2025 FINAL EXAM PHYS 204 Mechanics notes + examples 2025 Concordia
University
Notes
On
MECHANICS
(Physics 204)
, 2
Table of Contents
. ......................................................................................................................................................... 3
Chapter 1 Review of Vectors ...................................................................................................... 4
1.1 Mechanics ............................................................................................................................ 4
1.2 Physical Quantities .............................................................................................................. 4
1.3 Addition of Vectors ............................................................................................................. 4
1.4 Multiplication of vectors ..................................................................................................... 6
1.5 Component of a Vector ....................................................................................................... 6
1.6 Relationships Among Unit Vectors ˆi, ˆj, kˆ ........................................................................ 7
1.7 Solved Examples .................................................................................................................. 9
Chapter 2 Units and Significant Figures ................................................................................. 12
2.1 Introduction ........................................................................................................................ 12
2.2 Length ................................................................................................................................ 12
2.3 Mass ................................................................................................................................... 12
2.4 Time ................................................................................................................................... 13
2.5 Volume, Density, and Angle............................................................................................. 13
2.6 Significant Figures ............................................................................................................ 14
Chapter 3 Kinematics .............................................................................................................. 16
3.1 Position Vector and Displacement.................................................................................... 16
3.2 Newton’s Laws of Motion ................................................................................................. 19
3.3 Friction ............................................................................................................................... 21
3.4 Weight ................................................................................................................................ 22
3.5 Equilibrium of Forces ....................................................................................................... 22
3.6 Conditions of Equilibrium of a Particle ........................................................................... 22
3.7 Solved Examples ................................................................................................................ 23
Chapter 4 Rectilinear Motion ................................................................................................. 27
4.1 Rectilinear Motion with Constant Acceleration .............................................................. 27
4.2 Solved Examples on Rectilinear Motion ......................................................................... 28
5.1 Newton’s Law of Gravitation: It states .......................................................................... 33
5.2 Free-Fall under Gravitational Force............................................................................... 33
5.3 Gravitational Field ............................................................................................................ 34
5.4 Solved Examples ................................................................................................................ 34
Chapter 6 Motion in a plane................................................................................................... 39
6.1 Motion of a projectile ........................................................................................................ 39
6.2 Solved Examples on Projectiles ........................................................................................ 40
6.3 Solved Examples on Motion in a Circle .......................................................................... 46
Chapter 7 Work and Energy .................................................................................................. 51
7.1 Work .................................................................................................................................. 51
7.2 Power ................................................................................................................................. 51
7.3 Energy and its Conservation ............................................................................................. 52
7.4 Potential and Kinetic Energy ........................................................................................... 53
7.5 Transformation of Potential and Kinetic Energy ........................................................... 54
7.7 Gravitational Potential Energy ........................................................................................ 55
7.8 Elastic Potential Energy.................................................................................................... 55
7.9 Solved Examples on Work and Energy ........................................................................... 55
Chapter 8 Momentum and Impulse ........................................................................................ 61
8.1 Momentum ......................................................................................................................... 61
8.2 Conservation of Momentum ............................................................................................. 61
, 3
8.3 Center of Mass .................................................................................................................. 63
8.4 Collision Phenomena ......................................................................................................... 65
Chapter 9 Equilibrium............................................................................................................. 73
9.1 Translational Equilibrium................................................................................................. 73
9.2 Rotational Equilibrium ...................................................................................................... 73
9.3 Rotational Motion ............................................................................................................. 74
9.4 Motion of a Point on a Rotating Body............................................................................. 75
9.5 Moment of Inertia .............................................................................................................. 76
9.6 Kinetic Energy of Rotation ............................................................................................... 78
9.7 Dynamics of Pure Rotation ............................................................................................... 78
9.8 Solved Examples ................................................................................................................ 79
Chapter 10 Dynamical Systems ................................................................................................. 87
10.1 Momentum of a Dynamical System ................................................................................. 87
10.2 Conservation of Momentum ............................................................................................. 88
10.3 Angular Momentum of System of particles ..................................................................... 88
10.4 Rolling Motion of a Rigid Body ....................................................................................... 88
10.5 Solved Examples ................................................................................................................ 89
Chapter 11 Elasticity and Periodic Motion ................................................................................ 95
11.1 Elastic Properties ............................................................................................................... 95
11.2 Spring ................................................................................................................................. 96
11.3 Solved Examples on Elasticity .......................................................................................... 97
11.4 Periodic Motion ................................................................................................................. 98
11.5 The total energy of a particle executing simple harmonic motion ................................ 98
11.6 Simple Pendulum ............................................................................................................... 99
11.8 Examples on SHM .......................................................................................................... 100
, 4
Chapter 1 Review of Vectors
1.1 Mechanics
It is an area of physics which is concerned with the study of
motions of material particles. Mechanics is highly successful in
describing the motions of material bodies, such as planets, rockets
and baseballs.
Motion implies change in position. A change in position of
a point body is called ‘displacement’. Position of a particle is
specified by coordinates in a reference frame. A simple reference
frame is the Cartesian frame shown in Fig. 1.1, where the three
axes OX, OY, OZ are at right angles to each other.
1.2 Physical Quantities
The characteristics of a moving body are described in terms of some parameters
called “physical quantities”, like mass, length, speed, momentum, velocity, energy, etc.
Generally, we come across two types of physical quantities in physics.
Type 1: A physical quantity in which only ‘magnitude’ (a number with unit) is required to
specify it. Examples are: mass, length, speed, work, energy, power, etc. Such a physical
quantity is called ‘scalar’. A scalar is denoted by an algebraic quantity (number) like, A, B,
C, D etc.
Type 2: A physical quantity in which both ‘magnitude ’and the ‘direction’ are required to
specify it. Examples are: displacement, velocity, momentum, acceleration, etc. Such a
physical quantity is called ‘vector’.→ A vector is denoted by putting an arrow on top of an
algebraic number (magnitude) as in A, B,→ C→.
- Geometrically, a vector is represented by drawing a line whose
length is proportional to the magnitude of the vector and the
direction is indicated by drawing the line in the direction of the
vector with an arrow. The direction of the vector can be specified by
an angle it makes with the reference line OX. (Fig. 1.2)
→
- A vector A at a point P can be moved to any point Q by drawing a line
parallel to the line at P and of the same length as the line at P. (Fig. 1.3).
1.3 Addition of Vectors
→ → →
Vector→ A + Vector→ B = Vector→ R (resultant)
This can be written as, A+B=R (1.1)
→ →
- When vector A and vector B are parallel to each other, the resultant is simply,
→ → →
R = A + B, and the direction of Vector R is the same as that of vector A or vector B .
University
Notes
On
MECHANICS
(Physics 204)
, 2
Table of Contents
. ......................................................................................................................................................... 3
Chapter 1 Review of Vectors ...................................................................................................... 4
1.1 Mechanics ............................................................................................................................ 4
1.2 Physical Quantities .............................................................................................................. 4
1.3 Addition of Vectors ............................................................................................................. 4
1.4 Multiplication of vectors ..................................................................................................... 6
1.5 Component of a Vector ....................................................................................................... 6
1.6 Relationships Among Unit Vectors ˆi, ˆj, kˆ ........................................................................ 7
1.7 Solved Examples .................................................................................................................. 9
Chapter 2 Units and Significant Figures ................................................................................. 12
2.1 Introduction ........................................................................................................................ 12
2.2 Length ................................................................................................................................ 12
2.3 Mass ................................................................................................................................... 12
2.4 Time ................................................................................................................................... 13
2.5 Volume, Density, and Angle............................................................................................. 13
2.6 Significant Figures ............................................................................................................ 14
Chapter 3 Kinematics .............................................................................................................. 16
3.1 Position Vector and Displacement.................................................................................... 16
3.2 Newton’s Laws of Motion ................................................................................................. 19
3.3 Friction ............................................................................................................................... 21
3.4 Weight ................................................................................................................................ 22
3.5 Equilibrium of Forces ....................................................................................................... 22
3.6 Conditions of Equilibrium of a Particle ........................................................................... 22
3.7 Solved Examples ................................................................................................................ 23
Chapter 4 Rectilinear Motion ................................................................................................. 27
4.1 Rectilinear Motion with Constant Acceleration .............................................................. 27
4.2 Solved Examples on Rectilinear Motion ......................................................................... 28
5.1 Newton’s Law of Gravitation: It states .......................................................................... 33
5.2 Free-Fall under Gravitational Force............................................................................... 33
5.3 Gravitational Field ............................................................................................................ 34
5.4 Solved Examples ................................................................................................................ 34
Chapter 6 Motion in a plane................................................................................................... 39
6.1 Motion of a projectile ........................................................................................................ 39
6.2 Solved Examples on Projectiles ........................................................................................ 40
6.3 Solved Examples on Motion in a Circle .......................................................................... 46
Chapter 7 Work and Energy .................................................................................................. 51
7.1 Work .................................................................................................................................. 51
7.2 Power ................................................................................................................................. 51
7.3 Energy and its Conservation ............................................................................................. 52
7.4 Potential and Kinetic Energy ........................................................................................... 53
7.5 Transformation of Potential and Kinetic Energy ........................................................... 54
7.7 Gravitational Potential Energy ........................................................................................ 55
7.8 Elastic Potential Energy.................................................................................................... 55
7.9 Solved Examples on Work and Energy ........................................................................... 55
Chapter 8 Momentum and Impulse ........................................................................................ 61
8.1 Momentum ......................................................................................................................... 61
8.2 Conservation of Momentum ............................................................................................. 61
, 3
8.3 Center of Mass .................................................................................................................. 63
8.4 Collision Phenomena ......................................................................................................... 65
Chapter 9 Equilibrium............................................................................................................. 73
9.1 Translational Equilibrium................................................................................................. 73
9.2 Rotational Equilibrium ...................................................................................................... 73
9.3 Rotational Motion ............................................................................................................. 74
9.4 Motion of a Point on a Rotating Body............................................................................. 75
9.5 Moment of Inertia .............................................................................................................. 76
9.6 Kinetic Energy of Rotation ............................................................................................... 78
9.7 Dynamics of Pure Rotation ............................................................................................... 78
9.8 Solved Examples ................................................................................................................ 79
Chapter 10 Dynamical Systems ................................................................................................. 87
10.1 Momentum of a Dynamical System ................................................................................. 87
10.2 Conservation of Momentum ............................................................................................. 88
10.3 Angular Momentum of System of particles ..................................................................... 88
10.4 Rolling Motion of a Rigid Body ....................................................................................... 88
10.5 Solved Examples ................................................................................................................ 89
Chapter 11 Elasticity and Periodic Motion ................................................................................ 95
11.1 Elastic Properties ............................................................................................................... 95
11.2 Spring ................................................................................................................................. 96
11.3 Solved Examples on Elasticity .......................................................................................... 97
11.4 Periodic Motion ................................................................................................................. 98
11.5 The total energy of a particle executing simple harmonic motion ................................ 98
11.6 Simple Pendulum ............................................................................................................... 99
11.8 Examples on SHM .......................................................................................................... 100
, 4
Chapter 1 Review of Vectors
1.1 Mechanics
It is an area of physics which is concerned with the study of
motions of material particles. Mechanics is highly successful in
describing the motions of material bodies, such as planets, rockets
and baseballs.
Motion implies change in position. A change in position of
a point body is called ‘displacement’. Position of a particle is
specified by coordinates in a reference frame. A simple reference
frame is the Cartesian frame shown in Fig. 1.1, where the three
axes OX, OY, OZ are at right angles to each other.
1.2 Physical Quantities
The characteristics of a moving body are described in terms of some parameters
called “physical quantities”, like mass, length, speed, momentum, velocity, energy, etc.
Generally, we come across two types of physical quantities in physics.
Type 1: A physical quantity in which only ‘magnitude’ (a number with unit) is required to
specify it. Examples are: mass, length, speed, work, energy, power, etc. Such a physical
quantity is called ‘scalar’. A scalar is denoted by an algebraic quantity (number) like, A, B,
C, D etc.
Type 2: A physical quantity in which both ‘magnitude ’and the ‘direction’ are required to
specify it. Examples are: displacement, velocity, momentum, acceleration, etc. Such a
physical quantity is called ‘vector’.→ A vector is denoted by putting an arrow on top of an
algebraic number (magnitude) as in A, B,→ C→.
- Geometrically, a vector is represented by drawing a line whose
length is proportional to the magnitude of the vector and the
direction is indicated by drawing the line in the direction of the
vector with an arrow. The direction of the vector can be specified by
an angle it makes with the reference line OX. (Fig. 1.2)
→
- A vector A at a point P can be moved to any point Q by drawing a line
parallel to the line at P and of the same length as the line at P. (Fig. 1.3).
1.3 Addition of Vectors
→ → →
Vector→ A + Vector→ B = Vector→ R (resultant)
This can be written as, A+B=R (1.1)
→ →
- When vector A and vector B are parallel to each other, the resultant is simply,
→ → →
R = A + B, and the direction of Vector R is the same as that of vector A or vector B .
