Introduction

and Rakhi Kaila2



(1)
School of Physics, University of New South Wales, Sydney, NSW, Australia

(2)
School of Medicine, University of New South Wales, Sydney, NSW, Australia

 



Abstract

The human brain has evolved over many millenniums. But over the last three centuries or so the analytical power of mind has accelerated many fold and at a much faster rate. There is a gradual increase in the desire to understand the science of the human mind and of the brain. The brain science in action we need to know has to be experienced at the miniaturized the atomic and molecular level. Newton around three centuries ago was amazed to see that an apple from a tree falls to the ground and not the other way around. Our earth is like a sphere hanging (due to gravitational forces among our star the Sun and other planets revolving round it) in space. All points on earth are equally exposed to the space around it and yet the apple is attracted toward the earth. This is despite apple’s density is higher than the density of air space surrounding it. This thought about the act of science was just a curiosity in Newton’s mind. Science is curiosity and curiosity is science. The science in its mathematical analytical aspect was little developed in Newton’s time. Soon the art of thinking became the art of scientific creation. The science became a tool, of production of technologies those followed. It changed the humanity forever. This was the beginning of the discovery of the forces of gravity which keep the sun, earth, moon, the galaxies, etc. in a dynamic equilibrium. In fact the universe we live in makes the human mind create science all the time.



1.1 The Science of the Macroscopic World Around Us


The human brain has evolved over many millenniums. But over the last three centuries or so the analytical power of mind has accelerated many fold and at a much faster rate. There is a gradual increase in the desire to understand the science of the human mind and of the brain. The brain science in action we need to know has to be experienced at the miniaturized the atomic and molecular level. Newton around three centuries ago was amazed to see that an apple from a tree falls to the ground and not the other way around. Our earth is like a sphere hanging (due to gravitational forces among our star the Sun and other planets revolving round it) in space. All points on earth are equally exposed to the space around it and yet the apple is attracted toward the earth. This is despite apple’s density is higher than the density of air space surrounding it. This thought about the act of science was just a curiosity in Newton’s mind. Science is curiosity and curiosity is science. The science in its mathematical analytical aspect was little developed in Newton’s time. Soon the art of thinking became the art of scientific creation. The science became a tool, of production of technologies those followed. It changed the humanity forever. This was the beginning of the discovery of the forces of gravity which keep the sun, earth, moon, the galaxies, etc. in a dynamic equilibrium. In fact the universe we live in makes the human mind create science all the time.

The mathematical techniques e.g., the calculus, algebra, trigonometry, coordinate geometry, etc. then evolved to quantify the dynamics of planetary motion. It created the technology of aviation industry, space travel and created many other inventions we take for granted. The technologies that have evolved over the last century have created wonderful applications in our, day today life. To mention a few are aviation, satellite communication, microwaves as source of energy in household cooking, mobile phones, TV, etc. Mind imagines out of curiosity sometimes impossible but wonderful things. In today’s environment of economics human mind is restricted to limit its scope of thinking. This is to limit imagination now. It is to fit according to the survival in the materialistic world we live in. The scope of the thinking today is restricted to the art of money making as the main priority. Money drives science and the science is restricted in its horizon to simply making more money. So the scope of science is not unlimited now in the environment we live. The Newtonian’s art of thinking was free and, unlimited. It was on finding the secrets about the universe around us. The science and, the technology that developed then was about the dynamics of large physical bodies like planets, aeroplanes, cars, etc. The society today cannot live without the technologies that have been developed. Think of the indoor and outdoor comforts we enjoy in our day to day life. They are not the results of restricted art of thinking. So far the major developments that have taken place are about the macroscopic world which we can see, and feel around us. But there is another world the microscopic world. This is the world of ensemble of nano-scale atoms and molecules which make the world of living beings e.g., our body.

The science of the dynamic functions of the atoms and molecules happening in a living body keeps a living being going. The nature has worked out the art of the control of life quite well. Science has yet to work out many mysteries. Even god likes to keep many secrets for a scientist to find out. The activities of atoms and molecules on a nano scale in our body cannot be seen by a naked eye. To view their behavior in small volumes we use sophisticated tools, like the electron microscope, atomic force microscopy, magnetic resonance imaging (MRI) etc. They are expensive tools. But they expose the secrets of nature which would not be known otherwise. The wealth they generate e.g., a good health system, industries, etc., more than compensate, the cost of the art of the creativity of the human mind. Human mind has the power of the art of thinking to inherit the science of the universe we live in. The microscopic secrets of the human brain science are not externally visible. They are the invisible, science of the nature. To work them out would be a science of our survival.


1.2 The Internal Science of the Microscopic Brain


The human body is made of the microscopic world of the atoms, molecules, electrons, etc. These tiny particles control the day to day and moment to moment functions of our brain, mind, and the body. The dynamics (Figs. 1.1 and 1.2) of these particles is processed in our brain without us knowing how it works. How does the molecular dynamics coordinates about what is happening in our brain? It also controls the other parts of the body? There comes the art of thinking. In today’s science we call it as art of modeling about the unknown things. In this work an effort is made to expose the art of imaging the activities happening in the human brain. The technology we have at present about the magnetic resonance imaging (MRI) is called as the conventional MRI. It started about half a century ago as a curiosity tool to understand the phenomena of nuclear magnetic resonance (NMR) in solids and liquids. But as the time progressed, it slowly found its way into applications in imaging in plants, animals and the humans. Human brain is made of soft matter including tissues, molecules, electrons, atoms, nuclei, etc. The electrons with their negative charge are orbiting the nuclei due to coulomb attraction from the nucleus with its positive charge. Due to the quantum mechanical art of nature an electron can only stay in a fixed quantized stationary orbit. The outer slightly loosely bound electrons around the nucleus and the coordinated geometry of the atoms and molecules establishes the chemistry of events happening in the brain.

A300799_1_En_1_Fig1_HTML.gif


Fig. 1.1
The Gravitational spinning Top (GST). Suppose that the toy GST is set into circular motion as the kids do by wrapping around a string around the rim of the top, pulling the string, and let the Top go landing on the ground vertically. Initially the Top rotates around the vertical axis. There is an initial angular momentum of magnitude L of the Top along Z-axis and acting through the centre of the Top. It is produced by rotational motion about the vertical Z-axis. Soon the angular momentum L changes to, $$ \text{L}{^{\prime} = \text{L + {dL}}}$$ with its direction slightly away from the vertical axis. This change is produced by the force the Top’s weight mg (m is the mass of the Top and g the acceleration due to gravity) as if there were a torque at a point on the periphery around the X-axis. The Top is slightly tilted from the vertical axis. This makes a change in the magnitude and direction of the angular momentum L initially pointing in the Z-direction, by dL, towards Y-direction. There is thus now an additional rotational component (around X-axis) on the Top, L′ being side-way from the Z-axis. This additional component rotation of the tip of the top is as if to rotate the Top around the X-axis and let it fall ultimately towards the Y-axis. The motion at an instant of time is now more complex. There are now two types of motions. The top is still executing circular motion instantly around its new axis with angular momentum axis, L′. But the Top, as a whole body, also starts rotating around the Z-axis. The tip of L′ is now executing circular motion with larger and larger radius in time around the Z-axis. The Top eventually, falls towards the ground


A300799_1_En_1_Fig2_HTML.gif


Fig. 1.2
The Gyro Magnetic Ratio (GMR) is the ratio of the μJ = the total dipole magnetic moment to the resultant angular momentum, J and is, GMR = μJ/J, J = L+S+… Here μJ = i A, i = electrical current created by the orbital motion of the electron of charge, e, around the nucleus and i = ev; A = area of the orbit = πr2; L = angular momentum = mvr, pointing perpendicular to the plane of the orbit and acting through the centre of the circle; m = mass of the electron; v = tangential velocity; r = radius of the orbit; μJ/J = eћ/2m = (e/2m)/ћ = μb/ћ (the atomic orbital case only); μb = (eћ/2m) = Atomic Bohr Magneton = 0.927 × 10−23 amperes meter2 (Am2) = the fundamental unit of the dipole magnetic moment. The electron orbiting around the nucleus generates an angular momentum L. It is directed in a direction perpendicular to the plane of the obits and acts through the centre of the orbit

Electrons simultaneously can be part of many molecules in a region. There is like a smeared out cloud of the electrons shrouding the much deeper secrets of the nuclei. MRI through the magnetic spins of the electrons, atoms and molecules spread locally over a small region called as a voxel images a small region in time. Scanning the small regions (voxels) one at a time in a sequential order over the whole brain by the use of RF radiation and collating them one can develop a planar image of the whole brain. This is a broad the simplest and the crudest mode of imaging. It is referred to as the chemical shift imaging (CSI). The electrons buried deep inside the atoms, molecules, in tissues are stable and are not electrically active. They however provide the necessary background in controlling the science of the brain over all. Near the surface of the macromolecules, charges form the basic equilibrium structure of a tissue. The charges create a collective reactive protected equilibrium for the brain. The equilibrium among the molecules is over distances varying from nanometers (nm) to millimeters (mm) to centimeters (cm). The common molecular electron charges have a magnetism associated with them. It is loosely called as the spin magnetism.

Inside each molecule the atoms act like small disc shaped magnets of the size of around nm. The rotation of the electrons around it own axis plus around the nuclear axis constitutes the total magnetism of the atom. The atoms add up to produce the resultant magnetic moment for the molecule. It is the overall effect over many tens and hundreds of nm that is the key to the performance of the brain. There is some kind of local magnetic order in the brain in the short range. The human brain is overall electrically and magnetically neutral. But over small regions say hundreds of nm there is an effective local magnetic field at any point created by the local group of nuclei. This magnetic field can be of the order of 1 Tesla (somewhat lesser due to the medium of the brain). It polarizes the region around it into a magnetic order along its direction. Every small (around a micron wide) region all over the brain has an independent magnetic order of its own. But these orders are randomly distributed with an overall effect being zero. There is interaction or say communication between all these small regions to make the moment to moment activities e.g., neurotransmission, metabolism, etc. to continue. The surrounding mobile fluids like water, blood, etc. keep the overall all activities of the brain and the body alive and thus keep us going. The overall simplified picture of the brain has in fact to be quantified to enable understand the functions of the brain. It amounts to understanding the order–disorder processes on nm and ns (nanosecond) scales. This is an impossible situation. There now comes the art of thinking.

One needs to make a machine which can probe the secret dynamics of the brain. The new machine will have to do quantum imaging with minimum disturbance to the brain. The present MRI machine strives to achieve wonderful things. But it still has long way to go. The more the knowledge is disseminated about MRI in a simplified manner to the masses the better is the chance for its further development. This is the aim behind this exposition. MRI allows us to work out the dynamics of the invisible processes happening in the brain by using non-invasive RF (radio frequency) electromagnetic radiation. The MRI technology tries to simulate an order of its own in the brain by external means. It mimics a probe by external means to find out the local magnetic order in the brain created by nature. The external means stimulate the local order and it is measurable in the laboratory with minimum disturbance to the internal order. Since the internal order is on a nm scale, simulating a superficial internal situation in the brain, in the laboratory, can be a formidable task. The MRI machine tries to achieve a close to the real situation in the brain. A static magnetic field larger than 1 Tesla is first applied through the skull. This direction in the MRI literature is referred to as the Z-direction. What this field does is that all the tiny magnets i.e., the spins bound to the molecules, tissues, and the free ions as part of the moving fluids etc. are oriented magnetically along the Z-direction. The RF radiation applied in the perpendicular direction to the static field allows to manipulate the spins in space and time for imaging.


1.3 The Nature’s Invisible Science of Magnetic Order in the Brain


There is some kind of regional as well as overall magnetic order in the brain. It is experienced through a local distribution in the size and the extent of the arrangement of the spins. This order produced is static as well as dynamic. It changes over space as well as time. This is because the atoms, molecules, tissues, fluids, etc. are reactive and interact with each other. The dynamics of the atoms and molecules is governed by the rules of quantum science. In quantum science the statics and the dynamics of the atoms and molecules is a totally different matter. The tiny molecules not only move in space and time but also react chemically with each other. They also communicate electronically with each other and also exchange a few quantum’s of energy in the process. These objects are guided by the rules of quantum mechanics (QM). This is in contrast to the dynamics of the larger bodies like the aeroplanes, space shuttle, etc. The large body’s dynamics is a continuous one and is controlled by the Newtonian Mechanics (NM). The reactions between atoms and molecules take place in a discrete manner rather than in continuous manner. What it means is that when the amount and direction of interaction is just right a quantum or two of energy is exchanged and so is the information for a particular activity. The most fundamental (smallest) quantum of energy is expressed as ђω. The ђ is the familiar angular Planck’s constant $$\grave{O}={\text{h} { 2 {\uppi}}} $$ of electromagnetic radiation and ω is the angular frequency of the RF radiation. RF energy is in a narrow band selected in MRI. It is out of the much wider spectrum of the total electromagnetic (EM) radiation spectrum. This is called as the RF band of the EM radiation. The band is selected because the reactions between atoms and molecules in our body take place in this band of energy.

The MRI machine uses RF radiation for its operation. The RF radiation extends over frequencies in the MHz (~106 Hz) range and in wavelength over the meters range. To probe the brain the RF radiation is applied in a perpendicular direction to the applied static field. This direction is referred to as the X-direction (refer to Fig. 1.1). The RF field is applied in pulses in time in the X-direction so as to project the spins in the X–Y plane. In the time in between the pulses called as the evolution time the spins decay and produce electrical signals in the X–Y plane. The spins are used to collect the naturally happening events in the brain in a controlled manner. The signals collected carry the signatures of the locally happening events. This information is collected by a receiver as a field induction decay. There is an asymmetric distribution of the field induction decay with a particular characteristic time. The characteristic decay time of spins in the X–Y plane is called as the transverse relaxation time T2. This becomes the source of the so called T2, weighted image. Alternately the spins can be projected from +Z to −Z direction by applying the 180° RF pulse along the X-direction. The decay of the electrical signals induced by the spins in this case can be measured as the spins move from −Z towards the +Z direction. The inversion of spins method collects information about the decay time which is called as the longitudinal relaxation time T1 and is present in the X–Z and Y–Z planes.

The asymmetry due to chemical and physical events happening, in the X–Z and Y–Z planes becomes the source of image called as the T1 weighted image. The broader T1 and T2 weighted imaging restricted to mm3 to cm3 voxels with little insight into atomic and molecular level intricacies is the major source of information in the conventional MRI. The modulations of the incident RF radiation received back as echoes over selected voxels in a sequence all over the brain are later coordinated in a computer program and produced as an image. The data collected provides a detailed bank of the scan of the events over space and time. One can analyze data much further rather than just looking it as source of an image to understand the human brain science deep to the roots. The nano science of the brain is a quantum one. One needs to slowly approach the quantum chemistry of the natural events and image them. The emphasis as multi-quantum magnetic resonance imaging (MQMRI) rather than just MRI in this work has two prong purpose. Firstly it helps advance imaging on the real natural scale that is the nano scale. Secondly it induces education about quantum science over a much wider population. So far it has remained within the domain of physics, chemistry and mathematics (PCM) experts. Further it has only been a part of the university level education so far. It did not form secondary school level part of the education system. This is the status all over the world today. It is high time for a change. The UNESCO (United Nations Educational Scientific and Cultural Organization) owes this initiative to the education system on an international scale.


1.4 The Familiar Gravitational Spinning Top and the Not so Familiar Atomic Magnetic Spinning Top



1.4.1 The Gravitational Spinning Top (GST, The Gyro)


It would be very instructive to understand the dynamics of the familiar toy spinning Top which basically represents a Gyro used in satellites in space. It is used for fixing the direction of the axis of the satellite towards the centre of the earth. The basic principle involved in the rotation of the familiar gravitational spinning (GST) is similar to that of an electron orbit around the nucleus i.e., the atomic spinning top (AST). In AST the electron motion around the nucleus is such that the electron stays in stable orbit around the nucleus and does not collapse into the nucleus. On the principles of the quantum mechanics (QM) one would expect that there are n electron orbits around the nucleus and are quantized i.e., the area of the orbits would have quantized values in terms of the area of the closest orbit (A) expressed as $$ {\text{A}}_{\text{n}} = 1{\text{A}},2{\text{A}},3{\text{A}}, \ldots , $$etc. In the atom there is a precise balance between coulomb attraction on the orbiting electron due to the positive charge of the nucleus at the centre and the repulsive centripetal force. The balance is a quantum one that is the radius of the orbit changes in quantum numbers, $$ {\text{n}} = 1,2,3, \ldots ,\;{\text{etc}} . $$ and not in continuous manner. Nature has cleverly produced the quantization of the angular momentum of the electron in orbits around the nucleus. These orbits have stationary energy states. Why has this happened? Is quantization the nature’s science? In fact this is the science of the basis of the quantized nature of atomic orbits and the possible inter-molecular interactions in an ensemble like the brain.

It is not the right place to go into the quantum science of the atoms and molecules in detail. An undergraduate or perhaps a secondary school text book in physics may be very helpful to gain some conceptual insight. The Figs. 1.1 and 1.2 are included as a pictorial illustration about GST and AST respectively. The illustrations draw an analogy between rotational motion of the familiar gravitational spinning Top and the motion of the electron orbiting around the nucleus. The simplified comparison depicted is purely on the educational concept involved and not an exact quantitative estimate of the physical quantities involved.

In summary the initial rotational motion of the Top along the Z-axis is soon taken over by the torque due to the weight milligram (mg) (arising due to frictional forces, m being the mass of the GST) till it falls to the ground. But in-between the Top is still executing a circular motion around its own axis which gets weaker and weaker with time till the whole thing comes to a sudden halt. This is as if one motion (pointing towards the ground) takes over the other (trying to keep rotation as whole around the Z-axis) as time progresses. Imagine there were no frictional force (between the bottom tip of the GST and the ground, air, etc.). Will the Top keep on rotting around the Z-axis? The second slower motion of the Top as a whole around the Z-axis is called as the precession. The change in L is due to mg (the weight of the GST) is in the direction outward, from the Z-axis. The magnitude of L around the Top’s own axis in fact reduces as its initial angular velocity, $$ {{\upomega}} = {{\text{v}} \mathord{\left/ { {{\text{v}} {\text{r}}}} \right. \kern-\nulldelimiterspace} {\text{r}}} $$ (v the tangential velocity around the rim if the GST and r its radius), gradually reduces. The rotational speed of the Top around its own axis gradually slows down due to frictional forces e.g., air, contact at the floor, etc. Momentarily the tip of the Top is in the new position, with changed angular momentum L′. The result is that while the Top is spinning around its own axis with new frequency say ω′ it is as well rotating around as a whole around the Z-axis.

This additional rotation around Z-axis is with a different frequency, Ω. This is the precessional frequency. Gradually the Top increases its tilt from the Z-axis till it falls to the ground. The precession frequency is controlled by the gravity and is given as $$ {{\Upomega}} = \left( {{{\text{mgr}} \mathord{\left/ { {{\text{mgr}} {\text{L}}}} \right. \kern-\nulldelimiterspace} {\text{L}}}} \right) ,\;{\text{g}} = $$ the acceleration due to gravity. The force (weight) mg acts and produces a torque around X-axis. The precession frequency, Ω, is different from the angular frequency of rotation of the Top, ω. The angular acceleration of any point on the rim of the Top is towards the centre of the plane of Top. It is,  $$ {{\upalpha}} = {{{\text{v}}^{2} } \mathord{\left/ {
 {{{\text{v}}^{2} } {{\text{r}} = {\text{r}}{\omega }}^{2} .}} \right.
 \kern-\nulldelimiterspace} {{\text{r}} = {\text{r}{\omega }}^{2} .}}
 $$ Here $$ {\text{v}} = $$ the tangential velocity at a point on the rim of the Top and $$ {{\omega = }} $$ the angular velocity towards the centre of the circle of the top and r is the radius of the Top. One should note that the precessional frequency Ω is a result of the twist produced by the weight mg of the top around the point distant r from the point where the new centre of gravity of the top lies.

The precessional frequency Ω of the tip starts as small value increase gradually and suddenly comes to zero as the Top hits the ground. One can find that $$ {{\Upomega}} = \left( {{{\text{mgr}} \mathord{\left/ { {{\text{mgr}} {\text{L}}}} \right. \kern-\nulldelimiterspace} {\text{L}}}} \right) = {{\text{g}} \mathord{\left/ { {{\text{g}} {\text{v}}}} \right. \kern-\nulldelimiterspace} {\text{v}}} $$ initially. The change in angular momentum is the product of the linear momentum mv at a point on the rim of the Top multiplied with twist say R around the X-axis and is $$ {\text{dL = mvR}} $$ (note R increases with time). The g has constant enough value on the surface of the earth therefore the only variable left is v. But $$ {\text{v = R}}{\omega} $$ (the twist circle around X-axis). The change in initial angular velocity ω with time controls the precessional frequency Ω. The ω gets slower and slower with time and Ω gets faster and faster with time till it suddenly comes to zero as the Top hits the ground. The two processes compensate each other controlled by the frictional forces. Now think of the case of an electron orbiting around the nucleus. Why does it go on executing that motion forever? What are the forces responsible for it? Are there any frictional forces involved in the AST?


1.4.2 The Atomic Precessing Top: Case of Single Electron Orbiting Around Nucleus in an Atom


One should note that although the AST and the GST conceptually behave the same way the forces responsible for the precessional motion are different in the two cases. In the atomic case it is the quantum energy structure of orbital motion of an electron around the nucleus which is under consideration. In an ensemble of atoms and molecules like in our brain it is the angular momentum of a bunch of spins (tiny molecular magnets) say in a voxel, that spin around the applied static magnetic field, Hz, applied in the Z-direction, that is important. There is an associated magnetic dipole magnetic moment for the orbital electron pointing in the opposite direction to the angular momentum. The result is due to the negative charge of the electron. Similar kind of things are happening inside the nucleus due to protons and neutrons. The frequency of precession of an electron or a proton is given as


$$ {{\Upomega}} = {\text{g}}_{ 1}({\upmu}_{\rm b}/\text{h})\text{H}_{\rm z}$$

Here, μb, is the magnetic moment of the disc shaped magnet created by the orbital motion of the electron, around the nucleus, h, the Planck’s constant, and gl, the spectroscopic splitting factor. This factor allows to predict (calculate) various modulations in Ω which arise in an ensemble like the brain due to the interactions between the electron angular momentum (intrinsic S around its own axis), atomic orbital (L), the nuclear (I) and the intermolecular, etc. The value of gl is 1 for the orbital case that is if we only have spin due to the orbital motion of the electron around the nucleus and nothing else. It becomes 2 for the pure (intrinsic) electron precession, spin S, around its own axis and is represented by symbol gs. Thus in the electron intrinsic (around its own axis) case only one can write, $$ {\text{g}}_{ 1} = {\text{g}}_{\text{s}} = 2. $$ Spin S (electron) and orbital (atomic) angular momentum in fact interact and produce a resultant J. Each orbital energy spectral line thus will be split into two due to gs. Please see Fig. 2.3 for details about spin–orbit (SO) interactions. The precessional frequency of an atom or a nucleus is the source of resonance for the applied RF radiation for imaging. One should note it is the applied static magnetic field which produces planar quantization that leads to a concrete measurable result. Without this field the spins are randomly distributed and cancel each other.

In order to see a good example as to how the angular momentum operates one should consider the case of a helicopter. Normally the motion of the helicopter is vertically upward while its blades rotate in a plane perpendicular to the vertical axis, i.e., in the plane of the ground. When the helicopter wants to move sideways it simply twists the axis of rotation sideways. The angular momentum then points in the side direction of the helicopter. The helicopter thus flies side way. The laws of circular motion are analogous to that of the laws of motion of a body moving in a straight line. The linear motion generates a linear momentum, $$ {\text{p = mv}}, $$ in the direction of motion. Here, m, is the particle’s mass and v is its liner velocity. In a linear motion the force acts in the direction of the motion. In a circular motion of the electron there is a force that acts in turning the electron at every moment. This force is at right angle to the momentary linear motion at a point. The applied force acts in changing the direction of motion and not the distance from the centre (i.e., the radius R). The distance is fixed and is equal to the radius of the circle R traced. The angular momentum at a point in the circle can be worked out from the momentary linear momentum at a point. The circular motion is a result of a kind of angular force which turns the object at constant angular rate $$ \omega \,=\, {{\text{v}} \mathord{\left/ { {{\text{v}} {\text{R}}}} \right. \kern-\nulldelimiterspace} {\text{R}}}. $$ Here ω is the turning velocity at any instant of time t, and at any point, on the circle, and is constant. The rotational force is the result of a torque $$ {\tau\,=\,FR}$$ at a point as if trying to produce a couple or a twist at the point around the centre of the circle.

The angular momentum can be written as $$ {\text{L}} = {\text{pR}} = {\text{mvR = mR}}^{ 2} {\text{w = I}{\omega }} . $$ It is analogous to the linear momentum equation, $$ {\text{p = mv}} . $$ In rotational motion, $$ {\text{I = mR}}^{ 2} , $$ takes the place of mass m. I is called as the moment of inertia of the object around the centre of rotation. The direction of the angular momentum L is perpendicular to the plane of the orbit. The angular momentum is taken as acting through the centre of the orbit. Thus L plays the same role in rotational motion as does the linear momentum p in a linear motion. In rotation the speed of rotation, ω, is the rate of change of angle θ with time i.e., $$ {{\omega}} =  {{{\text{d}}} {\theta } \mathord {\left/ { {{{\text{d}}}{\theta} {\text{dt}}}} \right. \kern-\nulldelimiterspace} {\text{dt}}} . $$ In linear motion the distance l changes with time t and we deal with linear velocity $$ {\text{v = }}{{\text{dl}} \mathord{\left/ { {{\text{dl}} {\text{dt}}}} \right. \kern-\nulldelimiterspace} {\text{dt}}} . $$ The linear force is given by $$ {\text{F = ma,a}} $$ being the linear acceleration; $$ {\text{a = }}{{\text{dv}} \mathord{\left/ { {{\text{dv}} {\text{dt}}}} \right. \kern-\nulldelimiterspace} {\text{dt}}} . $$ In a rotating electron we deal with a torque τ. It changes the angle θ with time. It is equal to the rate of change of angular momentum with time and is written as $$ {{\tau  }} = {{\text{dI}} \mathord {\left/ { {{\text{dI}}
{\text{dt}}}} \right. \kern-\nulldelimiterspace}
{\text{dt}}}{=}{{{\text{d}}\left( {{\text{I}}} {\omega} \right)}
\mathord{\left/ { {{{\text{d}}\left( {{\text{I}}} {\omega}
\right)} {\text{dt}}}} \right. \kern-\nulldelimiterspace}
{\text{dt}}}{\text{ = I}}\left( {{{{\text{d}}}{\omega}
\mathord{\left/ { {{{\text{d}}} {\omega} {\text{dt}}}}
\right. \kern-\nulldelimiterspace} {\text{dt}}}} \right){\text{ =
I}}{\alpha, } $$ assuming that I is constant in time. Here, α, is now the angular acceleration rather than the liner acceleration for a straight line motion. Thus in rotational motion the liner force equation, $$ {\text{F = ma,}} $$ is replaced by an analogous torque equation $$ \tau = {\text I }{\alpha} . $$

Only gold members can continue reading. Log In or Register to continue

Stay updated, free articles. Join our Telegram channel

Aug 21, 2016 | Posted by in GENERAL RADIOLOGY | Comments Off on Introduction

Full access? Get Clinical Tree

Get Clinical Tree app for offline access