
It is observed that if a mass, m, causes a given spring (behaving elastically)
to extend x, then 2m causes extension 2x and 3m causes extension 3x etc. 

We conclude that the force due to gravity
exerted by a suspended mass m is directly proportional to its mass.


This observation will be used in the method outlined below. 

Newton's second law can be demonstrated using a linear air track as shown
here. 





The mobile floats on air pumped through the holes in the track (a metal tube
of rectangular cross section). 



The law involves three
variables: mass, acceleration and force. 

For this reason, the experiment must be done in two
steps 



Part 1: Relation between acceleration and force
with mass constant 

In order to keep the total mass being accelerated constant, we add some
extra masses to the mobile for the first measurement. 



Allow the mass m to fall, causing an acceleration, which will be calculated
knowing the speed with which the mobile was moving after it had travelled a
known distance (that is the speed with which it passes through the infra red
gate). 

For the next reading, move one of the masses, m from
the mobile (where is contributes to the total mass but not to the
force causing the acceleration) to the hanging mass, as shown in the
next diagram. 



Similarly for a third reading 



To calculate the acceleration, we use the familiar equations of motion. 

The motion started from rest so we can write: 



Eliminating t gives: 



Note that the speed we calculate (v in the above equation), using the time taken
by the mobile to pass through the I R gate, will be an average speed
(the system is accelerating). 

This will be the speed the mobile had in the middle of the period of
its passage through the gate. 

Also note that we do not need to know the magnitudes of the forces but just
how the force varies 



The results will (hopefully !) be something like this 

force 
acceleration 
F 
a 
2F 
2A 
3F 
3A 


Results like this lead to our first conclusion, namely that the acceleration
is directly proportional to the force, when mass is constant, written as 





Part 2: Relation between acceleration and mass, with
force constant. 

It is easy to keep the force constant by simply having the same mass hanging
on the hook. 

It would be convenient if we could have extra masses to add to the mobile
each equal in mass to the mobile itself (in what follows, we will
assume this to be the case). 

The procedure is similar to the previous part except that now we pull three
different masses using the same magnitude of force. 



The results this time should be something like this: 

mass 
acceleration 
m 
a 
2m 
a/2 
3m 
a/3 


which gives us our second conclusion; that the acceleration of a body is
inversely proportional to the mass, when the force is constant, written as 



We can combine our two conclusions in one statement: 



or 



where k is a constant...
but, what (I hear you ask) is the value of this constant? 

Well, what would you like it to be ? 

How about "1" ? That would be convenient... 



Up to now we haven't mentioned anything about the units of
the quantities in this equation. 

However, the units of mass and acceleration have already been
specified elsewhere (kg and ms^{2} respectively). 



We will now use this equation to define a unit for measuring force: 

1 Newton is the force which can give a mass of 1kg an acceleration of 1ms^{2}. 



Defining the Newton in this way is equivalent to putting k = 1 in the equation
above. 



Thus, we have verified Newton's second law of motion (and, at the same time
defined a unit for force). 



The final conclusion is usually written as 

force = mass × acceleration 
