Force and Newton's three Laws
Force and Newton's three Laws
We have all known from a young age that an acceleration is caused by a push or a pull. Today we will express this more qualitatively in 3 laws which are called Newton's Laws.
Newton's first Law.
Newton's first law really harks back to the early 17th century. It was Galileo who expressed what he called the law of inertia, he stated:
"A body at rest remains at rest and a body in motion continues to move at constant velocity along a straight line unless acted upon by an external force."
Now you can read Newton's own words from his famous book Principia:
"Everybody perseveres in its state of rest or of uniform motion in a right line unless it is compelled to change that state by forces impressed upon it."
The problem is that Newton's 1st law goes clearly against our daily experiences; things that move don't move along a straight line, nor do they continue to move for ever. The reason for this is gravity; and there is another reason too, even if you remove gravity there is still friction, and there is air drag. So things will always come to halt.
But we believe, though, that in the absence of any forces, that an object, if it had a velocity, would continue along in a straight line forever, and ever, and ever.
Warning advanced ideas may be found in the following ...
However Newton's first law, this profoundly fundamental law, does not hold in all reference frames. For instance, it doesn't hold in a reference frame which itself is being accelerated. Let me try to explain.
Imagine I am being accelerated whilst typing this - I'm on my Acme physics-o-computer-cycle perhaps - you, sitting in lab 6 watch me being accelerated in a known direction, say directly towards you. You immediately say to yourself, 'Aha! His velocity is changing; therefore, according to the Newton's first law, there must be a force acting on him! You can't contain yourself and shout, "Hey there Mr Dey, can you feel the force?" I would reply, "Yeah I do, I really do. I feel something pushing me through the seat." This is consistent with the first law. Perfect, the first law works for you observing from lab 6.
Now the observation switches, I'm watching you looking at me, and am still being accelerated in the same direction. You all appear to come towards me being accelerated in the opposite direction. I think 'Aha! The 1st law should work so you people should feel a push.' I'm terribly excited by this breakthrough and shout, "Hey there! Do you feel the force?" You reply "I feel nothing! There is no push, there is no pull."
Therefore the 1st Law doesn't work for my frame of reference if I'm being accelerated towards you.
So the obvious question is; when does the first law work?
Well the first law works when the frame of reference is what we call an 'inertial frame of reference.' An inertial frame of reference would then be a frame in which there are no accelerations of any kind. Is this possible? Is the room you are sitting in, lab 6, an inertial reference frame?
Well firstly, the Earth rotates about its own axis and your room goes with it; that gives you something called a centripetal acceleration (which you'll learn about at A21).
Secondly, the Earth goes around the sun; that too gives a centripetal acceleration, ...
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Well the first law works when the frame of reference is what we call an 'inertial frame of reference.' An inertial frame of reference would then be a frame in which there are no accelerations of any kind. Is this possible? Is the room you are sitting in, lab 6, an inertial reference frame?
Well firstly, the Earth rotates about its own axis and your room goes with it; that gives you something called a centripetal acceleration (which you'll learn about at A21).
Secondly, the Earth goes around the sun; that too gives a centripetal acceleration, which includes the Earth, includes you, and includes the room you are sitting in.
Thirdly, the Sun goes around the Milky Way. You can on and on! You get the gist.
So, clearly, the room you are sitting in is not an inertial reference frame.
So if lab 6 isn't an inertial reference frame how come we use Newton's laws?
We can start to explain this by trying to estimate how large these accelerations are that we experience, let's start with the one that is due to the Earths rotation. With a bit of A2 wizardry we find that the centripetal acceleration at the equator is approximately 0.034 ms-2. This is way, way less than gravity, in fact its three hundred times smaller than the gravitational acceleration that you experience here on Earth. Now if we also take the motion of the Earth around the sun then there is an additional factor of five times lower!
In other words, these accelerations, even though they are real and can be measured - easily with today's high-tech instrumentation - they are much, much lower than what we are used to experiencing, which is the gravitational acceleration. Therefore, and in spite of these accelerations, we can accept lab 6 as a reasonably good inertial frame of reference in which Newton's first law then should hold.
Can Newton's Law be proven?
The answer is no, because it is impossible to be sure that your inertial reference frame is without any accelerations.
Do we believe in Newton's Law?
Yes we do. We believe in it because it is consistent within the uncertainty of the measurements of all experiments that have been done so far.
Newton's second law
I have a spring, forget gravity for now, I'm doing this somewhere in outer space.
this is the relaxed length of the spring.
I extend the spring
I extend it by a certain amount over a certain distance, it unimportant how much, I know that when I do that there will be a pull - no negotiable!
I attach a mass to the end and I measure the acceleration, a1, that this pull causes on this mass immediately after I release it. It's easy to measure. Now I replace the object with mass, m2, but the extension is the same, so the pull must be the same - the spring doesn't know what the mass is at the other end, right! So, the pull is the same, m2 has replaced m1 and I measure the acceleration, a2, of m2.
It is thus an experimental fact that;
m1 x a1 = m2 x a2
And this product, ma, we call the force. Starting to make sense yet? Ma is our definition of force.
So the same pull on a ten-times larger mass would give a ten-times lower acceleration.
This is Newton's second law stated for you to read;
"A force action on a body gives it an acceleration which is in the direction of the force and has a magnitude given by ma."
So lets write this in its full glory. It's perhaps the most important law in the whole of physics!
Force = mass x acceleration
F = ma
The units of this force are kg m/s2, and in honour of the great man we call this one Newton.
Like the first law the second law only holds in inertial reference frames.
Can Newton's second law be proven?
No.
Do we believe in Newton's second law?
Yes.
Why do we believe in it?
We believe in it because all experiments and all measurements, within the uncertainty of the measurements, are in agreement with the 2nd law.
Now you may object and you may say, this is strange what you have been doing. How can you ever prove a mass if there no force somewhere? Because if you want to determine the mass maybe you have to put it on a scale, and when you put it on a scale to determine the mass you make use of the gravitational force!
Isn't this 2nd law spring-mass idea some kind of circular argument? The answer is no.
I can be somewhere in outer space where there is no gravity. I have two pieces of cheese that are identical in size, they are cheese without holes by the way, the sum of the two has double the mass of one. Mass is determined by how many molecules, by how many atoms I have. You don't need gravity to have a relative scale of masses - cheesy units in this case. So you see, I can determine the relative scale of masses m1 and m2 without ever using a force. Thus my 2nd law treatment is a very legitimate way of checking out the second law.
Since all objects on Earth fall with a constant acceleration which we call, g, we can write down that the gravitational force, Fg, equals the mass of the object (relative to Earth) multiplied by g.
Fg = mg
So you see that the gravitational force, due to the Earth on a particular mass, is linearly (directly) proportional with the mass; if the mass becomes ten-times larger then the force due to gravity goes up by a factor of ten.
Imagine I am holding a ball in my hand, in the reference frame of lab 6 - which we will accept to be an inertial reference frame. If I just place it on my palm it isn't being accelerated in our reference frame which means the force on it must be zero.
Above is the ball, and we know that if it has mass m, in this case 0.5 kg, then there must be force here called, mg, which is about 5 N. But the net force is zero. Therefore it is clear that I, N K Dey, must push up, from my hand on to the ball, with a force of about 5 N. Only now there is no acceleration. So I can write down:
Fnkd + Fg = 0
Because it's a one-dimensional problem you could say:
Fnkd = -mg
Let's go back to F = ma.
Notice there is no statement made on velocity or speed. As long as you know F and as long as you know m, a is uniquely specified. No information is needed on the speed.
So that would mean, if you take the gravity on an object that was falling down at 5 m/s, that the law would hold. If it were falling at 5000 m/s it would also hold.
Will the law always hold?
No. Once your speed approaches the speed of light then Newtonian mechanics no longer works and you have to use Einstein's theory of Special Relativity. So this law is only valid when have speeds that are substantially lower then the speed of light.
Newton's 3rd Law
"If one object exerts a force on another, the other exerts the same force in the opposite direction on the one."
I normally summarise the above statement as follows:
Action = - reaction
The minus sign indicates that it opposes. So you sit on your seats and you're pulled down onto your seats because of gravity and the seats will push back on you with the same force. Action = - reaction!
I hold a cricket ball in my hand, the ball pushes on my hand with a certain force and I push on the ball with the same force.
I push against the wall with a certain force, the wall pushes back with exactly the same force.
The 3rd law always holds. Whether the objects are accelerated or still, it makes no difference. At all moments in time, the force, which we call a contact force, between two objects; one on the other is always the same as the other on one but in the opposite direction.
Let's take a look at a very simple example.
We have an object, 1, which has a mass, m1 = 5kg, and attached to it is an object, 2, which has a mass, m2 = 15 kg. There is horizontal force, F = 20 N.
What is the acceleration of this system?
F = ma
Clearly the mass is the sum of the two objects as the force acts on both, so we get;
F = (m1 + m2)a
Substituting, we find that a = 1 m/s2 in the same direction as F. So the whole system is being accelerated at 1 m/s2.
Now watch closely,
I single out object 2.
Remember object 1, while this acceleration takes place, must be pushing on object 2 otherwise object 2 could never be accelerated. I'll call this force F12, the force that 1 exerts on 2.
I know that object 2 has an acceleration of 1 m/s2, that's a give-me already. Here comes F = ma;
F12 = m2a
We know a = 1 m/s2, and m2 = 15 N; so we see that the magnitude of the force F12 is 15 N.
Now I'll isolate object 1.
Object 1 experiences a Force F = 20 N, AND it must experience a contact force from object 2. Somehow object 2 must be pushing on object 1 if 1 is pushing on 2! I'll call that force F21.
I know that object 1 is being accelerated and I know the magnitude is 1 m/s2. It's non negotiable. So we have;
F + F21 = m1a
With simple substitution we find that F21 = -15 N
So you can see that F21 is opposite in direction to F12 but equal in magnitude!
So 1 is pushing on 2 with 15 N and 2 is pushing back with 15 N, meanwhile the whole system is being accelerated at 1 m/s2.
In these two examples, the one with the cricket ball in hand and the one above, both can be seen to be consistent with the 3rd law. The contact force of one on the other is the same as the other on one but on opposite sides.
So is this a proof?
No.
Can Newton's 3rd law be proven?
No.
Do we believe in it?
Yes. Because all measurements, all experiments within the uncertainties are consistent with Newton's 3rd law.
Action = -reaction. This is something that you experience every day.
I remember that as a child I used to play with the garden hose. On warm days I would turn on the tap fully and watch the hose snake backwards, drenching all spectators. Can you explain why?
Now lets play with some balloons. Can you demonstrate action = -reaction? The balloon pushes onto the air and the air pushes on to the balloon, so when you let it go the balloon will zoom off in the opposite direction of the hole. It's the basic idea behind a rocket.
What about firing a gun? The gun exerts a force on the bullet, the bullet exerts an equal and opposite force on the gun which is called the recoil.
DEMO: Hero's Engine (GK legend, priestess of Aphrodite, her lover Leander, would swim across Hellas pond to be with, but one night the poor chap drowned and Hero threw herself into the sea. Very romantic but not very smart! Ask.com = heros engine ala coka cola, nail at angle)
How does N3L help us explain walking?
Some bizarre consequences of Newton's Laws. (PiVot 26:42)
Centripetal simply means 'centre seeking' and refers to the force whose effect you feel pushing you towards the centre of a rotating circle. Imaging sitting on a chair facing the centre of a roundabout, when spinning you feel the back of the chair pushing you towards the centre, viola, its as simple as that.