UPSC Exam > UPSC Notes > Mathematics Optional Notes for UPSC > Dynamics of a Particle
Dynamics of a Particle | Mathematics Optional Notes for UPSC PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 Page 2 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 the negative sign on the right is due to the fact that the acceleration is towards ?? , hence negative when ?? is positive. Integrating and using initial conditions, when ?? =0,?? =?? , and ?? =0 we get ?? 2 =6?? 5/3 (?? 1/3 -?? 1/3 ) Therefore ?? =-v6?? ?? 5/3 v?? 1/3 -?? 1/3 At ?? =0,?? =-?? v6?? Integrating v6?? ?? 5/3 ?? =-? ?? =?? 0 ? ???? ?? 1/3 -?? 1/3 Therefore ?? = 8 15 v 6 ?? Example 2: A particle moves along a straight-line with an acceleration directed toward fixed ?? point O on it and inversely proportional to the square of the distance of the particle from O. If the particle is at rest initially at a distance 2a from ?? ,?? how that the time of motion from the distance 2?? to the distance ?? is to the time from distince a to O in the ratio ?? +2:?? -2. ?? ???? ???? =- ?? ?? 2 Integrating and using initial conditions. ?? 2 =?? ( 2 ?? - 1 ?? ) Page 3 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 the negative sign on the right is due to the fact that the acceleration is towards ?? , hence negative when ?? is positive. Integrating and using initial conditions, when ?? =0,?? =?? , and ?? =0 we get ?? 2 =6?? 5/3 (?? 1/3 -?? 1/3 ) Therefore ?? =-v6?? ?? 5/3 v?? 1/3 -?? 1/3 At ?? =0,?? =-?? v6?? Integrating v6?? ?? 5/3 ?? =-? ?? =?? 0 ? ???? ?? 1/3 -?? 1/3 Therefore ?? = 8 15 v 6 ?? Example 2: A particle moves along a straight-line with an acceleration directed toward fixed ?? point O on it and inversely proportional to the square of the distance of the particle from O. If the particle is at rest initially at a distance 2a from ?? ,?? how that the time of motion from the distance 2?? to the distance ?? is to the time from distince a to O in the ratio ?? +2:?? -2. ?? ???? ???? =- ?? ?? 2 Integrating and using initial conditions. ?? 2 =?? ( 2 ?? - 1 ?? ) or ???? ???? =-v ?? ?? v 2?? -?? ?? Therefore v ?? ?? ?? =-1v ?? 2?? -?? ???? ? -?? v2???? -?? 2 ???? or v ?? ?? ?? =v?? 2 -(?? -?? ) 2 +?? cos -1 ( ?? -?? ?? ) The constant of integration is zero as ?? =0 when ?? =2?? . Time ?? 1 to go from ?? =2?? to ?? = ?? is given by v ?? ?? ?? 1 =?? [1+ ?? 2 ] Time ?? 2 , to go from ?? =?? to ?? =0 is given by v ?? ?? ?? 2 =???? -?? (1+ ?? 2 )=?? ( ?? 2 -1) Therefore ?? 1 ?? 2 = ?? +2 ?? -2 2. Vertical motion under gravity with resistance (a) A particle falls freely from rest under gravity (assumed constant) in a medium whose resistance is proportional to the square of its velocity. Find the distance travelled and or the velocity of the particle at time ?? . When the particle has fallen a distance ?? in time ?? Page 4 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 the negative sign on the right is due to the fact that the acceleration is towards ?? , hence negative when ?? is positive. Integrating and using initial conditions, when ?? =0,?? =?? , and ?? =0 we get ?? 2 =6?? 5/3 (?? 1/3 -?? 1/3 ) Therefore ?? =-v6?? ?? 5/3 v?? 1/3 -?? 1/3 At ?? =0,?? =-?? v6?? Integrating v6?? ?? 5/3 ?? =-? ?? =?? 0 ? ???? ?? 1/3 -?? 1/3 Therefore ?? = 8 15 v 6 ?? Example 2: A particle moves along a straight-line with an acceleration directed toward fixed ?? point O on it and inversely proportional to the square of the distance of the particle from O. If the particle is at rest initially at a distance 2a from ?? ,?? how that the time of motion from the distance 2?? to the distance ?? is to the time from distince a to O in the ratio ?? +2:?? -2. ?? ???? ???? =- ?? ?? 2 Integrating and using initial conditions. ?? 2 =?? ( 2 ?? - 1 ?? ) or ???? ???? =-v ?? ?? v 2?? -?? ?? Therefore v ?? ?? ?? =-1v ?? 2?? -?? ???? ? -?? v2???? -?? 2 ???? or v ?? ?? ?? =v?? 2 -(?? -?? ) 2 +?? cos -1 ( ?? -?? ?? ) The constant of integration is zero as ?? =0 when ?? =2?? . Time ?? 1 to go from ?? =2?? to ?? = ?? is given by v ?? ?? ?? 1 =?? [1+ ?? 2 ] Time ?? 2 , to go from ?? =?? to ?? =0 is given by v ?? ?? ?? 2 =???? -?? (1+ ?? 2 )=?? ( ?? 2 -1) Therefore ?? 1 ?? 2 = ?? +2 ?? -2 2. Vertical motion under gravity with resistance (a) A particle falls freely from rest under gravity (assumed constant) in a medium whose resistance is proportional to the square of its velocity. Find the distance travelled and or the velocity of the particle at time ?? . When the particle has fallen a distance ?? in time ?? ?? ???? ???? =?? -?? ?? 2 or ?? ???? ???? +?? ?? 2 =?? Integrating and using initial conditions, we get ?? 2 = ?? ?? [1-?? -2???? ] … (1) It follows that ?? < v ?? ?? and ?? ? v?? ?? as ?? ?8 Integrating ???? ???? =?? -?? ?? 2 and using initial conditions we get ?? =v ?? ?? tanh (?? v???? ) Therefore Page 5 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 the negative sign on the right is due to the fact that the acceleration is towards ?? , hence negative when ?? is positive. Integrating and using initial conditions, when ?? =0,?? =?? , and ?? =0 we get ?? 2 =6?? 5/3 (?? 1/3 -?? 1/3 ) Therefore ?? =-v6?? ?? 5/3 v?? 1/3 -?? 1/3 At ?? =0,?? =-?? v6?? Integrating v6?? ?? 5/3 ?? =-? ?? =?? 0 ? ???? ?? 1/3 -?? 1/3 Therefore ?? = 8 15 v 6 ?? Example 2: A particle moves along a straight-line with an acceleration directed toward fixed ?? point O on it and inversely proportional to the square of the distance of the particle from O. If the particle is at rest initially at a distance 2a from ?? ,?? how that the time of motion from the distance 2?? to the distance ?? is to the time from distince a to O in the ratio ?? +2:?? -2. ?? ???? ???? =- ?? ?? 2 Integrating and using initial conditions. ?? 2 =?? ( 2 ?? - 1 ?? ) or ???? ???? =-v ?? ?? v 2?? -?? ?? Therefore v ?? ?? ?? =-1v ?? 2?? -?? ???? ? -?? v2???? -?? 2 ???? or v ?? ?? ?? =v?? 2 -(?? -?? ) 2 +?? cos -1 ( ?? -?? ?? ) The constant of integration is zero as ?? =0 when ?? =2?? . Time ?? 1 to go from ?? =2?? to ?? = ?? is given by v ?? ?? ?? 1 =?? [1+ ?? 2 ] Time ?? 2 , to go from ?? =?? to ?? =0 is given by v ?? ?? ?? 2 =???? -?? (1+ ?? 2 )=?? ( ?? 2 -1) Therefore ?? 1 ?? 2 = ?? +2 ?? -2 2. Vertical motion under gravity with resistance (a) A particle falls freely from rest under gravity (assumed constant) in a medium whose resistance is proportional to the square of its velocity. Find the distance travelled and or the velocity of the particle at time ?? . When the particle has fallen a distance ?? in time ?? ?? ???? ???? =?? -?? ?? 2 or ?? ???? ???? +?? ?? 2 =?? Integrating and using initial conditions, we get ?? 2 = ?? ?? [1-?? -2???? ] … (1) It follows that ?? < v ?? ?? and ?? ? v?? ?? as ?? ?8 Integrating ???? ???? =?? -?? ?? 2 and using initial conditions we get ?? =v ?? ?? tanh (?? v???? ) Therefore 1-?? -2???? =?? 2 ?? ?? =tanh 2 (?? v???? ) from (1), (2) ?? -2???? =sech 2 (?? v???? ) or ?? ???? =cosh (?? v???? ) or ?? = 1 ?? ln cosh (?? v???? ) (b) A particle is projected vertically upwards with velocity ?? in a medium whose its resistance is proportional to the square of its velocity. To find its motion, assuming gravity to be constant. We have ???? ???? =-?? -?? ?? 2 Integrating and using initial conditions, we get v???? =tan -1 (?? v ?? ?? )-tan -1 (?? v ?? ?? ) Again ?? ???? ???? =-?? -?? ?? 2Read More
|
387 videos|203 docs
|
FAQs on Dynamics of a Particle - Mathematics Optional Notes for UPSC
| 1. What is the equation of motion for a particle in dynamics? |  |
Ans. The equation of motion for a particle in dynamics is given by Newton's second law of motion, which states that the sum of all forces acting on a particle is equal to the mass of the particle multiplied by its acceleration.
| 2. How does velocity affect the dynamics of a particle? | |
Ans. Velocity plays a crucial role in the dynamics of a particle as it determines the rate at which the position of the particle changes with time. The velocity of a particle can affect its acceleration and ultimately its motion.
| 3. What is the difference between statics and dynamics in the context of particles? | |
Ans. Statics deals with the study of particles at rest or in equilibrium, where the sum of all forces acting on the particle is zero. On the other hand, dynamics deals with particles in motion and studies the forces that cause the motion.
| 4. How can we calculate the acceleration of a particle in dynamics? | |
Ans. The acceleration of a particle in dynamics can be calculated by dividing the net force acting on the particle by its mass. This is in accordance with Newton's second law of motion, which relates acceleration to force and mass.
| 5. What are the different types of forces that can act on a particle in dynamics? | |
Ans. In dynamics, particles can experience various types of forces such as gravitational force, frictional force, normal force, tension force, and applied force. These forces can influence the motion and behavior of the particle under study.
Related Exams
About this Document
|
4.79/5 Rating |
|
Oct 31, 2024 Last updated |
Document Description: Dynamics of a Particle for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Dynamics of a Particle have been prepared according to the UPSC exam syllabus. Information about Dynamics of a Particle covers topics
like and Dynamics of a Particle Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Dynamics of a Particle.
Introduction of Dynamics of a Particle in English is available as part of our Mathematics Optional Notes for UPSC
for UPSC & Dynamics of a Particle in Hindi for Mathematics Optional Notes for UPSC course.
Download more important topics related with notes, lectures and mock test series for UPSC
Exam by signing up for free. UPSC: Dynamics of a Particle | Mathematics Optional Notes for UPSC
Description
Full syllabus notes, lecture & questions for Dynamics of a Particle | Mathematics Optional Notes for UPSC - UPSC | Plus excerises question with solution to help you revise complete syllabus for Mathematics Optional Notes for UPSC | Best notes, free PDF download
Information about Dynamics of a Particle
In this doc you can find the meaning of Dynamics of a Particle defined & explained in the simplest way possible. Besides explaining types of
Dynamics of a Particle theory, EduRev gives you an ample number of questions to practice Dynamics of a Particle tests, examples and also practice UPSC
tests
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
past year papers
,Viva Questions
,mock tests for examination
,Previous Year Questions with Solutions
,ppt
,shortcuts and tricks
,MCQs
,Summary
,Free
,Sample Paper
,Dynamics of a Particle | Mathematics Optional Notes for UPSC
,Dynamics of a Particle | Mathematics Optional Notes for UPSC
,Objective type Questions
,Exam
,study material
,Semester Notes
,video lectures
,Extra Questions
,Important questions
,practice quizzes
,Dynamics of a Particle | Mathematics Optional Notes for UPSC
;
Additional Information about Dynamics of a Particle for UPSC Preparation
Dynamics of a Particle Free PDF Download
The Dynamics of a Particle is an invaluable resource that delves deep into the core of the UPSC exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Dynamics of a Particle now and kickstart your journey towards success in the UPSC exam.
Importance of Dynamics of a Particle
The importance of Dynamics of a Particle cannot be overstated, especially for UPSC aspirants.
This document holds the key to success in the UPSC exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Dynamics of a Particle Notes
Dynamics of a Particle Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Dynamics of a Particle.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Dynamics of a Particle Notes on EduRev are your ultimate resource for success.
Dynamics of a Particle UPSC Questions
The "Dynamics of a Particle UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Dynamics of a Particle on the App
Students of UPSC can study Dynamics of a Particle alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Dynamics of a Particle,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Dynamics of a Particle is prepared as per the latest UPSC syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup