The Force Equation: Understanding the Relationship Between Mass and Acceleration

[physio]

Why is force = mass x acceleration.. No, literally why is it mass multiplied by acceleration always? How did Newton come to this equation and how am I to believe it? Why is it always mass multiplied by acceleration? What is force??

 

[sophiecentaur]

Think of the equation as telling you what force is NEEDED to produce a given acceleration on a given mass. That may help.
The question "Why?" is one of the last ones that you can expect Science to answer because it all depends upon the depth to which you require the answer to take you.

 

[Simon Bridge]

Technically Newton defined force to be the rate of change of momentum.
This definition kinda follows from the other two laws. I think of it as a trick to make the math easier since we seldom use F directly, but as a stage in the calculation of something else like when one kind of motion is changed into or linked with another.

But that's the "how" - the "why"? For that sort of question you need a philosopher.

 

[spoirier]

Indeed the scientific quest does not systematically focus on the "why" questions, and does not worry if no absolute "why" can be found, however it did succeed quite well to find intermediate causes to phenomena, that is going some steps forwards in the chain of explanations : explain the laws that describe the most obvious phenomena, as consequences or approximations of deeper laws.

Here the laws of classical mechanics have been explained as derived from the laws of relativistic mechanics, which come as made of 2 principles.
One is that time is somehow just another dimension.
The other is the least action principle.
The least action principle is the space-time equivalent of the space-only law of equilibrium, which says that a state of equilibrium of a physical system (where nothing happens in the time dimension so that the expression of the law of mechanics is reduced to its aspects on the space dimensions here considered) if its potential energy is minimal.
We can also consider unstable equilibrium where the potential energy is not minimal but its variations are zero at the first order of approximation with respect to small movements of the system.

Consider a wire stretched between two poles. It is subject to gravity, but has to keep a constant length. So it forms a curve which is the one giving it the lowest center of mass among possible shapes with the same length.
Each part of the wire (say, each centimeter length that is distinguished by thought and seen as attached to the rest at its two ends) is subject to gravity, that is a force oriented downwards, so that the equilibrium is reached when that part is just a little below the middle between its two attachment points. Each of the ends also exerts a force of pulling in the direction of the wire at the respective points. And these forces are at equilibrium, that is, the variation of the potential energy of the whole with respect to any given possible movement (of this part towards any direction), that is expressed as the sum of variations of potential energies of each part (this part, the left hand part, the right hand part) does not vary at the first order.

Now think that this is a wire of electric supply in the countryside on the border of a railway, and you are in a train going along this wire. You look at it by the window, and what do you see ?
You see the wire falling (accelerating) upwards. And at each passing pole, you see Bump ! the wire that was falling up is bounced downwards by the pole and now goes down, then is accelerated up again until the next pole.
This way we recover the laws of classical mechanics in space-time by applying them in space then letting one space dimension play the role of time, but with some sign changes.

Namely, uh, we can say that the pulling force along the wire plays the role of the mass, but a negative one (so that the sign of the acceleration is opposite to that of the force).
The stronger the wire is stretched (stronger pulling force), the smaller the acceleration (curvature) for a given force (weight per length).

 

[K^2]

Why is force = mass x acceleration.. No, literally why is it mass multiplied by acceleration always? How did Newton come to this equation and how am I to believe it? Why is it always mass multiplied by acceleration? What is force??

You can look at F=ma as definition of force, because how else are you going to measure it? So then the real question becomes why forces are additive and why Newton's 3rd holds. Both of these are answered by conservation of momentum. If p=mv, and Ʃp is conserved, then 2nd and 3rd laws follow directly.

 

[Simon Bridge]

Indeed the scientific quest does not systematically focus on the "why" questions, and does not worry if no absolute "why" can be found, however it did succeed quite well to find intermediate causes to phenomena

The "causes of phenomena" is "how" not "why".

You also have to be careful about the "chain of causality" image - it leads to the idea of a "first cause" which gets tricky and way off topic.

@K^2: "...that's not the real question, the real question is..." and get give a diverting speech ... very Yes Prime Minister I love it :)

 

[denjay]

As stated, the why part is a question left to philosophers (it's still fun to think about).

A way to think about force is using the equation you stated.

Force = Mass x Acceleration

Using some standard units, 1 unit of Force called a Newton is the amount it takes to accelerate an object having a mass of 1 kg exactly 1 m/s^2 (assuming no friction).

1 Newton = 1 kg x 1 m/s^2

If you push a 1 kg object hard enough on a frictionless floor such that its velocity increases 1 meter/second every second, you are putting a force of 1 Newton on that object.

 

[NegativeDept]

Here's a recent, thought-provoking, and controversial answer: Erik Verlinde derives $\mathbf{F} = m \mathbf{a}$ as an entropic effect from a purely information-theoretic argument involving 't Hooft's holographic principle! He then goes on to derive Newtonian gravity and Einstein's GR equation.

On the Origin of Gravity and the Laws of Newton

It's published Open Access, so you don't have to pay to read it.

 

[Simon Bridge]

@denjoay: you are saying that's a definition - it is F=ma because it is F=ma ... but why make that particular choice of definition? I maintain it is because it made the math, that Newton wanted to do, easier... and the resulting relations more general. In this way it was simpler than the force/impetus type models that preceded it.

@NegativeDept: I'll see your link to a paper and raise you a link to the BadScience blog article discussing it. Link includes a bit of to-and-fro and the author links to Erik's defense of the criticisms he writes about.

I think it's useful for people to see this sort of thing happen in science - and compare with how it happens in pseudoscience.

However - even if we accepted Erik's paper, he has only described how it is that F=ma turns out to be so useful ... OP asks "why F=ma" at all? It turns out to be a choice of definition that, with a method of using it, that has withstood the test of time. It is useful for a variety of reasons which boil down to making the math easy.

You can make any definitions you like so why accept the ones which make life hard for you?

It can also be a bit sobering to contemplate that as difficult as students generally find the math of physics in general, it is no where near as difficult as it could have been. This is as simple as we've been able to get it so far. Could it be simpler? Working on it!

 

[NegativeDept]

@NegativeDept: I'll see your link to a paper and raise you a link to the BadScience blog article discussing it.

Excellent! I was going to post links to the entropic-gravity debate, but I was too lazy. I hope it's not too far off-topic... but this thread is getting pretty philosophical, so I think it's perfect.

 

[Groundupboxin]

Think of it this way! I am a Professional fighter. I am working in the ring with you and we are just sparing so I am not wanting to knock you out. My mass is 235 pounds. If I stick out my arm and just twist my body you get hit with 235 pounds. That is with no speed. So it is a easy punch to take. Now I am working in the ring with one of my opponents and my job as a Professional fighter is to win and one of the easiest ways and quickest ways it to knock the guy out. So to throw my right fist as fast as possible, I dig deep with my right foot pushing off the floor propelling my body forward, which increases the speed of my twisting 235 pound body, and I unleash my right hand as fast as possible creating a straight stiff arm until his chin connects with my knuckles. This means that the speed of my punch has greatly increased the power of my punch! Therefore, Force= Mass x acceleration... The faster I throw my punches the harder your going to feel every pound behind it!

 

[sophiecentaur]

Think of it this way! I am a Professional fighter. I am working in the ring with you and we are just sparing so I am not wanting to knock you out. My mass is 235 pounds. If I stick out my arm and just twist my body you get hit with 235 pounds. That is with no speed. So it is a easy punch to take. Now I am working in the ring with one of my opponents and my job as a Professional fighter is to win and one of the easiest ways and quickest ways it to knock the guy out. So to throw my right fist as fast as possible, I dig deep with my right foot pushing off the floor propelling my body forward, which increases the speed of my twisting 235 pound body, and I unleash my right hand as fast as possible creating a straight stiff arm until his chin connects with my knuckles. This means that the speed of my punch has greatly increased the power of my punch! Therefore, Force= Mass x acceleration... The faster I throw my punches the harder your going to feel every pound behind it!

Hi
The problem with this approach is that it doesn't actually simplify things, which is what Physics attempts to do. Your right hand is not just an accelerated mass; it is joined to your body by your great chunky, well trained, muscles, which have mass, in themselves and are also providing varying force throughout the punch. As you say, you use your body, arms and feet (plus friction with the ground) to optimise the grief you give someone. It's really too complex to analyse in a simple equation.
We've had a number of threads about weight lifting and trying to link what goes on with simple principles of Mechanics and even that is pretty well impossible. You are adding an extra dimension in the form of a collision at the end of it all.
Newton's Laws deal with single objects and forces acting on them. They can be expanded into more complex situations, of course, but they are best used to explain simple situations like rockets, falling stones and, colliding billiard balls. I'm not wimping out of this but just suggesting that a simpler model would be a better way into understanding this stuff.

I could ask you consider whether putting a horse shoe inside your glove would help and why. Then I could ask how massive could you make this horse shoe before the extra mass on your arm would mess up your fighting ability. It soon turns into a very hard problem to solve using simple principles.

 

[Simon Bridge]

We haven't actually heard back from OP yet either.
Though it can be fun testing out your own ideas by trying to answer these things.

 

[Dickfore]

The OP touched upon what I think is the essence of the Physical Law.

In my opinion, a physical Law is a mathematical relationship between physical quantities.

Viewed as a dimensional equality, this relationship enables one to define the dimension of one of the involved quantities through the dimensions of the others. In case of 2nd Newton's Law, it is customary to define the dimension of force through the dimensions of mass, and acceleration.

But, how do we define the dimensions of the other physical quantities? The answer is either through other physical laws, or by accepting the quantity as a base. Mass is a base quantity. Acceleration is defined through the "lesser" physical laws (or defining equations for the respective quantities on the l.h.s.):
$$\mathbf{a} = \dot{\mathbf{v}}, \ \mathbf{v} \equiv \dot{\mathbf{r}}$$
and both length and time are base quantities.

But, this is only part of the story. The beauty of the physical law is that it holds for all possible values of each physical quantity involved. This enables us to use it multiple times (for different numeric values of each physical quantity to avoid underdeterminacy), and use some symmetry of the situation to relate the quantities of the same kind between the separate cases. This enables us to set up experimental procedures for measuring a quantity. After all, measuring a physical quantity means to compare its value relative to an accepted value of a quantity of the same kind.

For example, it is an experimental fact that if a spring is stretched to a certain length, the stretching force is always the same (you may also call this a physical law, namely, that of an elastic deformation). Thus, if we apply the same force on two bodies and measure their accelerations, we may write:
$$F = m_1 \, a_1 = m_2 \, a_2 \Rightarrow m_2 = \frac{a_1}{a_2} \, m_1$$
This enables us to measure m₂ in units of m₁ by measuring the two accelerations.

This enables us to give an operational definition of the concept of mass through the use of the same law, and measuring acceleration.

 

[thomasradiel]

As stated, the why part is a question left to philosophers (it's still fun to think about).

A way to think about force is using the equation you stated.

Force = Mass x Acceleration

Using some standard units, 1 unit of Force called a Newton is the amount it takes to accelerate an object having a mass of 1 kg exactly 1 m/s^2 (assuming no friction).

1 Newton = 1 kg x 1 m/s^2

If you push a 1 kg object hard enough on a frictionless floor such that its velocity increases 1 meter/second every second, you are putting a force of 1 Newton on that object.

I think now there must be a 'further than this' argument. The answer to 'why' F=ma is in what happens after the object (in a frictionless environment) reaches 1 m/s. So a Force of 1 Newton is placed upon an object and that object is 1 kg. This object begins to move in the direction of the force at a rate of 1 m/s per second. Thus in one second the object will have begun to move 1 m/s and will have moved 1 meter. Thus the force required to move the object 1 meter during the time of one second is going to be 1 Newton. What happens after that object has been pushed by this unnamed force, does it stop moving? No, there is nothing to stop it, it will go on forever because there is absolutely no force required to keep an object in motion.

So then, what is force? It is something that has the ability to move another object up to any velocity. So long as no other forces are acting, that object will continue to 'accelerate' in the direction of the force. Therefore we have covered one factor pertaining into this mystery force we have not identified yet. There must be something else, however, since not all objects are pushed and pulled about by any constant force with the same ease. The other factor (this is inductive reasoning) is how big or heavy something is. In Physics, since the adjectives 'big' or 'heavy' mean nothing, we calculate the thing's mass, which holds true regardless of an object's size or weight. Now, we know from pushing a car or trying to lift a mountain, that the force required to do so is most often outside our ability, yet this is only because of gravity or friction. If neither of these forces were in place, one could potentially use force on any object, regardless of the size, except that the bigger the object, the more force would be required in order to get it moving 1 m/s. Force, then, is simply the thing that makes another thing move and the factors that go into motion are the mass and acceleration of an object. There are no other factors, as Isaac Newton noted and this can be proved through scientific experimentation. So long as other forces are accounted for, an object will behave exactly how the applied math of physics tell it too.

F=ma can also be deduced (whereas the above was an induction) For instance if one drops an apple and a car and shoots a bullet at 90 degrees all at the same time, the apple, the car, and the bullet will all hit the ground at the same time. Thus we can tell that there is a force acting upon all three (even something in motion itself) which is in all cases completely equal barring an obstruction or other force. This unknown force can be deduced through math.

Physics says that all objects have a gravitational force which pulls them together depending on the mass of the objects. Mathematically, this means that gravity is not truly independent of mass, only that the mass between the problems cancel out, since a bullet itself (having it's own gravitational mass) pulls less than a car, so the mass of a car is also pulling the Earth bringing it toward the Earth at an equal acceleration despite (and because) it has a greater mass.

While this is a riveting tangent, the greater understanding behind the why of F=ma comes through the deductive and deductive reasoning behind the behavior of a stationary object, a, moving object, and an accelerating object, which just so happens to be the first three laws of motion as understood by the expression which Newton derived.

If I were to have written the equation, I would have written that a=F/m. and I would have set m/s2 equal to N/kg. This would make the ratio more understandable. For example, when I say 9.8N/kg, then a 4,000kg object would have to be pulled by a force of 39,200N in order to reach an acceleration of 9.8m/s2. Then you get a better idea of the downward pull on an object and why it is pulled downward with a constant acceleration.

This website* does a good job of explaining that a force is interactive and gravity, for example is dependent on the mass of both objects affecting one another constructively, and the distance affecting it destructively.
*http://physics.weber.edu/amiri/phys...&Gravity/reviewofgravity/ReviewofGravity.html

 

[AlephZero]

The answer to 'why' F=ma is in what happens after the object (in a frictionless environment) reaches 1 m/s. So a Force of 1 Newton is placed upon an object and that object is 1 kg. This object begins to move in the direction of the force at a rate of 1 m/s per second. Thus in one second the object will have begun to move 1 m/s and will have moved 1 meter.

No, the object will have moved 0.5 meters in one second ...

Thus the force required to move the object 1 meter during the time of one second is going to be 1 Newton.

...and that hs also wrong.

 

[ModusPwnd]

I think the "F" is just a label. F=ma is a definition. ma is defined to be equal to F. Its when you equate things like -kx=ma that you get to physics rather than definitions.

 

[Devils]

I've said this before but never got an answer that satisifes me.

If we push a 1kg object at 1 m/s^2 on a frictionless surface we are using a force of 1 N. But assume that the 1kg object is pushed onto a boulder, so stops moving. The standard response is "when it stops, the force on the object is equaled to the force on the boulder pushing in the opposite direction" but i find this deeply unsatisfying. We are saying the some accelerating objects move but others do not, I just find the concept troubling. Maybe Einstein resolves this

 

[Damo ET]

Unless the 'Newton' has been re designated since I was at school, all the above calcs using 1kg as 1 Newton are wrong! 1 Newton is the approx weight of an apple (remember Issac and the orchard!), and I have never come across a 1kg apple!
1kg is equal to 9.8Newtons guys.Damo

 

[sophiecentaur]

I've said this before but never got an answer that satisifes me.

If we push a 1kg object at 1 m/s^2 on a frictionless surface we are using a force of 1 N. But assume that the 1kg object is pushed onto a boulder, so stops moving. The standard response is "when it stops, the force on the object is equaled to the force on the boulder pushing in the opposite direction" but i find this deeply unsatisfying. We are saying the some accelerating objects move but others do not, I just find the concept troubling. Maybe Einstein resolves this

It isn't just when it stops. The forces are equal and opposite all the time.
An object can still be accelerating but instantaneously stationary in some / any frame of reference. Just throw a ball in the air. That is the condition when it's at the top of its flight.

Lots of Science can be "deeply unsatisfying" until you start to grasp it well enough.