Physics with Calculus/Mechanics/Newton's Second Law

Newton's second law states: In an inertial reference frame, force is equal to mass times acceleration ($$\vec{F} = m \vec{a}$$). This also defines what force is quantitatively.

Second Law (Force)
Force can be a confusing term because it is an everyday term, as well as a physics term. When we use it in everyday language, we use it in a whole number of different contexts, for instance "He forced the door", "He was forced to take third semester Calculus", or "This justifies the use of force". There are a great many ways in which the word force may be used in casual conversation.

However, in physics, there is only one meaning to force, that which was given to us by Newton's Second Law. For now we will treat force as having been defined by:

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

where m represents the mass of the object in question and a its acceleration.

When you consider Newton's first law, you have to wonder how an object starts moving. Something has to happen to an object that is standing still to make it move. What happens is that an external object applies some sort of force, which can be summed up as a push or a pull. We know that, of course. Ever since we started observing the natural world around us we have been aware of the way in which things are moved around us.

What is important about this law is that force is now quantified, and can be shared between two objects. For instance I can design a weight that applies a force of thirty Newtons to the end of a lever. I can use this to calculate the force I apply to the other end of the lever. Using force it will become possible to solve complex physical systems for important variables.

It's interesting to note that if Newton's second law is a definition, then it's completely useless unless we know what the force is equal to. The actual physics comes in, for example, when you know that gravity gives a force inversely proportional to the square of the distance. Then you have a physical model and not before. Otherwise all you have is a definition, and you can't define how reality works! However, strangely enough, the concept of force seems to be a deep physical quantity that actually exists. Even though the ma side is wrong at speeds near light, the derivative of momentum is still force, but momentum is not mv. Also, the equations of physics are very elegant when written in terms of force. Maxwell's equations which govern electricity and magnetism can easily be translated into force, and gravity gives an elegant force. Many thing produce an elegant force law, so force seems to be a concept that is more than a definition.

Note: Newton studied the importance of the quantity $$ mv $$, now termed momentum. He stated the his Second Law as

$$ \vec F = \frac{d \vec p}{dt} $$

If the mass of the object is constant,

$$ \vec F = m \frac{d \vec v}{dt} = m \vec a $$

This form of the Second Law is used to prove the conservation of momentum and has significance in relation to Relativity Theory because this form holds true near the speed of light while F = ma breaks down.

There are quantities hidden Newton's second law which give rise to profound laws of physics. For example, the momentum is in there as well as energy. Both energy and momentum are conserved, which means that no matter what interaction takes place, they remain the same for an isolated system. In fact, both energy and momentum are locally conserved, which means that they not only remain the same, but they cannot teleport; it is possible to look at energy flow and momentum flow into and out of a system. In fact, in quantum mechanics, the concepts of energy and momentum remain, while the concept of force fades away to nothing -- energy and momentum are the deeper quantities. It is even possible (and extremely powerful) to reformulate the classical laws of physics in terms of energy and momentum with no reference to force at all!