Two Mathematical Innovations – one Greek, one Indian Pi & Zero

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From the first year of secondary school to postgraduate studies at universities around the world, there is nowhere to hide from Π. It is a nightmare for maths-hating teenagers and a never-ending mystery for professors who teach it in the daytime and dream about it at night. Since Archimedes tried to ‘square the circle’ in third century B.C.E. Sicily, at that time a part of the Greek Empire, mathematicians have wondered about this number. Nowadays, supercomputers can calculate Π to 1.4 trillion places, although we only need the first thirty-two digits to work out the size of the universe. So, why does Π continue to excite scientists and bore school kids more than two thousand years after Archimedes first played with the idea of this strange number?

The Mighty Atom

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The physicist Richard Feynman once said that if you had to reduce scientific history to one statement it would be “All things are made of atoms.”

Relativity and the Expanding Universe

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As the 19th century ended, scientists could be happy they’d solved most mysteries of the physical world: electricity, magnetism, gases, acoustics and statistical mechanics, all had fallen before them. They had discovered the X-ray, the electron, and radioactivity, invented the ohm, the watt, the Kelvin, the joule, and the amp. Many people believed there was nothing left for science to do.

Chapter 1.1: Rectilinear motion & Rotational motion (Mechanics)

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Rectilinear motion: Linear motion (also called rectilinear motion) is a motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration and non-uniform linear motion with variable velocity or non-zero acceleration. This type of motion describes the movement of a particle or a body. A body is said to experience rectilinear motion if any two particles of the body travel the same distance along two parallel straight lines. The figures below illustrate rectilinear motion for a particle and body. Fig. 1 shows (a) Rectilinear motion of a particle; (b) Rectilinear motion of a body.

Chapter 1.2 Laws of motion & Frames of reference (Mechanics)

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- It states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This is normally taken as the definition of inertia. The key point here is that if there is no net force acting on an object (if all the external forces cancel each other out) then the object will maintain a constant velocity.

Chapter 1.3 Introduction of torque, Moment of inertia & Application (Mechanics)

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Torque: In physics, a moment is a turning effect of a force. In principle, any physical quantity can be multiplied by distance to produce a moment; commonly used quantities include forces, masses, and electric charge distributions. For examples, the moment of force, or torque (τ), is a 1st moment: τ=d.F, or, more generally, τ=r×F, the electric dipole moment is also a 1st moment: p=qd, the moment of inertia is a 2nd moment: I=r^2 m for a point mass.

Chapter 1.4 Radius of gyration & Polar moment of inertia (Machines)

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Radius of gyration or gyradius refers to distribution of the components of an object around an axis. In terms of mass moment of inertia, it is the perpendicular distance from the axis of rotation to a point mass (of mass, M) that gives an equivalent inertia to the original object(s) (of mass, M). The nature of the object does not affect the concept, which applies equally to a surface, a bulk mass, or an ensemble of points. Therefore, radius of gyration is the distance from an axis at which the mass of a body may be assumed to be concentrated and at which the moment of inertia will be equal to the moment of inertia of the actual mass about the axis.

Chapter 1.5 Work, Energy, Power & Impulse-Momentum Change Theorem (Mechanics)

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Work: In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. The work W done by a constant force of magnitude F on a point that moves a displacement s in a straight line in the direction of the force is the product W = Fs.

Chapter 2.1 Different states of matter (Properties of Matter)

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Solid, liquid, gaseous and plasma are the four states for a given substance that can exist (Fig. 1). The chemical composition of the substance is the same in solid, liquid, and gaseous states. Properties of matter that can be changed with changing the state of a substance: density, elasticity, surface tension, viscosity, specific heat, conductivity, semi-conductivity, Ferro-magnetism, Para magnetism, reflecting properties of light etc.