrium to discuss subtalar joint biomechanics. By using an equilibrium situation to discus subtalar joint biomechanics, the mechanical model of the subtalar joint can be significantly simplified so that the goal of a more clear understanding of subtalar joint mechanics may result, without adding the complexity of inertial forces that may prevent some from understanding many important concepts of foot and lower extremity biomechanics (Kirby KA: Rotational equilibrium across the subtalar joint axis. JAPMA, 79: 1-14, 1989).

This thread should probably be split off into a new thread: Inertial Forces and Newton's Laws of Motion.
Click to expand...

Eric Fuller said:

What really helped me was reading an engineering statics textbook, specifically the section on free body diagram analysis. The first step in free body diagram analysis is defining the body in question. This is where a lot of the confusion comes from. Once you define the body you should only draw in the forces acting on that body. Take a human body in static stance. Which force do you draw to represent ground reaction force? (The force is drawn with arrow pointing upward at the bottom of the foot.) Imagine that person standing on a trampoline. The trampoline springs stretch because of which force(s)? So, if you define your body as the person, then the forces acting on the person are gravity acting downward at the center of mass and force from the trampoline acting upward on the bottom of the foot. There are two different things to look at here. There is equilibrium of the forces acting on the body and then there is looking at the equal and opposite forces and the foot trampoline interface. At rest, the force from gravity acting on the body and the force from the trampoline acting on the body are the same. This is not Newton's third law. This is Newton's second law. F= ma with a being zero. So the net force on the body has to be zero. Newton's third law is also working at the same time so the force acting on the trampoline from the body is equal to the force from the body acting on the trampoline. For Newton's second law we were looking at the forces acting on the body. For Newton's third law we were looking at the interface of two different bodies.

So, now start with the body falling toward the trampoline. The body is accelerating downward because of the force of gravity is unopposed. The body hits the trampoline and there is now a force from the trampoline applied by the body. Initially, the upward force from the trampoline applied to the body is still less than gravity, so the body will still be accelerating downward, but at slower rate. Eventually, the springs will stretch and start applying more upward force on the body and the force will equal body weight. At this time there is still some downward velocity, but the acceleration is zero. This leads to further stretch of the springs and further increasing force from the trampoline applied to the body. This will continue until the body is fully decelerated and its velocity is zero. However at this time, the force from the trampoline will be greater than the force of gravity and the body will start accelerating upward.

During that whole time the force from the trampoline acting on the body was variable, but the force from the trampoline acting on the body always equaled the force from the body acting on the trampoline because of Newton's third law.

Additionally, using inverse dynamics, we could measure the mass of the body and the acceration and then we could calculate the amount of force at any instant in time by using Newton's second law F = ma.

So, as I said before, you have to carefully label the forces you are analyzing and also know what situation you are analyzing them in. Yes, all the forces acting on a body add up to zero when a body is at rest. That is a special case of Newton's second law and should not be confused with Newton's third law.
Click to expand...

Dave Smith said:

If you goto this address http://www.foothouse.co.uk/access/inertial-force.htm I have posted up a link to and excel spreadsheet that shows the relationship between inertial force and acceleration when the applied force is constant and the external resisting is varying i.e. not equal to the applied force, in this case it is friction. As friction increases, acceleration and inertial force decrease proportionally and constant to each other i.e. their slopes are parallel. Therefore as frictional force decreases the inertial force increases to maintain the overall zero balance in the summation of forces acting opposite to each other.

Sorry you can't interact with the spreadsheet as it is read only but I am working on a way to make an interactive link. If you could you would see that however you alter the friction coefficient (and so alter resisting frictional force), mass or applied force the IF and acceleration slopes remain parallel.
Click to expand...

Dave Smith said:

Aha! I've done it, go here for a live interactive version of the XL spreadsheet
https://www.avgroom.com/browse/share...podiatryroom1#. I used AVG Livekive
Click to expand...

Isn't it nice that we can have nice debates/disagreements here on Podiatry Arena without any name calling and accusations that we are somehow stifling pet beliefs and patented/trademarked ideas........ as long as certain individuals aren't also involved in these discussions (i.e. D.S., E.G., B.R.)?!

HansMassage · Mar 1, 2012

OK I have a couple of clients that are engineers that I con discuss the problem of the moment of gravitational force being anterior to the fore foot. Gives more mystique to my calling myself an Antalgic Posture Pain Specialist

Kevin Kirby MVP

HansMassage Active Member

Forces Applied on the Foot Rest and Paddle by Elite Kayakers

Sensoria Health and DARCO Join Forces to Create Smart Diabetic Footwear

Proprioceptive Insoles decrease medial midfoot plantar forces

Vertical forces produce horizontal progression

low propulsive forces but high velocity progression in walking?

Vertical ground reaction forces and sprinting

Gluteal muscle forces and footstrike pattern during running

Share This Page