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## Newton’s 2nd law of motion, force and acceleration

Last week, I blogged about Newton’s 1st law of motion, and the concept of inertia. At the end I said that this week I would discuss what happens to an object if a force is applied. Or, to put it more correctly, an “external resultant force” is applied. This is what Newton’s 2nd law of motion is all about – the effect on a body of an applied force.

Newton’s three laws of motion appear in his masterpiece, The Principia, which was published in 1687.

If we apply a force to an object it will change its velocity, which means it will accelerate. As I have mentioned before, in physics acceleration has a more precise meaning than it does in everyday life. It doesn’t just mean an object is changing its speed, it can also be keeping a constant speed but changing its direction, such as an object moving at a constant speed in a circle. But, whether an object is changing its speed or changing its direction, it has to accelerate to do this, and so a force needs to be applied.

## The most important equation in physics

The relationship between force and acceleration is given by Newton’s 2nd law of motion, which states that

$\boxed{ F = m a }$

where $F$ is the force, $a$ is the acceleration, and $m$ is the mass of the body. From this equation, along with Newton’s 3rd law (which I will discuss next week), nearly all of mechanics can be derived. For example, this equation tells us that for the Moon to orbit the Earth, it must have a force acting upon it. That force is gravity, and Newton was also the first person to produce an equation to describe gravity. For example, in this blog I showed how we can derive the acceleration felt by a body travelling in a circle, which we call the centripetal acceleration.

Using calculus, this equation also allows us to derive the three equations of motion, equations like $v = u + at$ and $s=ut + \frac{1}{2}at^{2}$, as I did in this blog. It tells us that it is more difficult to accelerate a more massive object than it is a less massive one, which is why you need a more powerful engine in a large truck than you do in e.g. a small car. Along with Newton’s 3rd law, it explains why a bullet comes out at such a high speed from the nozzle of a rifle, but why the recoil of the gun moves much more slowly. As I said, the most important equation in physics.

Next week I will discuss the last of Newton’s laws of motion, his 3rd law.

## Inertia and Newton’s 1st law

This weekend I was helping my youngest daughter revise for her Christmas physics exam. She tells me she enjoys physics, but I’m not sure whether she’s just saying this to please me! Her brother has just started his physics degree this last September, and her elder sister seriously thought of doing physics for A-level before deciding against it; but I suspect my youngest is more on the languages and creative arts side than a scientist. We shall see, she is only 13.

The material we were going over was the basics of motion, or mechanics. She has been learning about forces, pressure, velocity and resistances to motion (friction and air resistance). I asked her if she had been learning Newton’s three laws of motion, and from her answer I wasn’t sure whether she had or not!

I distinctly remember my own first encounter with Newton’s three laws of motion. I was about my daughter’s present age, and due to a Horizon programme about cosmology and particle physics (that I mention in this blog), I had already decided I wanted to study astronomy and physics. It therefore came as a bit of a shock to me when we were presented with Newton’s laws of motion, and I found I could not remember them.

Despite seeming to have a very good ability to remember poems and song lyrics, I am terrible at remembering ‘facts’, and our physics teacher presented Newton’s three laws to us as a series of facts. After several days of trying and failing to remember them in the words he had used (or which the text book had used), I was beginning to have serious doubts that I could go into physics at all.

And then I had an epiphany. I realised that if I understood Newton’s laws of motion, I did not need to remember them. If I could understand them, I could just state them in my own words; and within less than an hour I felt I had understood them thoroughly (although I’d like to think I have a deeper understanding of them now than I did at 13!). I often say to my students that the only things they need to remember in physics are the things they do not understand. That, certainly, has been my own experience.

## Inertia

The concept of inertia is fundamental to our ideas of motion, and yet it is not the easiest concept to understand or explain. But, I will have a go! Galileo was the first person to think of the concept of what we now call ‘inertia’. He realised that a stationary object wants to stay stationary, and you have to do something to it (push it, pull it, or drop it) to get it moving.

Galileo was the first person to outline the concept of inertia.

He also realised that objects which are moving want to carry on moving. This is not obvious, as we all know if we give an object a push it may start to move but will slow down and stop. Galileo realised that objects which were moving stopped because of resistive forces like friction or air resistance, and in the absence of these an object would carry on moving. This is, essentially, the idea of inertia. The tendency a body has to remain at rest or to carry on moving.

## Newton’s 1st law of motion

Newton’s 1st law of motion is essentially a statement of the concept of inertia, and is sometimes called the ‘law of inertia’. If someone asks me to state Newton’s 1st law the wording I use will probably change slightly each time, but the key idea I make sure I try to get across is the concept of inertia. So, a way to state Newton’s 1st law is

A body will maintain a constant velocity unless a force acts upon it

This is, more or less, the most succinct way I can express his 1st law. But, by stating it so succinctly, there are hidden complications. The first is to realise that the term ‘velocity’ has a very precise meaning in physics. I was trying to explain this to my daughter over the weekend (except we were doing it in Welsh, so using the Welsh word ‘buanedd’).

For a physicist, velocity is not the same as speed, even though we may use them interchangeably in everyday language. Speed (‘cyflymder’ in Welsh 😛 ) is a measurement of how quickly an object is moving, but velocity also includes the object’s direction of motion. A stationary object has zero velocity.

Therefore, a more long-winded (but no less correct) way to state Newton’s 1st law is

A body will remain at rest, or carry on moving with a constant speed in a straight line, unless acted upon by a force

The last thing I should mention in relation to the wording of Newton’s 1st law is that, strictly speaking, I should say ‘an external resultant force’, because an object can maintain a constant velocity with more than one force act upon it, as long as those forces are equal in size and opposite in direction. A good every-day example of this is driving a car at a constant speed in a straight line. The car does not continue to do this if we take our foot off of the accelerator, because air resistance and friction will slow the car down. When it is moving at a constant velocity the resistive forces are balanced by the force the engine is transferring to the tyres on the road. An ice puck, once pushed, will take a long time to slow down. This is because the friction on ice is much less than on a rougher surface, so the puck comes closer to acting like Newton’s 1st law says objects should act.

I don’t think my daughter has learnt about Newton’s 2nd law yet, which I often tell my students contains the most important equation in physics. His second law tells us what happens to an object if there is a (resultant external) force acting up on it. I will blog about that next week.