Kinematics

Lecture Content Overview

Course Information

  • Course Title: Kinematics

  • Subject: Medical and Biological Physics

  • Lecturer: Associate Professor Andrianov A.A.

  • Contact Emails: aaandrianov@oiate.ru, aketaledu@gmail.com

  • Location: Obninsk

Table of Contents

  1. Mechanics

  2. Kinematics

  3. One Dimensional Motion

  4. SUVAT

  5. Three-Dimensional Motion

  6. Curvilinear Motion

  7. Problem Set

Mechanics

Definition

  • Mechanics is the science that studies how objects move under the influence of forces.

  • It focuses on the how rather than the why of motion, answering the question: "Given the forces, how do objects move?"

  • The mechanics taught in this course applies to macroscopic objects moving at speeds significantly less than the speed of light (c = 3 x 10^8 m/s).

Idealizations in Mechanics

Mechanics utilizes certain idealizations to simplify the analysis:

  1. Particle: Idealized as having mass but negligible size, allowing mechanics to simplify problems by ignoring body geometry.

  2. Rigid Body: Considers a collection of particles maintaining fixed distances regardless of external forces, simplifying the interaction analysis.

  3. Concentrated Force: Models the effect of loads acting at a specific point on a body, facilitating representation of loads in calculations.

Basic Quantities in Mechanics

The following concepts are foundational in mechanics:

  • Length: Used to denote position in space and describe physical system size.

  • Time: Seen as a succession of events occurring.

  • Mass: Quantifies the amount of matter, allowing comparison of the actions between different bodies.

  • Force: Defined as a push or pull exerted by one object on another, either through direct contact or at a distance.

Vectors and Scalars

Definitions

  • Vector: A quantity defined by both magnitude and direction (e.g., force, velocity).

  • Scalar: A quantity defined by magnitude alone (e.g., speed, mass).

Examples

  • Vectors include: Displacement, Velocity, Acceleration, Momentum, Force.

  • Scalars include: Distance, Speed, Time, Mass, Energy.

Representation of Vectors

  • Vectors can be visually represented in several ways:

    • Boldface Letters: e.g., v

    • Letters with Arrows: e.g., ( \vec{v} )

    • The direction of arrows illustrates the vector and their length indicates magnitude.

Kinematics

Definition

  • The word "kinema" means movement in Greek. Kinematics is the branch of physics that deals with the mathematical description of motion, including:

    • Position

    • Time Interval

    • Displacement

    • Velocity and Speed

    • Acceleration

Coordinate System

To describe a point's position in space, a coordinate system consists of:

  1. An origin (point of reference).

  2. A set of axes with scales and labels.

  3. Positive direction choices for each axis using unit vectors.

  4. Can be Cartesian, Polar, or Spherical.

Types of Motion

  • Basic types include:

    • Translatory Motion

    • Rotatory Motion

    • Vibratory Motion

  • Other forms include:

    • Linear, Random, Circular, Uniform, and Non-uniform motions.

Distance vs. Displacement

  • Distance (d): Total path length traveled, direction irrelevant.

  • Displacement (Δx): Shortest distance from initial to final position with direction considered.

Speed vs. Velocity

  • Speed (v): Magnitude of distance over time (scalar).

  • Velocity (v): Speed with direction (vector).

  • Average speed: calculated as distance/time.

  • Average velocity is the rate of displacement over time.

Acceleration

  • Defined as the rate of change of velocity.

  • Uniform Acceleration: Object changes velocity uniformly, e.g., free-falling bodies.

  • Acceleration due to gravity (g) is approximately 9.8 m/s² near Earth's surface.

One Dimensional Motion

Position and Movement

  • In one-dimensional motion, the position vector is a function of time. For example, a vector from the origin to the position of the object.

  • Displacement is calculated as the change in position over a specified time.

Average and Instantaneous Velocity

  • Average velocity: Given by displacement over the time interval.

  • Instantaneous velocity: The limit of average velocity as the time interval approaches zero, represented as the derivative of the position with respect to time.

Average and Instantaneous Acceleration

  • Average acceleration is derived from the change in velocity over time.

  • Instantaneous acceleration represents the change per unit time at a specific moment, also defined as the derivative of velocity with respect to time.

Integrals in Motion

  • Velocity is the integral of acceleration with respect to time, and position is the integral of velocity with respect to time. The areas under the graphs of acceleration vs. time and velocity vs. time represent changes in velocity and position, respectively.

SUVAT Equations

Overview

The SUVAT equations relate the displacement, initial and final velocities, acceleration, and time of an object under constant acceleration, providing a framework for solving kinematic problems without specifying the force.

Three-Dimensional Motion

Definition

In three-dimensional kinematics, position, velocity, and acceleration are represented in vector form and can be decomposed into components along the x, y, and z axes.

Curvilinear Motion

  • Describes movement along a curved path and is characterized by changes in both velocity magnitude and direction. It can be analyzed by breaking acceleration into tangential and normal components.

  • Tangential Acceleration (at): Accurate along the path indicates acceleration speed changes.

  • Normal Acceleration (an): Points to the center of curvature and relates to direction changes.

Homework Problems

The set of homework problems involves different scenarios applying kinematics concepts:

  1. Kinematics of a car and a truck under various motion conditions.

  2. Projectile motion involving stones and rockets.

  3. Analyzing particle motion based on mathematical displacement laws.