Lecture Notes on Classical Mechanics

Contents

• 1. Kinematics

  • 1.1. Maths: a brief summary of vector algebra

  • 1.2. Radius vector, displacement, and velocity

  • 1.3. Acceleration

  • 1.4. Some basics of vector calculus, and formal solution of kinematical equations

  • 1.5. Summary

  • 1.6. Problems

• 2. Forces: First and Second Newton’s laws

  • 2.1. Newton’s First Law of Motion

  • 2.2. Newton’s Second Law of Motion

  • 2.3. Galilean invariance, inertial forces, and inertial frames

  • 2.4. Common phenomenological forces

  • 2.5. Summary

  • 2.6. Problems

• 3. Work and Energy

  • 3.1. Position-dependent force: motion in one-dimension

  • 3.2. Work-Energy Theorem

  • 3.3. Power

  • 3.4. Conservative forces and Potential Energy

  • 3.5. Conservation of energy

  • 3.6. Examples of Conservative Forces

  • 3.7. Non-conservative forces

  • 3.8. Energy Diagrams

  • 3.9. Summary

  • 3.10. Problems

• 4. Motion in plane using Cartesian and Polar coordinates

  • 4.1. Cartesian coordinates

  • 4.2. Polar coordinates

  • 4.3. Summary

  • 4.4. Problems

• 5. Third law of Newton, conservation of momentum, and centre of mass

  • 5.1. Newton’s Third Law of Motion

  • 5.2. Impulse

  • 5.3. Total momentum of a system of particles

  • 5.4. The Law of Conservation of Momentum

  • 5.5. Elastic and Inelastic Collisions

  • 5.6. Centre of mass

  • 5.7. Two body problem (central force motion) and reduced mass

  • 5.8. Collision energy

  • 5.9. Collisions in the centre of mass frame

  • 5.10. Summary

  • 5.11. Problems

• 6. Angular Momentum

  • 6.1. The Centre of Mass and Bodies in Motion

  • 6.2. Angular Momentum of a Point Particle

  • 6.3. Angular Momentum of a Particle in Circular Motion

  • 6.4. Angular Momentum of an extended body rotating about a fixed axis

  • 6.5. Orbital and Spin Angular Momentum

  • 6.6. Summary

  • 6.7. Problems

• 7. Torque

  • 7.1. Torque and angular momentum: point particle

  • 7.2. Torque and angular momentum: systems of particles and rigid bodies

  • 7.3. Centre of Gravity

  • 7.4. Dynamics of a Fixed Axis Rotation

  • 7.5. Kinetic Energy of a Body with a Fixed Axis of Rotation

  • 7.6. Summary

  • 7.7. Problems

• 8. Moment of Inertia

  • 8.1. Moment of Inertia of a Rigid Body

  • 8.2. The Parallel Axis Theorem

  • 8.3. Summation Rule for Moment of Inertia

  • 8.4. Motion involving translation and rotation

  • 8.5. Summary

  • 8.6. Problems

1. Kinematics

1.1. Maths: a brief summary of vector algebra

• Vectors: Defined by magnitude and direction.

• Position vector (𝒓), velocity (𝒗), acceleration (𝒂), and force (𝑭) are vectors.

• Unit vectors (𝒊̂, 𝒋̂, 𝒌̂) have a magnitude of 1 and are used for direction.

• A vector can be expressed as: 𝑨⃗ = 𝐴 ⋅ 𝐴̂

1.2. Radius vector, displacement, and velocity

• Radius vector (𝒓) represents the position of an object from a reference point.

• Displacement (Δ𝒓) is the change in position: Δ𝒓 = 𝒓₂ - 𝒓₁.

1.3. Acceleration

• Average velocity: 𝒗ₐᵥ = Δ𝒓/Δt

• Instantaneous velocity: 𝒗 = lim (Δt→0) (Δ𝒓/Δt).

• Acceleration is the rate of change of velocity: 𝒂 = d𝒗/dt.

1.4. Some basics of vector calculus, and formal solution of kinematical equations

• Derivatives of constant vectors are zero; derivatives of vector sums follow standard rules.

• Velocity obtained through integration of acceleration: 𝒗 = ∫𝒂(t) dt

1.5. Summary

• Displacement, velocity, and acceleration are time-dependent vectors.

• Velocity is tangent to the trajectory and acceleration defines how velocity changes.

1.6. Problems

• Example problems based on displacement, velocity, and acceleration relevant to kinematics.

2. Forces: First and Second Newton’s laws

2.1. Newton’s First Law of Motion

• A body remains at rest or in uniform motion unless acted upon by an external force.

2.2. Newton’s Second Law of Motion

• The rate of change of momentum of a body is proportional to the net force acting on it.

2.3. Galilean invariance, inertial forces, and inertial frames

• The form of Newton's laws is the same in all inertial reference frames.

2.4. Common phenomenological forces

• Contact forces, friction, tension, etc., are examples of phenomenological forces.

2.5. Summary

• Forces define motion according to Newton’s laws and describe the interaction between bodies.

2.6. Problems

• Example problems based on Newton's laws and common forces.

3. Work and Energy

3.1. Position-dependent force: motion in one-dimension

• Work done by a force as a function of the position of a particle.

3.2. Work-Energy Theorem

• Work done by net forces equals the change in kinetic energy.

3.3. Power

• Power is the rate of doing work: P = W/t.

3.4. Conservative forces and Potential Energy

• Potential energy is stored energy that can do work; derived from conservative forces.

3.5. Conservation of energy

• The total mechanical energy (kinetic + potential) is conserved in an isolated system.

3.6. Examples of Conservative Forces

• Potential energy examples and calculations.

3.7. Non-conservative forces

• Include friction and air resistance, which convert energy into non-recoverable forms.

3.8. Energy Diagrams

• Show the interplay between potential and kinetic energy.

3.9. Summary

• Differentiation of energy types and principles.

3.10. Problems

• Example calculations for work and energy.

4. Motion in plane using Cartesian and Polar coordinates

4.1. Cartesian coordinates

• Breaking down vectors into components makes solving problems easier.

4.2. Polar coordinates

• Useful for circular motion and related problems involving angles.

4.3. Summary

• Different coordinate systems allow efficient solutions to specific problems.

4.4. Problems

• Exercises based on motion in various coordinate systems.

5. Third law of Newton, conservation of momentum, and centre of mass

5.1. Newton’s Third Law of Motion

• For every action, there is an equal and opposite reaction.

5.2. Impulse

• The change in momentum is equivalent to the impulse applied to the object.

5.3. Total momentum of a system of particles

• Movement and interaction described through momentum conservation.

5.4. The Law of Conservation of Momentum

• Momentum remains constant in isolated systems.

5.5. Elastic and Inelastic Collisions

• Differences in energy conservation during collisions based on elasticity.

5.6. Centre of mass

• The average position weighted by mass.

5.7. Two body problem (central force motion) and reduced mass

• Simplifying two-body problems with central forces for analysis.

5.8. Collision energy

• Kinetic energy changes during collision events.

5.9. Collisions in the centre of mass frame

• Simplifying collisions by switching to the centre of mass reference frame.

5.10. Summary

• Recap of Newton's laws, momentum, conservation principles, and kinetic energy conservation.

5.11. Problems

• Example problems based on momentum and collision scenarios.

6. Angular Momentum

6.1. The Centre of Mass and Bodies in Motion

• Applying angular concepts to rigid bodies supports previous motion theories.

6.2. Angular Momentum of a Point Particle

• Defined as the cross product of position and linear momentum.

6.3. Angular Momentum of a Particle in Circular Motion

• Derivations and applications of angular momentum for circular scenarios.

6.4. Angular Momentum of an extended body rotating about a fixed axis

• Derivations for angular momentum calculations for extended objects.

6.5. Orbital and Spin Angular Momentum

• Understanding the contributions of different motion types to overall angular momentum.

6.6. Summary

• Recap of principles associated with angular momentum for systems.

6.7. Problems

• Exercises for angular momentum calculations and applications.

7. Torque

7.1. Torque and angular momentum: point particle

• Torque defined through angular momentum changes due to forces.

7.2. Torque and angular momentum: systems of particles and rigid bodies

• Understanding angular momentum within systems of particles; conservation rules apply.

7.3. Centre of Gravity

• The point about which gravitational torque balances out.

7.4. Dynamics of a Fixed Axis Rotation

• Establishing dynamics for rigid bodies rotating fixed axis; parallels with linear momentum.

7.5. Kinetic Energy of a Body with a Fixed Axis of Rotation

• Direct analogies between rotational and linear kinetic energy.

7.6. Summary

• Overview of torque, angular momentum, and energy relations.

7.7. Problems

• Exercises related to torque applications and consequences.

8. Moment of Inertia

8.1. Moment of Inertia of a Rigid Body

• Evaluating the moment of inertia, and summary of applications.

8.2. The Parallel Axis Theorem

• The moment of inertia for axes running parallel to a center of mass axis.

8.3. Summation Rule for Moment of Inertia

• Adding moments of inertia for multiple cylindrical sections in physical problems.

8.4. Motion involving translation and rotation

• Addressing the rolling motion concept in relation to translations.

8.5. Summary

• Summarizing key aspects of the moment of inertia and applications.

8.6. Problems

• Example evaluation based on moment of inertia applications.