Anurag Mishra Mechanics 1 with www.puucho.com www.puucho.com Anurag Mishra Mechanics 1 with www.puucho.com ·' !II Published by: 't§l SHRI BALAJI PUBLICATIONS (EDUCATIONAL PUBLISHERS & DISTRIBUTORS) 6, Gulshan Vihar, Gali No. 1, Opp. Mahalaxmi Enclave, Jansath Road, Muzaffarnagar (U.P.) Phone: 0131-2660440 (0), 2600503 (R) website : www.shribalajibooks.com email : [email protected] !II First edition 2009 !II Fourth edition 2012 !II Fifth edition 2013 !II Reprint 2014 !II © All rights reserved with author !II Price : { 428.00 !II Typeset by : Sun Creation Muzaffarnagar !II _printed at : Dayal Offset Printers Meerut (U.P.) !II All the rights reserved .. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the author and publisher. Any violation/breach shall be taken into legal action. www.puucho.com Anurag Mishra Mechanics 1 with www.puucho.com , · Preface I have been involved in teaching Physics for last 16 years. This book is an opportunity to present my experiences. During my interaction with UT-JEE aspirants. I realised that most feared topic is mechanics. Some oft he reasons put forward by students behind this thought were: @ No spontaneous thoughts.appear after reading a problem. Mind goes blank. Can not . proceed in a problem. @ How to ~roceed in a problem? Which law is applicable; that is a given problem will involve conservation of energy or momentum or both. @ If some one says solve the problem in non-inertial reference frame, horrible thoughts appear inmind. @ Total confusion about CM frame. @ Proper understanding of constraints. @ S)lort cut approach in relative motion. @ No single book available that gives large no. of solved examples with elaboration of concepts in asolµtion. This book will help the students in building analytical and quantitative skills, addressing keyl,Jlisconceptions and developing.confidence in problem solving. I sincerely wish that this book will fulfill all the aspirations of the readers. Although utmost full care has been taken to make the book free from error but some errors . - . . ina_dvertently may creep-in. Author and Publisher shall be highly obliged if suggestions. regarding improvement and errors are pointed out by readers. I am indebted Neeraj Ji for providing me an opportunity to write a book oft his magnitude. I am indebted to my father Sh. Bhavesh Mishra; my mother Smt. Priyamvada Mishra, my wife Manjari, my sister Parul, my little kids Vrishank and Ira for giving their valuable time which I utilized during the writing of this book and people of Morada bad, who supported . . ·me throughout my career. In the last, !also pay my sincere thanks to all the esteemed members ofM/s. ShriBalaji Publications in bringingo ut this book in"the present form. Anurag Mishra. www.puucho.com r Anurag Mishra Mechanics 1 with www.puucho.com Our Other Publications for (Engineering Entrance Examination)€ Er. Anurag Mishra Er. Anurag Mishra • Electrostatics Mechanics • Simple Harmonic Motion Electricity& , •, Electric Current j • !•JEE • Wave Motion Magnetism yapacitors • Solid and Fluids forJEE lI • The Magnetic Field ' • Gravitation 1 • Electromagnetic Induction Vol. II / and A.G. Circuits 'I-----~ Er. Anurag Mishra I' . )feat& 1 • Temperature, Heat & Optics . Thermodynamics tl)e. equation of State. l "G eometrical Optics' /•JEE Heat Transfer . foJ EE Wave Opti~ • Thermodynamics . I , ' . : : ·,-:' \ ~ www.puucho.com Anurag Mishra Mechanics 1 with www.puucho.com How to face the challenge ? Following are some doubts which arise in the mind of almost all the students but may face them by taking some care. 1. I can not solve numerical because my concepts are not clear. In fact numerical solvingitselfis an exercise t~ learn concepts. ·z. 1 can not study because I am in depression, I fell into it because I was not /' studying! Depression is escape mechanism of people afraid of facing failures. Failure is integral partoflearning. 3. I understand everything in class but can not solve on my own.WRITING work is · vital. It is a multiple activity, initially idea comes in mind then we put into language to express it, we are focussed in hand eye coordination, eyes create visual impression on brain which isrecorded there. WRITING WORKS ARE EMBOSSED ON BRAIN LIKE CARVINGS OF AJANTACAVES. 4. In exams my brain goes blank, but I can crack them at home. Home attempt is your second attempt! you are contemplating about it while home back. You do not behave differently in exam you replicate your instincts. Once a fast bowler was bowling no balls. His coach placed a-stump on crease, in fear of injury he got it right. CONCEPTUALIZATION, WRITING EQUATION, SOLVING, THEN PROBLEM GETS TO CONCLUSION! 5. I am an average student. It is a rationalization used by people afraid of hard work. In their reference frame Newton's first law applies "if I have a misconception I will continue with it unless pushed by an external agent even I will surround him in my web of misconception yielding zero resultant:' AVERAGE IS NOT DUE TO CAPACITY LACUANEBUT DUE TO LACK OF DETERMINATION TO SHED INERTNESS. 6. A famous cliche "/ do not have luck in my favour' PRINCIPLE OF CAUSALITY: CAUSE OF AN EVENT OCCURS IN TIME BEFORE OCCURRENCE OF THAT EVENT i.e., cause occurs first then event occurs. SHINING OF LUCK IS NOT AN iNSTANTANEOUS EVENT IT IS PRECEDED BY RELENTLESS HARD WORK. Sow a seed ofaspiration in mind, water it with passion, dedication it will bear fruit, luckcan give you sweeter fruit. www.puucho.com Anurag Mishra Mechanics 1 with www.puucho.com t. Do not take study as a burden actually )ts a skill like singing and dancing. It has to be honed by proper devotion and dedication. · 2. Withou't strong sen_se of achievement you can't excel. Before entering the cmppeti.tive field strong counselling by parents is must. Majority do not · know what for they are here. No strategic planning, they behave like a tail : ende_r batting in frontofSteyn's bouncers. 3. - 'science is not a subje,ct based on well laid dowh procedures or· based on it learning sonie facts, ipvolves very intuitive and exploratory approach. Unless their is desire and passion to learn you can not discover new ideas. It requires p'atience and hard work, whose fruits may be tangible later on. 4. Some students realize very late that they are studying for acquiring skills and , . honingthem. Their is a feeling that they can ride at the back of instructor and , · , achieve ~xcellence. Study comes as tqrtu·rous exercise enforced on them and ' their is some mechanism that can take this burden-of them. - 5. Science is not about gaining good marks, up toXth by reading key points good marks are achieved but beyond that only those survive who have genuine interest in learning and exp to ring. Self study habit is must. 6. · IF YOU WANT TO GAIN LEAD START EARLY. Majority of successful students try to finish .major portion-elementary part of syllabus before they enter Coaching Institute. Due to this their maturity level as comparecl to others is · more tliey get ample time to adj4st with th~ fast pace. They are less· ' :i. traumatized by the scientific matter handed over. For those who enter fresh must be counselled to not get bullied by ·early starters but work harder initially within first two months initial edge is neutralized.· 7. Once a sl:l,ldent lags behind due to scime forced or unforced errors his mind begins to play rationalization remarks like I am an average student, my mind is not sharp enough, I have low IQ etc: These words are mechanisms _used to · a\Toid hard work. These words a,re relative terms a person who has .early start may be intelligentr~lative t9y6u.' . . . 0lntelligence means _cu~ulative i-~sult of h_ard _work of previous years, that hard work has eventuallY. led to a developm~nt of instinct tci crack 'things - easily. _., · / www.puucho.com Anurag Mishra Mechanics 1 with www.puucho.com I ' CONTENTS - - (j UNIT AND DIMENSIONS (j I. DESCRIPTION OF MOTION I - Subject of kinematics (19), Vector notation (20), Displacement vector (20),Parallelogram law of vector addition (21)., Component ofvector(23), Unitvector(24), Expressing a vector in unit vector notati_on (25), Position vector (27), Rectangular resolution of a vector in three dimensions (28), Vector multiplication (31), The scalar product of two vectors (32), The vector product of two vectors (35), Rectilinear motion of a particle (37), Calculus supplementary ( 41), Rectilinear motion ( 44), Instantaneous velocity ( 45), Integration ( 48), Interpretation of graphs (53), Average velocity & Average speed (58), Two-dimensional motion with constant acceleration (69), Projection on an inclined plane (82), Relative motion (87), Application of advanced concept of relative motion (88), Equation of motion for relative motion (98), Projection of a particle in an accelerated elevator (101), Projection of a ball in horizontally moving trolley (101 ), Closest distance of approach between two moving bodies (102), Problems: Level-1 (106), Level-2(113), Level-3 (117),Answers(123), Solutions (124). Q 2. FORCE ANALYSIS ~~~~~p=--=~-=::;~ The concept of force (138), Reference frame (139), Ideal string (143), ideal pulley (143), Contact force (143), Concept of external and internal force (144), Pulley system (145), Tension in a hanging rope (145), Constrained motion (148), Pulley constraint (154), Normal constraint (155), Elastic force of spring (158), Parallel combination (159), Friction (165), The laws of sliding friction (165), Direction of kinetic Friction (171), FBD when arm is in_ deceleration (185), Circular motion (186), Angular velodty vector (187), Concept of , pseudo force (192), Non-inertial reference frame (193), Whirling rope (195), Lift Force on an airplane (196 ), Non-uniform circular motion on horizontal plane I {196), Problems: Level-1 (210), Level-2 (221),Level-3 (230),Answers (238), Solutions (240). 1 I ______________ ~ \_ ---------------- www.puucho.com --------·· Anurag Mishra Mechanics 1 with www.puucho.com --· ,· -CJ 3. WORK AND ENERGY Work done (265), Unit of work (266 ), Conservative and non-conservative force (268), Concept of potential energy (269), Classical work energy theorem (270), Conservation of mechanical energy (271), Work done by friction (271), Work done by spring force (271), Work depends on the frame of reference (272), Work due to internal force (friction) (272); Work energy theorem in a non-inertial reference frame (273), How to apply coriservation of energy_ equation (273), Vertical circ_ular motion (283), Power (293), lntemai energy so!-lrces & work (296), Problems Level-1 (299), Level-2 (306), Level-3 (310),Answers (316), Solutions (318). CJ 4. IMPULSE AND MOMENTUM Impulse (328), Conservation of momentum (328), Conservation momentum for a two particle system (329), Relative velocity and the conservation of momentum (330), Recoil disintegration, explosions (335), Impulsive force (336 ), Centripetal acceleration revisite~ (338), Centre of mass (340), Position of COM oft wo particles (340), Centre of gravity (341 ), Motion of the centre of mass (34,1), Kinetic energy of a system of particles (342), Most important concept (343), Finding the centre of mass by integration (353), Collisions (361), Models for elastic & inelastic collisions (362), Oblique impact (365), The velocity of the centre of mass for collisions (370), Elastic collisions in the CM reference frame (371), Inelastic collisions in CM reference frame (372), System of variable mass; Rocket propulsion (380), Problems Level-1 (383), Level-2 (395), Level-3 (399), Answers (4 06), Solutions (4 08). · (J 5. RIGID BODY MOTION What is rigid body(4 21), General rigid body motion (4 21 ), Rotation about centre of mass (4 22), Kinematics of fixed axis rotation (4 22), Vector representation of rotational quantities (4 25), .Torque (4 27), Newton's second law for rotation (4 29), Rotational kinetic energy and moment of inertia (4 30), Rotational kinetic energy of a collection of particles (431), Perpendicular axis theorem (4 37), Dynamics of a rigid body(4 41), Angular momentum (4 47), The ladder (450), Work done due to torque (457), Angular momentum of a projectile ,, (4 62), Angular momentum of an i[]verted conical pendulum (4 62), Angular '=--~='-~=-~=~-- impulse angular momentum theorem (4 64), Two bodies rotatory system 0 (4 66), Kinematics ofrigidbody rotation (4 75), Total kinetic energy of body (4 84), Dynamics of rigid body in plane motion (486), Torque on the rotating skew rod (506), Problem: Level-1 (511); Level-2 (519), Level,3 (525), Answers (531), Solutions (533), Exercisg_,_advanced problems (544), Comprehension based pro bl ems ( 553), Assertion and reason type problems (562). I ________________ I ,. .,..,.,,-/ www.puucho.com Anurag Mishra Mechanics 1 with www.puucho.com i UNIT AND DIMENSIONS . Physics is that branch of science in which we observe, (a) Magnitude of Physical Quantity= measure and describe natural phenomena related to matter Numerical value x Unit and energy. Like all the science, physics is ultimately based th us for a given physical quantity when the unit will on observation. To ~ssemble the relevant observations into a change, numerical value will also change, e.g. density of coherent picture by, constructing a logical framework is water =lg- cc-1 = 103kg- m-3and not lkg- m-3. called theory. Theory enables the physicist to account for (b) Larger the unit smaller will be the magnitude and past observations and to decide how new ones should be vice-versa, e.g., 1 kg= 1000 gm then as 1000 is greater than made. Nearly all physical observation are quantitative; they 1, gm is smaller unit than kg of mass. require measurement. Every measurement is a comparison of a Criteria for Standards quantity with a standard quantity that is, an The choice of the standard is arbitrary. However, several agreed upon quantity of the same kind. To measure a criteria must be met if a standard is to be as useful as length for example, you adopt as your standard a convenient possible. measuring rod, whose length you use as the unit of length. 1. Stability : The standard should not vary with time. You count the number of times that the rod fits into the If this criterion is satisfied, measurements made at different length to be measured. This number given the length in time, using the same standard, can be meaningfully terms of the chosen unit. compared. Physical Quantities 2. Reproducibility: The standard should be accurately The quantities by means of which describe the laws of reproducible so that copies, ideally identical with the physics are called Physical Quantities. A physical quantity is standard itself can be used elsewhere. If this criterion is complete specified if it has - satisfied, measurements made at different places can be ~---e>(A) Numerical value only ratio e.g., compared. refractive index, dielectric constant, etc. 3. Acceptability : The standard should be universally or accepted so as to eliminate clumsy and possibly inaccurate Magnitude only Scalar e.g., comparisons among measurement made with separate mass, charge,current etc. standard. or 4. Accessibility : As nearly as possible, the standard should be readily accessible to everyone ·vho needs to use it. Magnitude and direction Vector e.g., displacement, torque, etc. 5. Precision : It should be possible to measure the In expressing the magnitude of a physical quantity we standard itself with a precision at least as great as the choose a unit and then find physical quantity how many times precision with which any comparable measurement can be that unit is contained in the given physical quantity, i.e. made. www.puucho.com