How force is mass times acceleration?

Question

My teacher said that the force is mass times acceleration. But, how are mass and acceleration related to force?

Answer 1

And that is precisely the big, big question!

Unfortunately, we don't know everything. And the relationship you mention is one of those things - we know this is how the world works; but we don't know why.

The formula you mention is usually called Newton's 2nd law:

$$\sum F=ma$$

Newton "discovered" and formulated this law by doing many, many experiments. If you lift your pen and let go, it falls. It also falls when you do it again. And again. And 1000 times. And also when 1000 people do it 1000 times. In the end you start trusting this as something that will always happen - you can't prove it, but you still trust it to happen again next time you try.

Newton saw in this way that this just happens to be how the world works. It isn't an explanation, just an observation of the nature of the world.

We call it a law of nature; it can't be proven, but we trust it to work because it has done so many times before. Therefore there is no answer to a question about why this law is the case. We don't know and can't explain it - we just know that this is how it all works.

Answer 2

This is a special case of Newton's second law of motion:

Imagine an object of mass $m$ acted upon by a net force $F$. The force will change the momentum of the object. According to Newton's second law of motion we have $$F=\frac{\Delta p}{\Delta t}=\frac{mv-mu}{t}$$ where $u$ is the initial velocity of the object, $v$ is the final velocity of the object and $t$ is the time taken for the change in velocity.

If the mass $m$ of the object is constant the above equation can be rewritten as $$F=m\left(\frac{v-u}{t}\right)$$

The term in brackets is the acceleration $a$ of the object and if the mass $m$ of the object is not changing (unlike a rocket that consumes fuel as it accelerates) then $F=ma$ (a special case of Newton's second law where the mass is constant).

Answer 3

I think you might be asking why mass and acceleration are related. If so, I can help. Force is a measure of mass and acceleration that humans have agreed to use. The only reason the two properties are related is because physicists have defined force as being mass times acceleration.

The reason I didn't use Newton's second law in my answer like most others is this:

The use of Newton's Second Law as a definition of force has been disparaged in some of the more rigorous textbooks, because it is essentially a mathematical truism.

The equation is just a restatement of the actual definition.

Wikipedia article on Force

Answer 4

Newtons second law states that change in linear momentum is proportional to the external force acting on the body and us in the ditection of the external force. Therefore if $p$ is linear momentum of body and $f$ is the force applied on the body then $$\frac{dp}{dt}=f$$ Further, the velocity of a body is the rate of change of its position $$P=mv=m \cdot \frac{dr}{dt}$$ $r$ is position vector of the particle. Therefore $$\frac{dp}{dt}=f$$ yields $$m \cdot \frac{d^2r}{dt^2}=f$$ The quantity $$\frac{d^2r}{dt^2}$$ is the acceleration of the particle. Thus force is mass $\times$ Acceleration This equation is called an equation of motion --mk verma book introduction to mechanics

Answer 5

Acceleration is a change in velocity. Newton found that an unbalanced force is required to change an object's velocity.

Newton's Second Law of Motion defines force in this way: Acceleration is produced when a force acts on an object. The 2nd law provides the definition of force: F = m a, where F is force, m is the mass, and a is acceleration.

This relationship between mass and acceleration provides a useful way to define and measure forces that act upon objects and change their velocities.

Answer 6

Newton was the first one to quantify physical (physica means nature in some language I believe) observations. He built on Galileo's concepts of inertia.
What you must understand that the quantitative aspects of any physical entity are things which are created by us to 'gauge' their qualitative counterparts. In other words the numbers associated to entities and the units assigned to them are something we do and not nature.

Since Newton was the first to quantify stuff, he was at liberty to assign them any numerical value. His laws are simply axioms which agree with what physical experiments show.

That a ball will continue to move with the same velocity if there exists no external agency to oppose its motion is what makes Newton's first law

That it takes more force to stop a truck than a car (moving with the same velocity) or that it takes more force to stop a bullet fired by a gun than a bullet flung by you like a dart (more velocity in the former case) is what is contained in Newton's second law: The force 'required' is proportional to the rate of change of momentum (Force times acceleration is simply a special case of this law; a case in which the mass remains constant).

Remember, quantifying is always something we do.

Analogy: Think that the concepts of weighing are non existent. In case you and your friend would like to compare your weights without a reference standard, it'll simply be impossible. You know that you weigh more than your friend. But by how much? For that you need a reference standard. Maybe you'll try to compare yourselves to the weight of a boulder using a primitive balance you find somewhere. Then you're able to quantify your results as you like. Maybe you weigh enough to tip the balance (which has the boulder on one side) twice as much as your friend is able to. Voilà. Now you're able to quantify your weights. You 'weigh' twice as much as your friend.

I hope that adds to your intuition.

Answer 7

Imagine an object moving with a constant velocity in one direction.

Let's say you want the moving object to change direction by touching, i.e., pushing it by hand to some other direction. If you are trying hard to make it change it's direction then the object has something which it is resisting to change it's direction.

This tendency of the object to not change its state is called "Inertia".

But the question is "Can we MEASURE Inertia?". It turns out we can only approximate a comparison between different objects. A coefficient of Inertia, we call mass.

If we compare two objects moving with the same velocity, the object with more mass will be harder for us to change its direction. That something we feel when we try to change the object's course is, we call it force.

A mathematical expression, a product of mass and velocity "$mv$" is understood to be possessed by a moving body. This quantity, we called "momentum" possessed by a moving body is acted by something and let to change in direction or in quantity of the velocity per unit time. So, this action require to change the momentum per unit time is called force. i.e., if there is more momentum, we require more force to change it's state.

So, one of two things happen, if we compare two objects of different mass but the same momentum, the smaller object must have faster velocity than the bigger object. Therefore, if we wish to stop them than we will require the same force.

To know whether or not a body has more mass than another,we imagine, if we were to move the object at rest, the one with harder to change its state has more mass than the other, i.e., more inertia coefficient.

And thus the formulation of the Newton's laws.

And your question how is it related to mass and acceleration, that is answered by the reasoning that if we wish to change the direction or the magnitude of velocity of a moving object per unit time then by definition of acceleration it is by default related.