, Problems and Solutions in
Particle Mechanics
Pradeep Kumar Sharma MSc, MBA (IIT), Mtech, MPhil, PhD, AMIE,
CEng(I) MIET, MInstP, SMIEEE
Consultant physicist and researcher, (India)
IOP Publishing, Bristol, UK
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, Contents
Preface xi
Acknowledgments xv
Author biography xvii
1 Vectors 1-1
1.1 Introduction 1-1
1.2 Definition of vectors 1-2
1.3 Representation of vectors 1-3
1.4 Types of vectors 1-5
1.5 Multiplication of a vector by a scalar 1-9
1.6 Addition of vectors 1-10
1.6.1 Rectangular components of a vector in three dimensions 1-20
(optional)
1.7 Subtraction of vectors 1-21
1.8 Scalar product of two vectors 1-23
1.9 Vector product of two vectors 1-25
1.9.1 Applications of vector product 1-28
1.10 Some fundamental laws of vector algebra 1-29
1.10.1 Associative law 1-29
1.10.2 Distributive law 1-30
→ →
1.10.3 Condition of collinearity of vector A and B 1-30
1.11 Condition for coplanar vectors 1-31
1.12 Derivative (differentiation) of a vector 1-32
1.13 Integration of a vector 1-33
2 Motion in one dimension 2-1
2.1 Introduction 2-1
2.2 Position, displacement and distance 2-2
2.2.1 Distance 2-6
2.3 Velocity and speed 2-7
2.3.1 Instantaneous velocity 2-9
2.3.2 Acceleration 2-12
2.3.3 General kinematical equations for straight-line motion 2-14
2.3.4 Uniform accelerated motion 2-15
2.3.5 Kinematics of freely falling bodies 2-18
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vii
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, Problems and Solutions in Particle Mechanics
2.3.6 Use of kinematical equations 2-21
2.3.7 Uniformly accelerated motion 2-24
2.3.8 Graphs and graphical solutions 2-26
3 Motion in two and three dimensions 3-1
3.1 Introduction 3-1
3.2 Position, velocity and acceleration 3-2
3.3 Tractory and its equations 3-3
3.4 Uniform accelerated motion in a plane 3-4
3.5 Projectile motion 3-6
3.5.1 Projection onto horizontal plane 3-6
3.5.2 Projection onto an inclined plane 3-10
3.6 Relative motion 3-13
3.6.1 Galilean transformation of velocities and accelerations 3-14
3.7 Kinematics of circular motion 3-17
3.7.1 Constant angular acceleration 3-18
3.7.2 Radial acceleration 3-19
3.8 Curvilinear motion 3-21
3.9 Relative angular velocity 3-22
3.10 Equation of trajectory of one point relative to the other 3-24
3.10.1 Time of meeting of particles 3-26
4 Newton’s laws of motion of a particle 4-1
4.1 Introduction 4-1
4.2 Concept of force 4-1
4.2.1 Definition 4-1
4.2.2 Types of forces 4-2
4.3 Newton’s laws of motion 4-3
4.3.1 Linear momentum 4-3
4.3.2 First law (law of inertia) 4-3
4.3.3 Second law (law of accerleration) 4-4
4.3.4 Alternative explanation 4-4
4.3.5 Third law (law of action–reaction) 4-5
4.4 Types of forces 4-5
4.5 Application of Newton’s laws of motion 4-8
4.6 Constraint equations 4-17
4.7 Spring–particle system 4-24
4.7.1 Introduction 4-24
viii
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Particle Mechanics
Pradeep Kumar Sharma MSc, MBA (IIT), Mtech, MPhil, PhD, AMIE,
CEng(I) MIET, MInstP, SMIEEE
Consultant physicist and researcher, (India)
IOP Publishing, Bristol, UK
@seismicisolation
@seismicisolation
, Contents
Preface xi
Acknowledgments xv
Author biography xvii
1 Vectors 1-1
1.1 Introduction 1-1
1.2 Definition of vectors 1-2
1.3 Representation of vectors 1-3
1.4 Types of vectors 1-5
1.5 Multiplication of a vector by a scalar 1-9
1.6 Addition of vectors 1-10
1.6.1 Rectangular components of a vector in three dimensions 1-20
(optional)
1.7 Subtraction of vectors 1-21
1.8 Scalar product of two vectors 1-23
1.9 Vector product of two vectors 1-25
1.9.1 Applications of vector product 1-28
1.10 Some fundamental laws of vector algebra 1-29
1.10.1 Associative law 1-29
1.10.2 Distributive law 1-30
→ →
1.10.3 Condition of collinearity of vector A and B 1-30
1.11 Condition for coplanar vectors 1-31
1.12 Derivative (differentiation) of a vector 1-32
1.13 Integration of a vector 1-33
2 Motion in one dimension 2-1
2.1 Introduction 2-1
2.2 Position, displacement and distance 2-2
2.2.1 Distance 2-6
2.3 Velocity and speed 2-7
2.3.1 Instantaneous velocity 2-9
2.3.2 Acceleration 2-12
2.3.3 General kinematical equations for straight-line motion 2-14
2.3.4 Uniform accelerated motion 2-15
2.3.5 Kinematics of freely falling bodies 2-18
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vii
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, Problems and Solutions in Particle Mechanics
2.3.6 Use of kinematical equations 2-21
2.3.7 Uniformly accelerated motion 2-24
2.3.8 Graphs and graphical solutions 2-26
3 Motion in two and three dimensions 3-1
3.1 Introduction 3-1
3.2 Position, velocity and acceleration 3-2
3.3 Tractory and its equations 3-3
3.4 Uniform accelerated motion in a plane 3-4
3.5 Projectile motion 3-6
3.5.1 Projection onto horizontal plane 3-6
3.5.2 Projection onto an inclined plane 3-10
3.6 Relative motion 3-13
3.6.1 Galilean transformation of velocities and accelerations 3-14
3.7 Kinematics of circular motion 3-17
3.7.1 Constant angular acceleration 3-18
3.7.2 Radial acceleration 3-19
3.8 Curvilinear motion 3-21
3.9 Relative angular velocity 3-22
3.10 Equation of trajectory of one point relative to the other 3-24
3.10.1 Time of meeting of particles 3-26
4 Newton’s laws of motion of a particle 4-1
4.1 Introduction 4-1
4.2 Concept of force 4-1
4.2.1 Definition 4-1
4.2.2 Types of forces 4-2
4.3 Newton’s laws of motion 4-3
4.3.1 Linear momentum 4-3
4.3.2 First law (law of inertia) 4-3
4.3.3 Second law (law of accerleration) 4-4
4.3.4 Alternative explanation 4-4
4.3.5 Third law (law of action–reaction) 4-5
4.4 Types of forces 4-5
4.5 Application of Newton’s laws of motion 4-8
4.6 Constraint equations 4-17
4.7 Spring–particle system 4-24
4.7.1 Introduction 4-24
viii
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