# Type of force of $m\vec{a}$

As there are types of forces such as Normal, Tensional, Gravitational, etc. Suppose a block of mass $$m$$ kept on a table and a man is pushing it and a acceleration $$\vec{a}$$ is produced.

Is $$m\vec{a}$$ really a force? If yes, then please tell what would be it's type?

• At an atomic level, there are only two types of forces that humans interact with. These are the electromagnetic force and the gravitational force. – David White Sep 21 '20 at 19:44
• You can think of it as a cause-response pair. $F_{net}$ is the cause for an object’s $ma$ response. – Superfast Jellyfish Sep 24 '20 at 10:47

$$m\vec a$$ is not a force, and this is an important distinction to note. It's just what it says: the product of mass and acceleration, with no reference to any forces.

This confusion often arises from a misunderstanding of Newton's Second Law, which relates this quantity to force. $$\vec F_{\text{net}} = m\vec a$$ Here, we have two separate quantities, $$\vec F_{\text{net}}$$, the sum of all forces acting on an object, and $$m\vec a$$, which gives us some information about the motion of the object. Thus, Newton's Second Law is not merely an equation relating forces, but rather an equation telling us how to physically interpret how forces affect objects.

Since the object of mass $$m$$ has an acceleration of $$\vec a$$, we conclude not that $$m\vec a$$ is a force, but rather that the net force acting on the object (i.e. the force from the man pushing on it plus etc) is equal to $$m\vec a$$.

• an equation telling us how to physically interpret how forces affect objects precisely why it is better to express the equation as $F=m\ddot x$ reminding us that it’s actually a differential equation. – Superfast Jellyfish Sep 24 '20 at 10:50

Fundamentally there are only four types of forces namely Electromagnetic force ; Gravitational force ; Strong force ; and the Weak forces . The last two forces are at atomic scales and are very short ranged forces and have negligible influence at long separations .

Now coming to your question, all the forces like Normal, tension, and friction are due to one of these four fundamental forces, i.e. electromagnetic force mainly (also influenced by Pauli exclusion principle).

When you are pushing the block, (let's assume frictionless surface for simplicity) the atoms of your hand come closer to the atoms of the block and, due to the electron clouds surrounding the nucleus, a net repulsive force acts on the electrons of the block as well as of your hand. This repulsive force is what causing the block to accelerate.

So $$ma$$ is not a new force.

From Newton's second law of motion

$$F_{net}$$$$= ma$$

The normal force on the block is what serving here as the $$F_{net}$$ of the equation (assuming no friction by the table on the block). $$F_{net}$$ is not a new force. It just represents the vector sum of all the forces acting on it . But the origin of all the forces is still the four above mentioned fundamental forces.

Hope it helps ☺️.

So $$m\vec a$$ is not a force. There is a subtle difference.

A force is defined as a disposition to accelerate, not an acceleration itself. This is a fancy word that just means “if nothing else intervenes, then the thing accelerates.” Furthermore it is indeed weighted by this amount of stuff parameter $$m$$ which says “if the same disposition is applied to twice the stuff, and nothing else intervenes, then the thing accelerates half as much.”

# Some counter-examples

But for example sometimes things are in a state of force balance. Gravity is pulling down on me, but my chair is pushing up on me, and as a result I am not accelerating. But the forces are still there. Gravity still disposes me to want to go downward and the force of the chair still disposes me to want to go upward, it’s just that they happen to be perfectly balanced against each other so that I do neither. (I must pause to say that this has nothing to do directly with Newton’s third law. Many newcomers make the mistake of confusing force-balance with the third law, they are also different.)

One really clever thing about this definition of forces is that, if you imagine someone moving past me on a train, maybe they are juggling balls on the train, I see all of these balls move past me with some great velocity of around 90 km/hr or however fast the train is going: but we both agree on any change in velocity of the balls, and our clocks both agree on how long a second is (until we get to relativity but let’s ignore that), so we both agree on any accelerations we see, so we both agree on the forces we see on the balls, even though we disagree about the actual momentum and energy that the balls are carrying. Very handy!

And like sometimes forces are not balanced, and you see an acceleration in some direction. There might not be any forces actually pointed in that direction! So for example sailboats can sail into the wind by cleverly coordinating both the wind (which obviously pushes away from the wind) with a sail and a rudder and a mainboard anchored in the water, which each pushes in some completely other direction. You sum up these wildly different forces in wildly different directions and you get a "net force", and a resultant acceleration, in a totally different direction from any of them. I cannot say that such an $$m \vec a$$ is a force because there is no force pointed in the appropriate direction. But it is the vector-sum of forces on the boat.

# Are there types of forces?

Now beyond this you are asking whether there are substantially different types of forces, and the answer is “not at the level that you would like: but there are substantially different reasons we care about forces, and we label these forces by the reasons we care about them.” So for example the thing you are calling the “normal force” is caused by the fact that my chair (and the floor beneath it and the planet beneath that) has not yet broken, so I cannot fall through it. It is what we call a constraint force, it is created by a constraint on the motions of the system. We call it “the normal force” so that you know that it is the force that is normal (fancy old word meaning “orthogonal to” or “perpendicular to”) to the constraints of the system. But it is ultimately caused by electromagnetic and Pauli repulsion between electron clouds of atoms, so if we were classifying it by type we would have called it a “quantum force of Pauli repulsion between electron clouds because electrons cannot be in the same state as each other”. But we don’t classify it that way, we classify it by some sense of why we care about it, what it is doing in our equations: it is stopping me from falling through the floor, it is enforcing a constraint that I do not think is going to happen in the physics of this system. Similarly a tension force is expressing that there is a stretchy thing that has been stretched out past its equilibrium length and it wants to return back to its equilibrium length: that’s not the underlying type which is again an “electromagnetic force of atoms pulling on nearby atoms, plus probably an entropic force of long stringy molecules being forced to straighten up but thermally they really want to go back to being in complicated squiggles”. But we call it “tension” because we don’t care why the rope or spring wants to return back to its equilibrium length, it is sufficient that it does want to, and that is the reason why we care about it.

$$m\vec a$$ is not a force. It is the sum of all forces:

$$\sum \vec F=m\vec a$$

By pushing the object on the table sideways, there are at least three forces acting on it:

1. Its weight/the gravitational force,
2. the normal force holding it up, and
3. your pushing force.

It is possible that some of these cancel each other out, which would be the case if you push the object sideways over a horizontal surface. It is also possible that more forces, such as kinetic friction as an obvious next choice, are involved.

There is not necessarily a dedicated conventional name for all forces - there is i.e. no universal name for a random pushing force like here. We can just choose to call it a "pushing force" if we want.**

** It of course exists due to electromagnetic repulsion from the atoms that make up your finger tips - but I guess that's not the level you wish to ask about.

See the Newton's Second Law can be stated as: $$\sum \vec{F} =m \vec{a}$$ In the case of block,the 'type' of force the man is applying on the block is called Normal force and it is defined as the force which stops two bodies from occupying the same place. So let us represent Normal Force on block as N . Then: $$N=ma$$

Note here the force is N which happens to be numerically equal to $$ma$$ and not the other way around.

• What do you mean with "it doesn't work the other way around"? – Bernhard Sep 21 '20 at 18:31
• It is not anything significant as such but the OP seems to regard ma, the net force on block as the force applied by person which is a little absurd. – Tony Stark Sep 21 '20 at 18:33

I think that the OP question points to how we know that there are forces on an object, and what are their magnitudes and directions.

I can see two types of forces, classified by how they are measured:

Forces that be be measured by a load cell, (generally based on elastic properties).

The normal force, static friction forces and the force that is applied to the block by the man can be measured that way.

If only that types of forces are present, the vectorial sum of all of them $$\mathbf F_{net} = m\mathbf a$$. Kinetic friction forces seem more difficult to measure, and can be taken by difference in the equation above.

Gravitational.

A body in free fall is accelerated, but the force that is causing the acceleration can not be measured by a load cell. On the other hand, if a load cell is measuring the weight of a static object, what is being measured is the normal force, but there is no acceleration.

In those cases, a gravitational force must be postulated to avoid acceleration without a net force, or a net force without acceleration.