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Where Eastern thought meets Western technology.

*Reference: Course on Subject Clearing*

The starting postulate of the Matrix Model is:

**THE UNIVERSE IS ONE.**

Actually, the universe, by its very definition, is ONE. The dictionary defines the origin of the universe as follows:

1325–75; Middle English < Old French *univers *< Latin *ūniversum,* noun use of neuter of *ūniversus* entire, all, literally, turned into one, equivalent to *ūni-* one + *versus* (past participle of *vertere* to turn)

From this starting postulate arise other postulates. For example, this universe is knowable because it is manifested. That which is not manifested is unknowable. All that is knowable is connected with each other. Wherever we notice discontinuity, inconsistency or disharmony, there is an anomaly. When there is an anomaly, something is missing from our knowledge and understanding; and so on.

Thus, from the starting postulate, there comes about a system of postulates. These postulates may be arranged in a matrix. Each postulate may be seen as an element of this matrix.

This procedure repeats itself with each postulate in this matrix. In other words, each postulate in this matrix may generate its own system of postulates, which may, in turn, be arranged in a sub-matrix. This recursive process may continue this way to an infinite number of levels of sub-matrices, until this system of postulates is able to describe the whole universe in a consistent fashion.

Does thus recursion go in the other direction as well, meaning, is the starting postulate itself is an element of a larger matrix? This is quite possible. It shall lead us into a path of tremendous discoveries.

**For now, we may look at the universe as consisting of an infinity of elements, and all these elements relate with each other to make a consistent whole.**

This is the matrix model in a nutshell.

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The Being or “I” is a system of postulates that may be conceived as a matrix. This system of postulates may contain some degree of inconsistency. The aim of “I” is to make itself “complete.” In other words, the aim of “I” is to remove all inconsistencies from its system.

That is what my aim as a being or “I” is. I am a system of postulates that may be conceived as a matrix. My aim is to spot and resolve all inconsistencies in myself. At that point I shall be able to merge smoothly into the universal system of postulates, or matrix.

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**HOLES**

- Factor both the numerator N(x) and the denominator D(x).
- Cancel any common factors and simplify the function.
- Equate the canceled factor to zero. This will give you the x-value of the hole.
- Plug this x-value in the simplified function to find the y-value of the hole.
- Plot that hole (or holes) on the graph.

**VERTICAL ASYMPTOTES**

- Plot the remaining zeros of D(x) on the x-axis. Draw vertical dotted lines through them. These are the locations for the vertical asymptotes.
- Find the sign of f(x), just before and after the dotted line. This you can do by finding the signs of the factors in the simplified function and resolving them. This will tell you if the graph is going asymptotic upwards or downwards near the dotted vertical line.
- Find remaining zeros of N(x) on the x-axis. These are points where the graph crosses the x-axis.

**HORIZONTAL** **ASYMPTOTES**

- Horizontal asymptotes occurs at either end of the graph as x goes to plus or minus infinity.
- For
*n < m*, the horizontal asymptote is*y = 0*(the x-axis). - For
*n = m*, the horizontal asymptote is*y = a*_{n}/ b_{m} - For
*n = m+1*, the asymptote is a slanted line,*y = kx*, found by dividing N(x) by D(x). - For
*n > m+1*, there are no asymptotes;

when*n – m*is even, both ends of the graph rise up

when*n – m*is odd, the left end goes down while the right end rises up.

From the above data you can sketch a rough approximation of the shape of the graph.

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- The constant term of the polynomial shows the y-intercept of the graph.
- Look at the highest degree of the polynomial. The corresponding graph will have one less bend. For example, if the degree of the polynomial is 3, the corresponding graph shall have two bends.
- If all the zeros of the polynomial are real, then the corresponding graph will cross the x-axis as many times as the degree of the polynomial. For example, if the degree of the polynomial is 3, the corresponding graph shall cross the x-axis 3 times.
- If the zeros of the polynomial are visible because the polynomial is factored; then, match the zeros to the values where the graph crosses the x-axis.
- When two of the zeros are the same; then, the corresponding bend will simply touch the x-axis at that value.
- When two of the zeros are imaginary; then, the corresponding bend will not cross or even touch the x-axis.
- When all the zeros are real, the constant term of the polynomial shall be the product of the zeros.
- When there are two graphs matching the polynomial, take the root that is not common in both the graphs, and plug it in the polynomial. You will know if that zero belongs to the polynomial or not.

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*Reference:** Beginning Physics I*

**CHAPTER 10****: RIGID BODIES II: ROTATIONAL MOTION **

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**Stress, Strain, Elastic, Elastic Limit, Hooke’s Law, Young’s Modulus, Ultimate Strength, Force Constant, Shear Deformation, Twisting Deformation, Pressure, Bulk Modulus, Compressibility**

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For details on the following concepts, please consult **CHAPTER 11****.**

**STRESS**

Force needed for certain stretch is proportional to the cross-sectional area of the rod. If we define *stress* as the ratio of the force to the cross-sectional area, we have a quantity that measures the effectiveness of the force in accomplishing a given stretch, independent of the cross-sectional area of the rod. The dimensions of the stress are force per area (pascal = 1 N/m^{2}). A given stress will give rise to a definite strain in a rod of a certain material irrespective of either the thickness or the length of the rod.

**STRAIN**

A given force will cause a stretch that is proportional to the length of the unstretched rod. We define *strain* as the ratio of the change in the length of the rod to the unstretched length of the rod. The strain due to a given force will be the same for any length of rod of the same material and cross-section. The strain is thus a measure of the stretch of the rod that is independent of the length of the rod. The strain is dimensionless.

**ELASTIC**

Any material that returns to its original shape after the distorting forces are removed is said to be *elastic*.

**ELASTIC LIMIT**

For a rod of any given material there is a stress beyond which the material will no longer return to its original length. This boundary stress is called the *elastic limit*.

**HOOKE’S LAW**

For stresses below the elastic limit it is found that, to a good approximation, the strain is proportional to the stress; for example, if we double the stress, the strain would double. This is called the *Hooke’s Law*.

**YOUNG’S MODULUS**

In the elastic region *stress/strain = constant*. The constant is called the *Young’s modulus (Y)*. Its value depends on the material. Young’s modulus has dimensions of stress, and can be measured in pascals.

If a force tends to compress a rod rather than stretch it, the relationship of stress to strain still holds with the same Young’s modulus. In that case, the change in length represents a compression rather than stretch.

**ULTIMATE STRENGTH**

If one applies stress to a rod beyond the elastic limit, the rod will retain some permanent strain when the stress is removed. If the stress gets too great, the rod will break. The stress necessary to just reach the breaking point is called the *ultimate strength* of the material.

**FORCE CONSTANT**

For a rod of definite cross section (A) and length (L), the applied force (F) is proportional to the elongation ( L), and can therefore be expressed as *F = kx, *where k is the *force constant* of the system.

**SHEAR DEFORMATION**

**TWISTING DEFORMATION**

**PRESSURE**

**BULK MODULUS**

**COMPRESSIBILITY**

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*Reference:** Beginning Physics I*

**CHAPTER 10****: RIGID BODIES II: ROTATIONAL MOTION **

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**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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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: *a _{t }= 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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