CM [011] Newton's 2nd Law

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SECOND LAWCM [011] Newton’s


Newton’s Second Law

Acceleration motion is described by Newton’s Second Law, which says:

When a particle is acted on by an external force, it accelerates. The acceleration is proportional to the magnitude of the applied force 𝑭𝑭 and is in the same direction as the applied force. If the particle has a mass 𝑚𝑚 and the applied force is represented by the vector 𝑭𝑭, then: 𝑭𝑭 = 𝑚𝑚𝒂𝒂.

State of rest Object accelerated to 𝑣𝑣Force Applied


Constant Force and Motion

As we have said the action of changing the state of motion of an object is tantamount to the application of force on the object. When a force is applied horizontally to an object resting on a frictionless surface, the object moves and attains a certain speed. Aristotle fully recognized that this force is the cause of motion. However, he did not know about friction working on the object. Thus he came to the observation that a constant supply of force is required to keep the object in motion.


Constantly Adding Force = Acceleration

In Galileo’s and Newton’s theory, if force is applied to an object constantly, there is constant increase in momentum and thus resulted in constatnt acceleration.

Acceleration


Investigations on Force

But this ‘force’ which is so central to Newton’s theory of motion, was not clearly explained by Newton himself. It seems to be a kind of action causing agent which exists mysteriously in nature which he did not explain at all. People worked hard for thousands of years trying to find what it is: pioneer scientists wrote books on it. Notable examples are ‘Concepts of Force’ (1957) by Max Jammer (1915-2010) and ‘Forces and Fields’ (1965) by Mary Hesse (born 1924).

Hundreds of other scientists also talk about it in thesis, papers and textbooks. But none can give us a satisfactory answer to the nature of force.


Momentum Change

Fortunately, in Newton’s theory, there is another definition of acceleration and force. This is the change in momentum 𝑝𝑝. Remember that acceleration is but the time change in velocity, that is:

𝑎𝑎 =𝑑𝑑𝑣𝑣𝑑𝑑𝑑𝑑

If 𝐹𝐹 = 𝑚𝑚𝑎𝑎, then:

𝐹𝐹 = 𝑚𝑚𝑎𝑎 = 𝑚𝑚𝑑𝑑𝑣𝑣𝑑𝑑𝑑𝑑

Which ends up as:

𝐹𝐹 = 𝑚𝑚𝑎𝑎 = 𝑚𝑚𝑑𝑑𝑣𝑣𝑑𝑑𝑑𝑑 =

𝑑𝑑𝑚𝑚𝑣𝑣𝑑𝑑𝑑𝑑

𝑚𝑚𝑣𝑣 is the momentum 𝑝𝑝 of the system. So we have:

𝐹𝐹 =𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑

𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭

=

𝑪𝑪𝑪𝑪𝒂𝒂𝑪𝑪𝑪𝑪𝑭𝑭𝒊𝒊𝑪𝑪

𝒎𝒎𝑭𝑭𝒎𝒎𝑭𝑭𝑪𝑪𝒎𝒎𝒎𝒎𝒎𝒎


Momentum Change

Thus we have a basic nature of force as the change in momentum. This is correct so far in view of the motion of an body by added momentum to it, equivalent to the application of force.

Receiving momentum and moves accordingly

Force Applied = momentum injection


Momentum is important not only because it is the agent that choreographs constant rectilinear motion. It is also associated with acceleration and deceleration. When an object is at rests, the addition of momentum will spurs it on to motion; on the other hand, when a travelling object losses momentum, it slows down and drop to a stop.

Momentum Changes Motion


Momentum & Acceleration

A particle at rest is one without any momentum (known previously as impetus). As said, it is not a special state. It is only a state being without any momentum.

Particle B at restWith velocity v = 𝑣𝑣𝐵𝐵 = 0The state being without momentum

Palm not moving, thus no momentum


Mechanism of Acceleration

The particle being stricken gets the momentum from the palm and move away with the momentum.

Particle B in motionWith velocity 𝑣𝑣 = 𝑣𝑣𝐵𝐵Momentum is doing all the work. The particle now is a free one.

Palm gathering momentum for a push.

Particle being push thus gaining momentum from the palm and starts to accelerate.

Momentum enters body of particle

Particle at rest❶ ❷ ❸


Adding Force = Acceleration

We can also see that momentum is transferred only when objects are in contact. If we can extend the space of contact as a large zone visible to the eye, we can see momentum happily migrating from the pushing host to the object.

Contact zone


Collisions – When Momenta Change HandsThe transfer of momenta may happen under many situations. But the essence is in the duration of transfer. The momentum will be small when the transfer time in long; and vice versa.

Particle being push thus gaining momentum from the palm and starts to accelerate. The touch is soft.

In collisions, the exchange of momenta is fast and strong’.


Modified Second Law

So we added Part 3 of Law One into Law Two, and change force into momentum at the same time, we have a modified version of Law Two, written as:

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑖𝑖𝑖𝑖 𝑑𝑑𝑡𝐹𝐹 𝐹𝐹𝑡𝑎𝑎𝑐𝑐𝑐𝑐𝐹𝐹 𝐹𝐹𝑜𝑜 𝑚𝑚𝐹𝐹𝑚𝑚𝐹𝐹𝑐𝑐𝑑𝑑𝑚𝑚𝑚𝑚 𝑖𝑖𝑐𝑐 𝑑𝑑𝑖𝑖𝑚𝑚𝐹𝐹.𝐹𝐹𝐹𝐹𝑇𝑇𝑡𝐹𝐹 𝐹𝐹𝑡𝑎𝑎𝑐𝑐𝑐𝑐𝐹𝐹 𝐹𝐹𝑜𝑜 𝑚𝑚𝐹𝐹𝑚𝑚𝐹𝐹𝑐𝑐𝑑𝑑𝑚𝑚𝑚𝑚 𝑖𝑖𝑐𝑐 𝑑𝑑𝑖𝑖𝑚𝑚𝐹𝐹 𝑐𝑐𝑖𝑖𝑣𝑣𝐹𝐹𝑖𝑖 𝐹𝐹𝑖𝑖𝑖𝑖𝐹𝐹 𝑑𝑑𝐹𝐹 𝑜𝑜𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹


Modified Second Law

The resultant corollaries are:

i. Acceleration is the result of velocity change:

𝑎𝑎 =𝑑𝑑𝑣𝑣𝑑𝑑𝑑𝑑

ii. Force is the result of momentum change:

𝐹𝐹 = 𝑚𝑚𝑎𝑎 = 𝑚𝑚𝑑𝑑𝑣𝑣𝑑𝑑𝑑𝑑

=𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑

iii. The direction of force is related to the direction of the momentum transfer.

FORCE?


Popularity of Force over Momentum

These are the main reasons why momentum is of priority in the consideration of particle collisions. Force is a compound concept which is more convenience for handling mechanical applications. So convenient that the origin of force is overlooked. Laymen are more familiar with the concept of force rather than momentum and will mistake the two as the same.


Momentum the Basic Ingredient.

Force is the manner momentum is transferred. So the basic ingredient of the Second Law of motion is also momentum.


NEWTON'S 3RD LAWTo be continued on CM [012]:

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CM [011] Newton's 2nd Law