MECHANICS Lecture notes for Phys 111 Dr Vitaly A Shneidman Dated November 27 2013 Abstract Department of Physics New Jersey Institute of Technology Newark NJ 07102 These notes are intended as an addition to the lectures given in class They are NOT designed to replace the actual lectures Some of the notes will contain less information then in the actual lecture and some will have extra info Not all formulas which will be needed for exams are contained in these notes Also these notes will NOT contain any up to date organizational or administrative information changes in schedule assignments etc but only physics If you notice any typos let me know at vitaly njit edu For convenience I will keep all notes in a single le each time you can print out only the added part Make sure the le is indeed updated there is a date indicating the latest modi cation There is also a Table of Contents which is automatically updated For convenience the le with notes will be both in postscript and pdf formats A few Graphics Some of the graphics is deliberately un nished so that we have what to do in class Advanced topics these will not be represented on the exams Read them only if you are really other things interested in the material Computer Mostly the use of a computer will not be required in the lecture part of this course If I need it e g for graphics I will use Mathematica You do not have to know this program but if you are interested I will be glad to explain how it works 1 Contents I Introduction A Physics and other sciences B Point mass C Units 1 Standard units 2 Conversion of units D Units and dimensional analysis E Vectors 1 Single vector 2 Two vectors addition 3 Two vectors dot product 4 Two vectors vector product II 1 dimensional motion A v const B v 6 const C a const III 2D motion A General B a const 1 a g projectile motion C Uniform circular motion 1 Preliminaries 2 Acceleration 3 An alternative derivation IV Newton s Laws A Force 1 Units 2 Vector nature 0 2 2 2 2 2 2 3 4 4 5 6 7 9 9 9 11 13 13 13 14 15 15 16 16 17 17 17 17 V Newton s Laws applications to friction and to circular motion B The Laws 1 Gravitational force 2 FBD normal force C Statics D Dynamics Examples A Force of friction 1 Example block on inclined plane B Centripetal force 1 Conic pendulum 2 Sattelite C Advanced Forces of inertia VI Work A Units B De nitions C 1D motion and examples VII Kinetic energy A De nition and units B Relation to work 1 Constant force 2 Variable force C Power VIII Potential energy A Some remarkable forces with path independent work IX Conservation of energy A Conservative plus non conservative forces X Momentum A De nition 1 17 18 18 18 19 23 23 23 25 26 27 28 29 29 29 30 30 30 30 30 31 31 32 32 33 33 34 34 B 2d Law in terms of momentum XI Center of mass CM A De nition B Relation to total momentum C 2nd Law for CM D Energy and CM XII Collisions A Inelastic 1 Perfectly inelastic B Elastic 1 1D 2 2D XIII Kinematics of rotation A Radian measure of an angle B Angular velocity C Connection with linear velocity and centripetal acceleration for circular motion D Angular acceleration E Connection with tangential acceleration F Rotation with const XIV Kinetic Energy of Rotation and Rotational Inertia A The formula K 1 2 I 2 B Rotational Inertia Examples 1 Dumbell 2 Hoop 3 Rod 4 Disk 5 Solid and hollow spheres C Parallel axis theorem 2 34 35 35 35 35 35 35 35 36 36 36 37 38 38 38 39 39 39 40 40 40 41 41 41 42 42 43 44 D Conservation of energy including rotation 1 Atwood machine 2 Rolling XV Torque A De nition B 2nd Law for rotation C Application of I 1 Rotating rod D Torque as a vector 1 Cross product 2 Vector torque XVI Angular momentum A Single point mass B System of particles 2 Rotating rod with a point mass m at the end 3 Atwood machine revisited 4 Rolling down incline revisited XVII Conservation of angular momentum C Rotating symmetric solid 1 Angular velocity as a vector D 2d Law for rotation in terms of L A Examples 1 Free particle 2 Student on a rotating platform 3 Chewing gum on a disk 4 Measuring speed of a bullet 5 Rotating star white dwarf XVIII Equilibrium A General conditions of equilibrium 3 46 46 47 48 48 48 49 49 49 50 51 53 53 53 53 53 53 54 54 54 55 55 55 56 56 56 57 57 57 B Center of gravity C Examples 1 Seesaw 2 Horizontal beam 3 Ladder against a wall XIX Gravitation A Solar system B Kepler s Laws 1 1st law 2 2nd law 3 3rd law C The Law of Gravitation 1 Gravitational acceleration 2 Satellite D Energy 1 Escape velocity and Black Holes E Deviations from Kepler s and Newton s laws XX Oscillations A Introduction Math 1 sin x cos x for small x 2 Di erential equation x x 0 B Spring pendulum 1 Energy C Simple pendulum D Physical pendulum E Torsional pendulum F Why are small oscillations so universal G Resonance 4 57 58 58 58 59 60 60 61 61 61 62 62 62 62 63 63 64 67 67 67 67 68 69 69 70 72 72 72 Dr Vitaly A Shneidman Phys 111 1st Lecture I INTRODUCTION A Physics and other sciences in class B Point mass electron planet etc C Units 1 Standard units In SI system the basic units are The art physics is the art of idealization One of the central concepts in mechanics is a particle or point mass i e a body the size or structure of which are irrelevant in a given problem Examples m meter kg kilogram and s second Everything else in mechanics is derived Examples of derived units may or may not have a special name m s m s2 no name kg m s2 Newton kg m2 s2 Joule etc 2 Conversion of units Standard path all units are converted to SI E g length 1 in 0 0254 m 1 f t 0 3048 m 1 mi 1609 m 2 Examples 70 70 mi h 1609 m 3600 s 31 3 m s 3 cm2 3 10 2 m 2 3 10 4 m2 D Units and dimensional analysis Veri cation of units is useful to check the math More interesting however is the pos sibility to get some insight into a new problem before math is done or before it is even possible E g suppose we do not know the formula for displacement in accelerated motion with v0 0 Let us guess having at our disposal …