Author Archives: vinaire

I am originally from India. I am settled in United States since 1969. I love mathematics, philosophy and clarity in thinking.

Physics I: Chapter 10

Reference: Beginning Physics I

CHAPTER 10: RIGID BODIES II: ROTATIONAL MOTION 

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KEY WORD LIST

Rotational Motion, Angular Displacement, Angular Velocity, Angular Acceleration, Period, Frequency, Torque, Moment of Inertia, Linear and Angular Relationships, Table of Analogs, Conservation of Angular Momentum, CM Frame

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GLOSSARY

For details on the following concepts, please consult CHAPTER 10.

ROTATIONAL MOTION
In the following sketch, a body in x-y plane is rotating around the z-axis. The orientation of the rigid body can be completely specified by giving the orientation angle  of a single chosen line segment etched in the body.

ANGULAR DISPLACEMENT
The angle  is called the angular displacement of the rigid body. By convention, the angle  is considered positive when it is measured counterclockwise from the x-axis.

ANGULAR VELOCITY
To get an idea of how fast the body is rotating, we define the average angular velocity in a given time interval as follows:

The instantaneous angular velocity is defined as the limit of average angular velocity as follows:

The angular velocity is positive for counterclockwise rotation. For constant angular velocity, we have

ANGULAR ACCELERATION
The average angular acceleration is the rate of change of the angular velocity.

The instantaneous angular acceleration is,

Thus, we have for constant acceleration,

PERIOD
The time to make one complete revolution is called the period of the motion. For constant angular velocity, the period stays the same from one revolution to the next.

FREQUENCY
The frequency is the number of revolutions per second.

TORQUE
We consider the axis of rotation fixed in the z-direction. Then the torque is along the z-axis, and the forces causing this torques and their displacements lie in the x-y plane. All the internal torques in a rigid body add up to zero. Thus, the only torque left is due to external forces,

MOMENT OF INERTIA
We define the moment of inertia of a body about the z-axis as,

LINEAR AND ANGULAR RELATIONSHIPS
At any instant, the angular and linear properties are related as follows:

DISPLACEMENT:                   s = R             and              s = R

VELOCITY:                              v = R            and              v = R

ACCELERATION:                    at = R          and              ar = 2 R

TABLE OF ANALOGS
Work done in rotation a rigid body, Kinetic energy in rotation, Work-energy theorem applied to a rotating object, the power of rotation, angular impulse, and angular momentum are all rotation analogs of the definitions for linear motion.

CONSERVATION OF ANGULAR MOMENTUM
If the resultant external vector torque (about the origin) for a system of particles is zero, then the vector sum of the angular momenta of all the particles stays constant in time.

For the special case of objects rotating about a fixed axis: If the total external torque about the axis is zero, then the total component of angular momentum along that axis does not change.

CM FRAME
The CM Frame is a coordinate system whose origin is fixed at the CM (Center of Mass) of the object. The CM Frame moves with the object, but its axes remain parallel to the axes of a coordinate system fixed in an inertial frame.

The translation of the object is the same as the translation of the CM. The rotation of the object is about an axis that passes through the CM. If the direction of this axis of rotation remains fixed, then all the laws of rotation hold.

The total kinetic Energy of an object in the inertial frame is given by,

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Emptiness and the Matrix Model

Reference: Course on Subject Clearing

The “emptiness” of Buddha points to the nature of the universe. The same concept of “emptiness” points to the nature of anything that makes up this universe. This is because things are recursive. In other words, things are made up of things, which are also made of things, ad infinitum. Therefore, when you are describing the nature of a thing, you are describing the nature of all things, whether simple or complex.

So, emptiness is a concept that applies to the nature of a thing, which means, it applies to the nature of everything through recursion.

When you look at the nature of a thing, you find that there is a configuration of relationships. There is no permanence among those relationships. There is only a continual flux. You cannot put your finger on something and say that it is going to remain that forever. There is nothing that can be said to be fixed permanently.

We assign a label, for example, “Universe,” to such a configuration of relationships. This label may seem to provide a permanence of some sort. For example, we find the “Universe” to be existing forever. But the same label may mean a different thing to different people. It may even mean a different thing to the same person at a different moment. This is because the relationships represented by the label are in a continual flux.

When Buddha said that a thing is empty, he simply meant that it has neither substance nor sense that could be said to last forever.

Any sense of permanence is an illusion. There is nothing fixed. Any consideration of fixation is an aberration. This is explained in the THE DOCTRINE OF NO-SOUL: ANATTA

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Absolute and Relative

Buddha declared.

“The Absolute Truth is that there is nothing absolute in the world, that everything is relative, conditioned and impermanent, and that there is no unchanging, everlasting, absolute substance like Self, Soul, or Ātman within or without.”

DEFINITION: Absolute means, “Viewed independently; not comparative or relative; ultimate; intrinsic.”

This postulate may appear self-contradictory to some, but it essentially says, “There are no absolute certainties; all certainties are relative.” This statement does not degrade any certainty we have. It simply means that one can always come up with a better certainty.

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Theories

We devise theories to fully understand what is there. When we propose a theory, there is always a starting postulate. For example, Einstein’s theory of relativity starts with the postulate that the speed of light is a universal constant. 

This first postulate may appear to be “absolute” because it supports a whole system of postulates. Things make sense to us only because of the consistency of this system of postulates. If the first postulate falls apart our whole sense of reality is shaken. 

The first postulate of religions is the postulate of “God.” The division between Eastern and Western religions exist because the postulate of “God” has different meaning in the East and in the West. An atheist also has a first postulate of “God,” except that his God is not a “Super Being” but some ultimate principle of nature.

Different religions and realities come about because of the differences in the first postulate of “God”; and in the system of postulates arising from it.

The first postulate, and the system of postulates it supports, determines our reality. But even these postulates are empty in the sense that none of them can be said to be permanent. They can be questioned. They can be changed. They can be improved upon.

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The Matrix Model

The Matrix Model starts with the postulate of a MATRIX composed of PERCEPTUAL ELEMENTS, where each perceptual element, in turn, is composed of a matrix of perceptual elements recursively. It represents an infinite number of levels, with each level consisting of an infinite number of dimensions, and each dimension representing an infinity of values.

In a way, EMPTINESS is built into this model because the matrix and perceptual elements represent relationships; and these relationships are changing in infinity of ways like in the universe.

The method of Subject Clearing is based on this matrix model, for the clarity you can obtain from the application of this method has no limits.

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Starting Postulate

Reference: Course on Subject Clearing

Any theory must have staring postulates. It is the consistency among staring postulates that keep the theory consist. Furthermore, the lesser is the number of starting postulates, the more consistent the theory would be. That is the concept underlying the principle known as Occam’s Razor.

OCCAM’S RAZOR
The principle (attributed to William of Occam) that in explaining a thing no more assumptions should be made than are necessary. The principle is often invoked when solving problems.

Recently, I was having a tough time in trying to understand the Description Theory as described at: Description Theory Playlist. So, I applied the approach of SUBJECT CLEARING. I started to locate the KEY WORDS and how they were defined in this theory. The first word was description, which was defined in terms of the contrast between a foreground and a background. The foreground was described relative to the background; and the background was defined as the environment which was not the foreground. There were other starting postulates but something was not making sense, until I realized that the background was left as something arbitrary.

Then I realized that the pure description of something would be to contrast it from nothing. So, the ultimate background is nothing or emptiness. Since we understand the universe of things through our postulates, then emptiness would be defined as,

EMPTINESS
Emptiness is absence of all postulates.

I realized that an empty background has to be the first postulate if one were to describe anything in its purity. But the originator of the Description Theory looked at the description in relative terms only. His theory had no concept of pure description. This made the Description Theory very complicated. Once I realized this, I could move through the study of this theory very fast, but I kept on running into the arbitrary associated with the concept of background.

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NATURE OF STARTING POSTULATE

The relative nature of descriptions in the Description Theory, make the theory fundamentally inconsistent. The subject of Physics has the same problem. It is fundamentally inconsistent.

In physics, we have relative motion only; and no concept of “absence of motion.” Newton approximated the “absence of motion” by postulating the background of stars to be fixed in space. This worked perfectly for motion in most cases, but it could not predict the orbit of mercury with precision. This anomaly could not be resolved until, couple of centuries later, Einstein came along.

Einstein examined the problem of calculating mercury’s orbit using the equations derived from Newton’s theory. There was the phenomenon of the aberration of light that showed that celestial measurements were being affected by the observer moving with the Earth relative to Mercury. The measurement of the speed of light was found to be unaffected by Earth’s motion. When Einstein used the speed of light to account for the observer’s motion he could predict the motion of mercury’s orbit correctly. Einstein then proposed his revolutionary Theory of Special Relativity using the starting postulate of a universally constant speed of light.

Einstein’s Theory of Special Relativity has been highly successful, but it has given rise to anomalies in the very concepts of space and time. Moving the “Frame of Reference” from Earth (or matter) to Light has certainly improved the accuracy of celestial measurements. This means that the “reference frame of light” is more accurate than the “reference frame of matter.”

The anomaly of space and time reduces to the anomaly of how we measure motion. The very concept of motion depends on the concept of an infinitesimal “particle” in which “substance” is concentrated as mass. From matter to light, the concentration of “substance” in the “particle” reduces greatly. Matter has mass, but light has no mass. However, light still has some momentum, so light still has some substance. 

Space and time exist because substance exists—space is the extent of substance; time is the duration of substance. Space and time shrink or expand depending on the concentration of substance.

When the “reference frame” frees itself from its dependency on “substance” we can expect to have the purest “reference frame” to measure substance, space and time.

And this brings us back to the concept of emptiness.

The starting postulate for any universally applicable theory has to be the postulate of “emptiness.” 

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Physics I: Chapter 9

Reference: Beginning Physics I

 CHAPTER 9: RIGID BODIES I: EQUILIBRIUM & CENTER OF GRAVITY

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KEY WORD LIST

Equilibrium, Torque, Moment, Laws of Equilibrium (Rigid Bodies), Equivalent Sets of Coplanar Forces, Center of Gravity, Couple

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GLOSSARY

For details on the following concepts, please consult CHAPTER 9.

EQUILIBRIUM
For introduction to translational and rotational equilibrium. See Chapter 4.

For a situation in which every slight tilt from equilibrium gives rise to a couple that restores equilibrium, we say that the equilibrium is stable.

Whenever a slight tilt of an object away from equilibrium gives rise to a couple that continues the motion away from equilibrium, we say the equilibrium is unstable.

TORQUE
The words torque and moment are synonymous. They are used interchangeably in the present context.

MOMENT
The moment of a force is a tendency to produce motion about an axis. It is the product of a force and its directed distance from an axis. The axis is selected as a point in the plane of the force, and rigidly linked to the body. A moment arm is defined as the perpendicular distance from this point to the line of action of the force. The moment of the force is defined as the product of the magnitude of the force and length of the moment arm.

The plus sign indicates that the force would tend to rotate the body counterclockwise about the point chosen as the axis. The negative sign indicates clockwise rotation.

LAWS OF EQUILIBRIUM (RIGID BODIES)
The following are the necessary and sufficient conditions for translational and rotational equilibrium for a rigid body acted on by any number of coplanar forces:

  1. The vector sum of the forces must vanish.
  2. The algebraic sum of the torques about a given point must vanish.

Mathematically,

EQUIVALENT SETS OF COPLANAR FORCES
One can always replace one set of forces acting on a rigid body by any other set of forces having the same vector sum and the same resultant torque (about any chosen point) to get the same effect on the motion of the body.

CENTER OF GRAVITY
The center of gravity is the point in the body where the total weight can be assumed to be acting. The center of gravity and the center of mass are one and the same point.

COUPLE
A couple is a pair of equal and opposite forces giving rise to a torque.

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Physics I: Chapter 8

Reference: Beginning Physics I

CHAPTER 8: IMPULSE AND LINEAR MOMENTUM

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KEY WORD LIST

Impulse, Momentum, Conservation of Momentum, Elastic Collision, Inelastic Collision, Coefficient of Restitution, Ballistic Pendulum, Center of Mass

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GLOSSARY

For details on the following concepts, please consult CHAPTER 8.

IMPULSE
Impulse is the product of the average force acting upon a body and the time during which it acts, equivalent to the change in the momentum of the body produced by such a force.

I = Ft

I may be resolved into x and y components. It is equal to change in momentum,

MOMENTUM
If an object of mass m is moving at a given instant of time with velocity v then,

The concept of linear momentum can be generalized to two or three dimensions.

CONSERVATION OF MOMENTUM
When there are no external forces acting on a system (or when the resultant of external forces acting on a system is zero), the total momentum of the system is conserved. In other words, total final momentum = total initial momentum.

ELASTIC COLLISION
An elastic collision is one in which the total kinetic energy of the colliding objects is the same just before and just after the collision.

INELASTIC COLLISION
An inelastic collision is characterized by a certain disappearance of kinetic energy in the collision process.

COEFFICIENT OF RESTITUTION
The coefficient of restitution (e) is defined as the ratio of the magnitude of the relative velocity after the collision to that before the collision.

For an elastic collision, e = 1. Generally speaking, the smaller the e value, the more thermal energy is generated and hence the more kinetic energy is lost.

BALLISTIC PENDULUM
A ballistic pendulum is a device that is used to measure the velocities of small swift projectiles such as bullets. See the sketch above.

CENTER OF MASS
The center of mass is the point at which the entire mass of a body may be considered concentrated for some purposes. It is defined as the position of the average displacement of the particles of the body, weighted according to mass.

For a rigid body, the center of mass of the body moves as if it was a particle having a mass equal to the total mass of the body acted on by the resultant force on the body. The center of mass is a geometric point fixed in relation to a rigid body, but it is not necessarily in the body. 

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