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Experimental Verification of Newton's Second Law of Motion
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.

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
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