Fundamental Physics Notes

Joseph E. Johnson, PhD

Professor of Physics

University of South Carolina

July 24, 2007 Version

© Joseph E. Johnson 2006 All rights Reserved

Fundamental Physics Table of Contents

Joseph E. Johnson, PhD © 2006

  1. Mechanics
  2. Newtonian Mechanics
  3. Introduction <CJ - 1 & SV 1
  4. Kinematics in One Dimension <CJ - 2 & SV 2
  5. Kinematics in Two & Three Dimensions <CJ - 3 & SV3
  6. Forces & Newtons Laws of Motion <CJ - 4 & SV 4
  7. Uniform Circular Motion <CJ - 5 & SV 7
  8. Work & Energy <CJ - 6 & SV 5
  9. Momentum and Impulse <CJ - 7 & SV 6
  10. Rotational Mechanics & Gravity
  11. Rotational Kinematics <CJ – 8 & SV 7
  12. Rotational Dynamics <CJ - 9 & SV 8
  13. Gravitation <CJ - 4.7, 9.3 & SV7
  14. Solids, Fluids, & Waves
  15. Elasticity <CJ - 10.1, 10.7, 10.8 & SV9
  16. Simple Harmonic Motion <CJ - 10 & SV 13
  17. Fluids <CJ - 11 & SV 9
  18. Mechanical Waves & Sound <CJ - 16 & SV 14
  19. Linear Superposition of Waves, Interference, & Music <CJ - 17 & SV 13, 14
  20. Thermodynamics
  21. Temperature & Heat <CJ - 12 & SV 10
  22. Transfer of Heat <CJ - 13 & SV 11
  23. Ideal Gas Law & Kinetic Theory <CJ - 14 & SV 10
  24. Thermodynamics <CJ - 15 & SV 12
  25. Electromagnetic Theory
  26. Electricity
  27. Electric Forces <CJ - 18.1-18.5 & SV 15
  28. Electric Field <CJ - 18.6-18.8 & SV 15
  29. Gauss’ Law <CJ - 18.9 & SV 15
  30. Electric Potential & Potential Energy <CJ - 19.1-19.4 & SV16
  31. Capacitance <CJ - 19..5-19.7 & SV16>
  32. Electric Current & Resistance <CJ - 20.1-20.7 & SV 17
  33. Direct Electrical Currents <CJ - 20.8-20.15 & SV 18
  34. Magnetism
  35. Magnetic Fields <CJ - 21.1-21.6 & SV19
  36. Magnetic Field Sources <CJ - 21.7-21.10 & SV 19
  37. Faraday’s Law <CJ - 22.1-22.6 & SV20
  38. Induction <CJ - 22.7-22.10 & SV20
  39. Alternating Electric Currents <CJ - 23 & SV 21
  40. Electromagnetism
  41. Maxwell’s Equations <CJ - 24.1-24.3 & SV21
  42. Solution in a Vacuum – EM Waves <CJ - 24.4-24.7 & SV 21
  43. Light & Optics
  44. Reflection of Light & Mirrors <CJ - 25 & SV22, 23
  45. Refraction of Light & Lenses <CJ - 26 & SV 23, 25
  46. Interference & Wave Nature of Light <CJ - 27 & SV 24
  47. Relativity
  48. Special Relativity <CJ - 28 & SV 26
  49. General Relativity & Astrophysics <CJ - 28.8 & SV 26
  50. Quantum Theory – Atomic, Nuclear, & Particle Physics
  51. Foundations of Quantum Mechanics – Particles & Waves<CJ - 29 & SV 27
  52. Atomic Theory <CJ - 30 & SV28
  53. Nuclear Theory & Radioactivity <CJ - 31 & SV 29
  54. Elementary Particle Theory <CJ - 32 & SV 30
  55. Mathematics Background <CJ - 1 & Appendix >
  56. Some Useful Numerical Value and Relationships
  57. How to best process this material as a Physics Course

CJ = Cutnell & Johnson: Physics 7th Edition 2007 ISBN 0-471-66315-8

SV = Serway & Vuille: Essentials of College Physics 2007 ISBN 0-495-11129-5

Preface

These notes have been compiled in order to summarize the core concepts, definitions, terms, equations, and relationships for an introductory Physics course. My objective is to provide the student with an outline of the very essentials which are to serve as a guide to my lectures and any of the very well written texts that are available and to keep the focus on the core ideas as it is easy for a student to become overwhelmed or lost in the more than one thousand page texts and the massive information that is conveyed in the lectures. These notes are the skeletal framework upon which one can attach the rest of the material.

I have separated each chapter or topic into a separate page thus allowing one to print these pages from the web for personal use with space for taking ones own notes during lecture and later with the text in hand. Each chapter or topic is further divided into three areas: (1) Descriptive, (2) Mathematical, and (3) Advanced. The ‘Descriptive’ part covers the non-mathematical parts that might be covered in a course such as ‘Physical Science’ or ‘Physics in the Arts’ that is generally devoid of algebra and trigonometry. The ‘Mathematical’ part covers introductory physics at the level of algebra and trig but without calculus. Such a course is customary for the health and biological sciences. Such a course naturally includes the descriptive level as well. Finally the ‘Advanced’ section of a chapter includes calculus at both the introductory and advanced level (of vector calculus) along with differential equations and some use of linear algebra and matrix theory along with both the descriptive and mathematical sections. I have found that almost all students today in the biological sciences (pre-med, pre-dental, …) have had calculus and thus I use the advanced concepts even in the non-calculus course for edification, but I do not test them at that advanced level as I do with the physics, chemistry, geology, mathematics, and engineering students who all take the Calculus level course.

I have used red fonts for equations and green fonts for numerical values and constants. This allows their rapid recognition. I often use web-available software which I have developed for UNITS conversion as an environment that allows one to mix units in any valid way thus providing an environment for very rapid computation. Finally, I also am testing an on-line (Internet) prompt-response system for tests, quizzes, homework, polls, and demographic data collection. Both the UNITS and Prompt-Response System are in Beta testing during the Spring of 2007 in conjunction with my teaching of the Physics 202 (second semester) course. I am likewise developing on-line lectures that can be used as a supplement to my regular class lectures for additional review and for students who missed the lecture. These video lectures are designed to capture a chapter in no more than 30 minutes as I have found that I am able to cover a one hour lecture in that time if there are no interruptions, and no repeated material. This time is devoted to rapidly covering just the core concepts allowing the student to replay the lecture as is necessary. A version of these lectures for the IPOD is being made available for downloads.

I intend to modify these lecture notes on a continuous basis using the Internet site for posting. Thus I can correct typos and make enhancements as are required to the material. I welcome comments and suggestions (at ) to the general framework of these componentsNotes, Video Lectures, UNITS software, and the Question-Response software all of which can be found at

Joseph E. Johnson, PhD

Professor of Physics

University of South Carolina

Columbia, SC, 29208

August 18, 2007

Fundamental Physics

Joseph E. Johnson, PhD © 2006

  1. Mechanics
  2. Newtonian Mechanics
  3. Introduction <CJ chap 1 >
  4. Discussion
  5. Units
  6. One Meter = distance that light travels in a vacuum in 1/299,792,458 s
  7. Scales of distance: quark-quark, atom, virus, human, earth, to sun, galaxy, universe
  8. One Kilogram = the mass of a platinum-iridium cylinder in Paris (mass of 1/1000 of m3 of water)
  9. Scales of masses: electron, proton, .. human, planet, star, galaxy
  10. One Second = the time of 9,192,631,770 vibrations of Cesium 133 radiation
  11. Scales of time; light across proton, cesium, lifetime of human, age of earth, universe
  12. Discuss derived units: m/s, kg/m, m2, m3
  13. Correct use of units +- only of same types, */ any kinds, transcendental functions (dimensionless)
  14. Unit conversion by forming unity with which one can * and /
  15. Powers of 10 & Prefixes
  16. Use of the Greek Alphabet as additional symbols
  17. Numerical Uncertainty
  18. Rules for addition and multiplication with numerical uncertainty
  19. Mathematical
  20. Vector Addition, Subtraction, & multiplication by a constant – Linear Vector Space
  21. Graphical method
  22. ijk method
  23. Component form: (x, y, z) = (x1, x2, x3) = xi
  24. Products
  25. Scalar Product A * B = AxBx + AyBy + AzBz = AB cos a scalar value)
  26. Cross Product (A x B)iijk Aj Bk = AB sin in magnitude with direction from RHR
  27. The dimension of a space is the number of numbers needed to specify a point.
  28. Advanced
  29. Vectors Addition, Subtraction, & multiplication by a constant – Linear Vector Space
  30. Ordered n-tupe Method
  31. Scalar Product AB cos Metric Space
  32. Cross Product AB sin - ijk symbol use

1.1.2.Kinematics in One Dimension <CJ chap 2 >

1.1.2.1.Discussion

1.1.2.1.1.Methodology:

1.1.2.1.1.1.A single mass moves in three dimensions of space over time

1.1.2.1.1.2.Motion in three dimensions can be understood as three independent one dimensional motions

1.1.2.1.1.3.The internal behavior of the single mass can be ignored – its position is at the center of mass

1.1.2.1.1.4.The ‘state of a particle’ is given by the position and velocity at one instant of time in its motion

1.1.2.1.1.4.1.Velocity is defined as v =  x / t with units of m/s

1.1.2.1.1.4.2.Acceleration is defined as a =  v / twith units of m/s2

1.1.2.1.1.4.3.Graphical view of v and a

1.1.2.1.1.5.We seek to predict its motion: given position and velocity at one time, find them in the future

1.1.2.1.2.Simple problems:

1.1.2.1.2.1.When velocity is constant

1.1.2.1.2.2.When acceleration is constant

1.1.2.2.Mathematical

1.1.2.2.1.1.A single mass moves in three dimensions of space over time x(t)

1.1.2.2.1.2.We seek to predict its motion: given x(0) and v(0) then what is x(t) and v(t)

1.1.2.2.1.3.Define average velocity v = (x(t) – x(0) ) / t

1.1.2.2.1.4.Define average acceleration a = (v(t) –v(0))/ t

1.1.2.2.2.Simple problems:

1.1.2.2.2.1.When velocity is constant v(t) = v(0) and x(t) = x(0) + v(0) t

1.1.2.2.2.2.When acceleration is constant v(t) = v(0) + at and x(t) = x(0) + v(0) t + ½ a t2

1.1.2.2.2.3.Another equation is obtained on eliminating time: v(t)2 – v(0)2 = 2 a d where d = x(t) –x(0)

1.1.2.2.3.Constant gravity problems

1.1.2.2.3.1.a = g = 9.8 m/s2 or = 32 f/s2

1.1.2.2.3.2.v(t) = 0 at top of motion

1.1.2.2.3.3.a(t) = a = g all the time

1.1.2.2.3.4.v(0) = v(t) when the object rises and then falls back to the same height that it originally had

1.1.2.2.4.Terminal velocity – of a human 140 mi/hr max drag (spread) and 240 mi/hr minimum drag (standing)

1.1.2.2.5.

1.1.2.3.Advanced

1.1.2.3.1.1.Define instantaneous velocity v = dx(t) / dt

1.1.2.3.1.2.Define instantaneous acceleration a = dv(t) / dt

1.1.2.3.2.Simple problems – derive:

1.1.2.3.2.1.When velocity is constant v(t) = v(0) and x(t) = x(0) + v t

1.1.2.3.2.2.When acceleration is constant v(t) = v(0) + at and x(t) = x(0) + v(0) t + ½ a t2

1.1.2.3.2.3.Another equation is obtained on eliminating time: v(t)2 – v(0)2 = 2 a d where d = x(t) –x(0)

1.1.3.Kinematics in Two & Three Dimensions <CJ chap 3 >

1.1.3.1.Discussion

1.1.3.1.1.Graphical view of projectile motion in two dimensions

1.1.3.1.1.1.Vertical motion is as in one dimension with constant a = g

1.1.3.1.1.2.Horizontal motion is as though a =0 and v = constant

1.1.3.1.1.3.Compare to view of one dimensional motion from a moving car or train

1.1.3.1.2.Graphical view of motion in a river or with an air current using vectors graphically

1.1.3.2.Mathematical

1.1.3.2.1.Projectile motion using vectors r(t) = (x(t) , y(t) ) and v(t) = (vx(t) , vy(t))

1.1.3.2.1.1.Vertical motion is as in one dimension with constant a = g

1.1.3.2.1.2.Horizontal motion is as though a =0 and had v = constant

1.1.3.2.1.3.Combined motion of vertical & horizontal

1.1.3.2.2.Graphical view of motion in a river or with an air current using vectors graphically

1.1.3.2.2.1.Compound motion by adding vectors of person relative to water and water to ground.

1.1.3.2.2.2.Determine angle of real motion, angle necessary to stay still, time across water etc

1.1.3.2.2.3.Combined velocity of airplane & wind velocity

1.1.3.3.Advanced

1.1.3.3.1.Derivation of constant acceleration equations using dv/dt = a (constant)

1.1.3.3.2.More complex projectile problems

1.1.3.3.2.1.Projectile which goes over a cliff

1.1.3.3.2.2.Projectile in moving air

1.1.4.Forces & Newton’s Laws of Motion <CJ chap 4

1.1.4.1.Discussion

1.1.4.1.1.Mass as a measure of inertia, the resistance to acceleration. - units of kg

1.1.4.1.2.Forces are vectors

1.1.4.1.3.Inertial reference frame

1.1.4.1.4.Newton’s Laws: Force measured in NewtonsNt = kg m/s2

1.1.4.1.4.1.First Law: F=0 implies a =0 and conversely

1.1.4.1.4.2.Second Law: F= ma

1.1.4.1.4.3.Third law F1->2 = - F2->1

1.1.4.1.5.Fundamental forces:

1.1.4.1.5.1.Gravitational (all masses and energy – infinite range) 10-39

1.1.4.1.5.2.Weak (involving leptons and neutrinos, very short range) 10-14

1.1.4.1.5.3.Electromagnetic (involving charged particles and currents – infinite range) 10-2

1.1.4.1.5.4.Nuclear (range of 10-15 m: p & n bound by pions) 1

1.1.4.1.5.5.Strong (quarks bound by gluons) 10

1.1.4.1.6.Frictional Force (static & dynamic)

1.1.4.1.7.Centripetal Force (from circular motion with only a change in direction)

1.1.4.1.8.Elastic force (system near equilibrium as with a spring) – Hooke’s law

1.1.4.1.9.Force of tension

1.1.4.2.Mathematical

1.1.4.2.1.Newton’s Laws

1.1.4.2.1.1.First Law: F=0 implies a =0 and conversely

1.1.4.2.1.2.Second Law: F= ma (for constant mass situations)

1.1.4.2.1.2.1.More accuratelyF= p/t

1.1.4.2.1.3.Third law F1->2 = - F2->1

1.1.4.2.2.Forces

1.1.4.2.2.1.Gravitational Force Fgrav = G m1 m2 / r2 and Near the earth’s surface Fgrav = W = mg

1.1.4.2.2.2.Electrical & Magnetic Force Fem= q E + q v x B where F = k q1 q2 / r2

1.1.4.2.2.3.Frictional Force (static & dynamic)Ffric = Fnormal

1.1.4.2.2.4.Elastic Force near equilibrium Felas = -kx where x is the distance from equilibrium

1.1.4.2.2.5.Centripetal force Fcen = m v2 /r where r is the radius of curvature

1.1.4.2.2.6.Force of tension is equal to the force with which the rope is pulling.

1.1.4.2.2.7.Equilibrium as Ftotal = 0

1.1.4.2.3.Resolution of forces & their vector nature

1.1.4.2.3.1.Atwood’s Machine

1.1.4.2.3.1.1.Force of tension

1.1.4.2.3.2.Incline plane

1.1.4.2.3.2.1.Without friction – one mass

1.1.4.2.3.2.2.With friction – one mass

1.1.4.2.3.2.3.With friction and two masses - tension

1.1.4.2.3.3.Problems with vector force resolution

1.1.4.2.3.3.1.Problem with rope stretched horizontally with weight

1.1.4.3.Advanced

1.1.4.3.1.Newton’s second law: F = dp/dt = d(mv)/dt or when m= const, F=mdv/dt =ma

1.1.4.3.1.1.Thus for each direction: Fx = dpx / dt , etc.

1.1.5.Uniform Circular Motion <CJ chap 5 >

1.1.5.1.Discussion

1.1.5.1.1.Definition of uniform circular motion with velocity v and radius r

1.1.5.1.2.Centripetal (means directed toward a center) acceleration

1.1.5.2.Mathematical

1.1.5.2.1.Period T of circular motion is defined by v = 2r / T

1.1.5.2.2.acen = v2 / r thus Fcen = m acen

1.1.5.2.3.Problem of balancing friction with centripetal forces of a car driving around a curve– flat road

1.1.5.2.4.Same problem of car on a curve but with a road that is angled

1.1.5.2.5.Problem of satellites in circular orbit GmM/r2 = m v2/r thus v = (GM/r)1/2

1.1.5.2.6.Artificial gravity using circular motion

1.1.5.2.7.Problem of pail of water rotated in a vertical plane

1.1.5.3.Advanced

1.1.6.Work & Energy <CJ chap 6

1.1.6.1.Discussion

1.1.6.1.1.Work requires energy and are often considered synonymous –

1.1.6.1.2.Energy is conveyed from one system to another exactly by the work done.

1.1.6.1.2.1.More precisely, an increase in energy is always equal to (and due to) work that is done

1.1.6.1.3.Work is defined as the force times the distance moved in the direction of work – push a lawn mover

1.1.6.1.4.The unit of work is the Joule (J) = 1 Nt acting through 1 m i.e. 1J = 1Nt*1m

1.1.6.1.5.Work and energy are scalar quantities with no direction and are not vectors.

1.1.6.1.6.Types of energy:

1.1.6.1.6.1.Kinetic – energy of motion

1.1.6.1.6.2.Potential – energy due to position or configuration

1.1.6.1.6.3.Chemical – stored for possible energy releasingchemical reactions of atoms and molecules

1.1.6.1.6.4.Nuclear – stored for possible energy releasing nuclear reactions

1.1.6.1.6.5.Solar & radiant – energy from light and more generally electromagnetic radiation

1.1.6.1.6.6.Heat – energy due to the random motion of molecules and constituents

1.1.6.1.7.Power is defined as the rate of doing work or expending energy

1.1.6.1.7.1.Energy is often defined in terms of power times time, e.g. KWHR = 1000 J/s *3600 s

1.1.6.1.8.Conservative and nonconservative forces – path independence of work & reversible

1.1.6.2.Mathematical

1.1.6.2.1.W = Fr = F r cos 

1.1.6.2.2.Kinetic Energy KE = W = Fr = m (v/dt) r = m v v thus calculus will lead to: KE = ½ mv2

1.1.6.2.3.Gravitational Potential Energy W = Fgravr = m gh or PE = mgh

1.1.6.2.4.Elastic Potential Energy W = Felasr = kx x thus calculus will lead to PE = ½ kx2

1.1.6.2.5.

1.1.6.3.Advanced

1.1.6.3.1.W = Fdr and is conservative if this integral is path independent (or zero for any closed curve)

1.1.6.3.2.Kinetic Energy KE = dW = Fdr = m (dv/dt) dr = m v dv thus KE = ½ mv2

1.1.6.3.3.Gravitational Potential Energy dW = Fgravdr = m g dh or PE = mgh

1.1.6.3.4.Elastic Potential Energy dW = Felasdr = kx dx thus PE = ½ kx2

1.1.7.Momentum and Impulse <CJ chap 7 >

1.1.7.1.Discussion

1.1.7.1.1.Impulse is defined as the change in momentum of an object such as a baseball when hit

1.1.7.1.2.Thus Impulse is a vector quantity and is often useful when the force is a complicated function of time

1.1.7.1.3.Momentum is conserved in a system that has no outside forces acting upon it.

1.1.7.2.Mathematical

1.1.7.2.1.Momentum p = m v

1.1.7.2.1.1.For any system of particles with momentum pi one has

1.1.7.2.1.1.1.P/t= pi)/t= jiFj on i + iFexti = 0 + Fexttotal because Fj on i= - Fi on j

1.1.7.2.1.1.2.Thus if there is no total external force on a system, the internal forces cancel

1.1.7.2.1.1.3. and thus the total internal momentum is conserved.

1.1.7.2.2.Impulse = p = <F> t = the average force times the time interval.

1.1.7.2.2.1.Problem of hit baseball, & of rain verses hail on car roof (twice the impulse due to recoil)

1.1.7.2.3.Elastic collisions: Total kinetic energy after collision is same as before collision

1.1.7.2.3.1.Problem: 1 dimension – must use cons. of both energy & momentum to compute v1 & v2 after

1.1.7.2.3.2.Example of superball – bounce is essentially to equal to the previous height

1.1.7.2.4.Partially Inelastic collisions: Some kinetic energy is lost to heat from the objects collisions

1.1.7.2.4.1.Example of a bouncing ball – loss of KE is exactly measured by mgh via loss in height

1.1.7.2.5.Totally inelastic collisions: Objects stick together after collision & the maximum possible loss of KE

1.1.7.2.5.1.When object stick together there is only one v after collision which is obtained by cons. of mom.

1.1.7.2.5.2.Ballistic pendulum (bullet into a block of wood – velocity is obtained by height)

1.1.7.2.5.3.Two football playerswhere one tackles the other

1.1.7.2.6.Center of Mass R = i miri / M where M = i mi= total mass of the system

1.1.7.2.6.1.Recall from above that P /t= pi) /t= jiFj on i + iFexti = 0 + Fexttotal

1.1.7.2.6.2.Thus P /t= miri / t) /t = Fexttotal =  (MV)/ t = M V where V = velocity of COM

1.1.7.2.6.3.It also follows that P = M V

1.1.7.3.Advanced

1.1.7.3.1.Momentum p = m v

1.1.7.3.1.1.For any system of particles with momentum pi one has

1.1.7.3.1.1.1.dP /dt= dpi) /dt= jiFj on i + iFexti = 0 + Fexttotal because Fj on i= - Fi on j

1.1.7.3.1.1.2.Thus if there is no total external force on a system, the internal forces cancel

1.1.7.3.1.1.3. and thus the total internal momentum is conserved.

1.1.7.3.2.Elastic collisions: Kinetic energy after collision is same as before collision

1.1.7.3.2.1.Problem: 2 dimensional – must use cons. of both energy & momentum to compute v1 & v2 after

1.1.7.3.2.2.Example of billiard balls

1.1.7.3.3.Center of Mass R = i miri / M where M = i mi= total mass of the system

1.1.7.3.3.1.Recall from above that dP /dt= dpi) /dt= jiFj on i + iFexti = 0 + Fexttotal

1.1.7.3.3.2.Thus dP /dt= dmidri /dt) /dt = Fexttotal = d (MV)/dt where V = dR/dt =velocity of COM

1.1.7.3.3.3.It also follows that P = M V

1.2.Rotational Mechanics & Gravity

1.2.1.Rotational Kinematics <CJ chap 8 >

1.2.1.1.Discussion

1.2.1.1.1.Definition of angle in radians = s / r where s is the arc length subtended & r is the radius

1.2.1.1.1.1.Thus cycle= 2  r / r = 2  radians = 360 degrees when considering the arc of an entire circle.

1.2.1.1.1.2.Circular motion restricts the distance to be a constant value of r from a given point

1.2.1.2.Mathematical

1.2.1.2.1.Define angular velocity  =  / t in units of radians per second or rad/s

1.2.1.2.2.Define angular acceleration  =  / t in units of radians per second squared or rad/s2

1.2.1.2.3.Since s = rit follows that s/t = v = r  and v/t = a = r 

1.2.1.2.4.If  is constant then it follows that t in analogy with v = v0 + a t for translational motion

1.2.1.2.5.Likewise it follows that t + ½  t2 in analogy with x = x0 + v0t + ½ a t2

1.2.1.2.6.Combining these equations by eliminating t we obtain 

1.2.1.2.7.Centripetal acceleration acen = v2/r = r 2

1.2.1.2.8.Rolling motion problems: the tangential velocity is equal to the velocity of the center of the circle

1.2.1.3.Advanced

1.2.1.3.1.Define angular velocity  = d / dt in units of radians per second or rad/s

1.2.1.3.2.Define angular acceleration  = d / dt in units of radians per second squared or rad/s2

1.2.1.3.3.Since s = rit follows that ds/dt = vtan = r  and dv/dt = atan = r 

1.2.1.3.4.If  is constant then d = dt thus t in analogy with v = v0 + a t for translational motion

1.2.1.3.5.Then using d/dt =  we get t + ½  t2 in analogy with x = x0 + v0t + ½ a t2

1.2.1.3.6.Combining these equations by eliminating t we obtain 

1.2.1.3.7.Vector nature of circular motion uses the RHR to get the directions of , and 

1.2.2.Rotational Dynamics <CJ chap 9

1.2.2.1.Discussion

1.2.2.1.1.Just as forces give acceleration in translational motion, torques give angular acceleration in rotation

1.2.2.1.1.1.Thus Torque is to rotations as Force is to translations

1.2.2.1.2.For solid objects and systems, we can generally express the motion in translation & rotation

1.2.2.1.2.1.The translation is of the center of mass while the rotation is about the center of mass or an axis

1.2.2.1.3.Just as translational equilibrium has a net force of zero, rotational equilibrium means no torque

1.2.2.1.3.1.So equilibrium problems can be solved by requiring that the total torque (and force) is zero

1.2.2.2.Mathematical

1.2.2.2.1.Torque defined

1.2.2.2.1.1.Imagine a system with one fixed point (an axis or center) and a force is applied a distance r away

1.2.2.2.1.2.Torque  is defined as the distance to the force application point times the normal force, F sin 

1.2.2.2.1.3.Thus torque is defined as r x Fwith the right hand rule governing the direction of 

1.2.2.2.1.4.Units of torque are Newtons x meters = Nm

1.2.2.2.1.5.Equilibrium is defined by i= 0 and Fi= 0

1.2.2.2.1.6.Problem: Opening a door

1.2.2.2.1.7.Problem: Using a lug wrench or screw driver

1.2.2.2.1.8.Problem: Force to support the end of a bridge – sum of several torques

1.2.2.2.2.Center of Gravity = Center of masswith weights replacing masses after multiplication by g –prove:

1.2.2.2.2.1.How to find the center of gravity of an object - hanging it from two points (intersection of verticals)

1.2.2.2.3.Moment of Inertia defined by I = i miri2 with units of kg m2

1.2.2.2.3.1.r x F = r Fnor = r ma (but a = r) thus  = m r2 which holds for each particle in a system

1.2.2.2.3.2.Thus for an ensemble of particles  = (imi ri2 ) = I

1.2.2.2.3.3.Problem: Moment of inertia for different objects

1.2.2.2.3.3.1.Solid Sphere I=2/5 MR2 ; Hollow Sphere I=2/3 MR2 ; Solid Cylinder I=1/2 MR2

1.2.2.2.3.3.2.Rod with axis perp to center I=1/12 ML2 ; Rod with axis perp to end I=1/3 ML2

1.2.2.2.3.4.Problem: Object rolling down a hill

1.2.2.2.4.Rotational Work (Energy) W = Fs =(Fnor r)  =  thus W= 