Great Ormond Street Hospital for Children

Overview of Section A

• Duration: Next 4 weeks

• Topics Covered: Particle dynamics, kinematics, kinetics, and Newton's laws of motion.

• Importance of Silence: To cover content efficiently and allow for examples and problem-solving.

Key Concepts in Dynamics

1. Kinematics

• Definition: The study of motion without considering forces.

• Focus: Analyzing the motion of particles.

2. Kinetics

• Definition: The relationship between motion and the forces that cause it.

• Components: Newton's laws of motion, impulse and momentum, work and energy principles.

• Approaches: Different methods (Newton's laws, impulse, work-energy) applicable based on the problem type.

Problem-Solving Strategies

• No Unique Approach: Different problems may require different methods based on familiarity and problem type.

• Selection of Method: Understanding various approaches helps speed up problem-solving.

Rigid Body Dynamics (Part B)

• Progression: Similar methods as kinematics and kinetics, but with increased complexity.

Vectors in Engineering Dynamics

1. Definition of Vectors

• Characteristics: Defined by two quantities - direction and magnitude.

• Notation: Represented by letters with an underscore (e.g., a_) or by arrows on top (e.g., (\vec{a})).

• Unit Vectors: Vectors with a magnitude of 1, defined as vector divided by its magnitude.

2. Scalars vs. Vectors

• Scalars: Quantities described solely by magnitude (e.g., temperature).

• Vectors: Include both magnitude and direction (e.g., velocity).

Vector Operations

1. Addition and Subtraction

• Vector Addition:

  • Parallelogram Rule: Construct a parallelogram with two vectors to find the resultant vector.

  • Triangle Rule: Place vectors tip to tail to find the resultant.

• Vector Subtraction: Treat subtraction as addition of the negative vector.

2. Multiplication of Vectors

A. Dot Product (Scalar Product)

• Definition: Dot product = magnitude of A × magnitude of B × cos(theta).

• Result: Scalar quantity.

• Applications: Used in calculating work done by forces.

B. Cross Product (Vector Product)

• Definition: Cross product = magnitude of A × magnitude of B × sin(theta).

• Result: Vector quantity that is perpendicular to the plane formed by A and B.

• Applications: Used to find torque in rotational dynamics.

Basis Sets in Vectors

1. Concept of Basis Set

• Definition: A set of vectors that can describe all vectors in a space.

• 2D Basis: Two non-parallel vectors define a plane.

• 3D Basis: Three non-coplanar vectors needed to define the space.

2. Right-Hand Rule and Standard Basis

• Right-Hand Rule: Determine direction of cross product.

• Standard Basis in 3D: Unit vectors (i, j, k) aligned with x, y, z axes, providing orthonormal basis.

Introduction to Kinematics of Particles

1. Study of Motion

• Focus: Analyze motion without reference to the forces causing it (e.g., distance, displacement, speed).

• Difference from Kinetics: Kinetics relates motion to forces acting upon particles.

2. Definition of a Particle

• Concept: A particle is considered an object whose dimensions do not significantly affect the motion.

• Examples: Identifying everyday objects treated as particles based on their movement dynamics.

Practical Applications

• Use of Concepts in Real Scenarios: Understand how vectors influence motion in real-world contexts, such as airplanes or crumpled paper, to illustrate principles of dynamics.