Newton's Third Law of Motion

Date: 05/08/2008 at 22:29:40
From: Ahmed
Subject: Newton's laws of motion

Hello Dr. Math,

I'm having trouble understanding the laws of motion, specifically the 
third law.  What I do not get is that when applying a force on an 
object, the object itself would apply equal but opposite force.  My 
question is why, in some situations, the object which the force 
was exerted on it does not stay in its place rather than moving in the 
same direction of the force.  Why don't the forces cancel each other?  

An example would be a car's tire and a road.  The tire exerts a force
on the road and at the same time the road exerts an opposite and equal 
force so if this is true why do cars move?  I hope you can answer my 
question in a very simple manner.  Thank you very much.

Regards,

Ahmed

Date: 05/09/2008 at 15:03:52
From: Doctor Achilles
Subject: Re: Newton's laws of motion

Hi Ahmed,

Thanks for writing to Dr. Math.

Essentially, the car's mass is roughly 1000 kg.  Let's say the car
accelerates at 10 m/s^2 (ten meters per second per second).

The force that the car is applying, then is:

  F = ma

  F = 1000 kg * 10 m/s^2 = 10,000 kg*m/s^2

So somewhere a force of 10,000 kg*m/s^2 is being applied to the car. 
Where does that force come from?

Well, the car is spinning its wheels which have tread on them that
grips the road.  The car is pushing backwards on the road with a force
of 10,000 kg*m/s^2.

But, the road is attached to the planet Earth.  And the planet Earth
weighs about 6,000,000,000,000,000,000,000,000 kg!

  Measurement: Raindrops, the Weight of the Earth
    http://mathforum.org/library/drmath/view/52367.html 

So we have a car pushing backwards on the Earth with a force of 10,000
kg*m/s^2.  Using the equation F = ma again:

  10,000 kg*m/s^2 = 6,000,000,000,000,000,000,000,000 kg * a

We can solve for acceleration:

  a = 10,000/6,000,000,000,000,000,000,000,000 m/s^2

  a = 1/600,000,000,000,000,000,000 m/s^2

  a = 0.00000000000000000000167 m/s^2

So, while the car accelerates forward at 10 m/s^2 it pushes back on
the Earth and causes the Earth to accelerate in the opposite 
direction.  Because the Earth weighs 600,000,000,000,000,000,000 times
as much as the car, it accelerates 600,000,000,000,000,000,000 times
less (given that the force is equal but opposite).  And therefore, the
car moves the Earth, just VERY, VERY, VERY slowly.

Hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/09/2008 at 19:07:45
From: Ahmed
Subject: Newton's laws of motion

Hello Doctor Achilles,

First I would like to thank you for your response.  I think it makes
more sense.  But I'm still a little confused about the equal and
opposite force.  If I push on a pendulum, and it pushes back with an
equal force, why does it move?  If the forces balance, why isn't it
impossible for me to move the pendulum?

Also, can you explain how they came up with the formula F = ma?

Date: 05/09/2008 at 19:32:44
From: Doctor Achilles
Subject: Re: Newton's laws of motion

Hi Ahmed,

Thanks for writing back to Dr. Math.

With regard to your pendulum question, let's consider first the 
simplest case of an astronaut whose mass is 100 kg floating in the
middle of space sitting on top of a small rock that also has a mass of
100 kg.  If the astronaut wants to move to the left, then he pushes
the rock away from himself to the right.  He now goes off to the left,
away from the point where he and the rock were sitting and the rock
goes with an equal acceleration off to the right.  Now they are both
floating away from their common center.

If he didn't push very hard, then the small gravitational pull of the
rock will begin to pull him back to the right (toward the center
point) with a force.  He will also be exerting an equal gravitational
force on the rock, pulling it back to the left.  The two will meet and
eventually collide back where they originally started, at which point
they will both exert an equal force of collision on each other.  Both
the astronaut and the rock will have moved the same total distance.

Now, let's say that we have the same system, but the rock weighs 200
kg.  When the astronaut pushes the rock to the right, he will go off
to the left with a certain acceleration.  The rock will go to the
right, but with HALF the acceleration because it has twice the mass. 
They will then be pulled back by their gravitational forces to the
center point again, but the rock will have moved half as much distance
as the astronaut.

Now, consider a case where the astronaut is sitting on a rock (like
Earth) that has a mass of 6,000,000,000,000,000,000,000,000 kg, which
is 60,000,000,000,000,000,000,000 times as much as his mass.  When he 
pushes off the rock (let's call it jumping), he accelerates away from 
the rock with a certain acceleration (let's say 6 m/s^2).  In doing 
so, he causes the rock to accelerate in the other direction
1/60,000,000,000,000,000,000,000 times as fast, or at a rate of
0.00000000000000000000001 m/s^2.

This acceleration on the rock is so small that it is impossible to
measure, but it does happen and the forces are equal.

In the case of the pendulum, as you push it to the left, it pushes
your hand to the right.  Your hand is attached to a large mass (your
body) and if you're not floating in space your body is anchored to a
huge mass (the Earth) via the friction of your shoes.  The pendulum's
mass is so small that you don't notice the force pushing back.

But, consider if you had a very massive pendulum (like a huge Gothic
church bell).  If you want to push on that, you'll have to impart a
lot of force.  In so doing, you will also have to be wearing good 
shoes that grip the Earth well, or you will find that your feet keep
slipping on the ground and you can't move the giant bell at all.

When you don't see the opposite force, in general, you should always
ask yourself whether you (or whatever is doing the pushing) is 
touching the Earth (or anything like a house, that is anchored to the
Earth).  If you are, then the Earth is probably absorbing the extra force.


Regarding the derivation of F = ma, I am not an expert on this matter,
but as I understand it, it wasn't derived so much as defined.  Simply
put, Newton defined the unit of force (which is called a "Newton") to
be the amount of force it takes to accelerate a 1 kg mass at 1 meter
per second per second.  In other words, if you push on a 1 kg mass 
such that after 1 second it is moving at a velocity of 1 meter per
second, then the force you applied was defined as "1 Newton".  1
Newton is equal to 1 kg*m/s^2.

It was then observed that if you apply an equal force to a 2 kg mass,
the acceleration is half as much.  After many measurements like this,
it was determined that the acceleration an object exhibits is equal to
the force applied divided by the mass of the object.  I.e.:

  a = F/m

Multiply both sides of the equation by m and you have F = ma.

I hope this helps.  Please write back if you'd like to talk about this
further.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/