Part 3. newton's principia

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LAWS OF MOTIONNewton’s


On the Shoulder of Giants

We now have established that in nature, there are different properties pertaining to motion: velocity, force, impetus and inertia.

By setting up the Laws of Mechanics, Newton established the relationship between the various elements. He showed that the same physical laws are universal and can be applied to all matter, whether living or nonliving, on earth or in space, thus revolutionizing our view of the universe.


Newton’s ‘Three Laws’

Strictly speaking, the three laws of Newton’s are actually not laws analogous to the man-made law in our common sense. They are only generalized descriptions of certain phenomena of motion. Descriptions such as the motional behaviours of objects in the first law, the relation of force and acceleration in the second law, and the action-reaction relation in the third law. That is why we draw them as columns, not truly foundations. However they are so general, so amicably applicable in practice that people just accepted them as laws – and it is convenient to do so.


Need to Re-examine the Laws

On the surface, they seem to have settled the millennia old problem of motion. But at a closer look, they are not quite what them seem to be appropriate for our discussion. Although they have created a concrete body to embrace almost all problems about motion, they have not provided an answer to the basic question – “what is impetus or momentum?”. So we have to subject the laws to a closer examination.


1ST LAWS OF MOTIONNewton’s


[and] a particle in uniform rectilinear motion will continue to move on forever at constant speed in the same manner.

Newton’s First Law in 3 Parts

A particle in rest will remain forever at rest

It will change its state of motion only and only when it is compelled to do so by forces impressed on it.

Law No. 1 can be more conveniently studied in three parts I , II, & III.


Newton’s First Law : Part 1

A particle will remain at rest and stays like that forever. Why is it not moving? Because it is in the situation when there no momentum. So a particle at rest is not a special case of motion. It is a motion with 0 momentum.


𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 0

So momentum:𝑝𝑝 = 𝑚𝑚𝑚𝑚 = 0

So I am not moving




This is the old description of impetus and conservation familiar to John Philoponus, Jean Buridan, Descartes and Galileo Galilee. This will be studied in the coming sections.



It will change its state of motion only and only when it is compelled to do so by forces impressed on it.

This part deals with force and its effect on the object’s state of motion. So it is also to be dealt with in a later section on force and acceleration.


Law 1 Part 3

Law 2 Law 3

We start from here!

Sequence of Analysing Newton’s Law




This is not an independent case. It is only a special case when momentum is equal to zero. So it can be incorporated into Part 2 as a subsidiary, saying:When a particle has no momentum, it will not move, thus remains at rest.

𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 0So

momentum:𝑚𝑚𝑚𝑚 = 0

So the object is not moving




This phenomenon had been demonstrated by Galileo before in his ramp experiment. When there is no friction and air resistance, the ball will move on forever.


Comparison

When we compare this modified law with Newton’s original one, it can be seen that the latter is a description of a physical phenomenon. The new law spells out the more basic elements that is affecting motion. First we have the motion, then we have the event described by Newton’s First Law.

Momentum

A particle in rest will remain forever at rest, and a particle in uniform rectilinear motion will continue to move on forever at constant speed in the same manner. It will change its state of motion only and only when it is compelled to do so by forces impressed on it.

An object is moved by its momentum.

Newton’s Modified First Law Newton’s Old Descriptive First Law


Newton’s First Law of Motion - Modified

By eliminating Part 3 and incorporating Part 1 into Part 2, we have a more basic law of motion – Law No. 1:

An object is moved by its momentum.

The other situations can be written as corollaries:

i. Momentum never fades away and so is conserved.

ii. When momentum is zero, the object will be rest.

iii. An object travels at the direction of its momentum.

iv. Since momentum is conserved, constant motion is for ever.


Object Carried by Momentum

Instead of saying an object is motivated by its momentum, it is more correctly to say that an object is carried by its momentum: like a man carried by a swan or a saint carried by angels.


. . . and the deeper foundation of the First Law of motion is momentum, not force.


ACCELERATIONGalilean


. . . it will change its state of motion only and only when it is compelled to do so by forces impressed on it.

Part 3 of Newton’s First Law

Part 3 of Newton’s First Law spells the relation between state of motion and force:

The right place for this part should be a part of the second law where it would probably read:

Force changes the state of motion [of a particle].

However, this is enough for us to start to discuss on a new state of motion.


The States of Motion

Rest = Motionless = Constant rest In constant Motion of velocity v.

The states of motion of an object can be summarized as rest and being in motion at any speed as described by Part 1 and 2 of Newton’s First Law.


The States of Motion

Velocity start from 0 to any speed 𝑚𝑚.Object accelerated.

In raising the object at rest to a state of motion, we have acceleration.

Time

Velo

city


AccelerationActually, when an object picks up speed, it accelerates. If it slows down it is said to decelerate. However, for convenience in general discussion, any motional change is said to be in acceleration.

Acceleration Deceleration


Acceleration due to Gravity

The most familiar event of acceleration is in free falling caused by gravity. The earth pulls the object and it falls straight to the ground. It can be seen that the rate of falling varies. It increase steadily with height. The object will fall faster and faster until it hits the ground. That is, the rate of acceleration increases on time, the longer an objects falls, the greater acceleration it will reach. The final velocity will be tremendous.


Rate of Free Fall

We know that things fall down because of the force of gravity. But Galileo did not yet have the concept of gravity at the time. He took falling for granted and concentrated on measuring how fast things can fall.


Galileo Measured Acceleration

Galileo used many devices to study constant motion and acceleration. One of the major gadgets he used was a ramp. The ramp would slow down the rate of fall to make measurement manageable. He could vary the slope of the ramp so that the speed of a ball rolling down the ramp can be adjusted. The less incline was the ramp, the slower would be the speed of the ball, making it easier for timing purpose.


Galileo Demonstrated his Ramp Experiment


Equation for Acceleration

Over a period of 20 years, Galileo observed the motions of objects rolling down in various inclination. By measuring the distance a ball rolled down the ramp in each unit of time with water clocks or other timing devices. Galileo concluded from his experiments that if an object is released from rest and gains speed at a steady rate, then the total distance, 𝑑𝑑, travelled by the object is proportional to the square of the time that it took in motion with g as the acceleration constant:

𝑑𝑑 ∝12𝑔𝑔𝑉𝑉2

Thus the first correct concept of accelerated motion was born.


Geometric Representation of Acceleration

Since in acceleration, the velocity changes from time to time. The graph of acceleration is different from that of velocity. The distance 𝑑𝑑 travelled by a particle at constant velocity 𝑚𝑚 is:

𝑑𝑑 = 𝑚𝑚Δ𝑉𝑉 =Δ𝑥𝑥Δ𝑉𝑉

× Δ𝑉𝑉 = Δ𝑥𝑥

The distance 𝑑𝑑 travelled by a particle at constant acceleration 𝑎𝑎 is:

𝑑𝑑 =12ΔvΔ𝑉𝑉

× (Δ𝑉𝑉)2

Obviously the distance is much greater than constant motion.

Dist

ance

(spa

ce)

Time

Δ𝑥𝑥

Δ𝑉𝑉𝑚𝑚 =

Δ𝑥𝑥Δ𝑉𝑉

𝑎𝑎

Science

Part 3. newton's principia