Kinetics and Motion

Kinetics

Introduction to Kinetics

  • Focuses on why we move, examining energy and its related concepts.

Fundamental Concepts in Motion

  • What generates or resists changes in motion:

    • Energy storage

    • Gravity

    • Inertia

    • Elasticity

    • Energy transfer:

    • Force of gravity

    • Contact and reaction forces

    • Energy dissipation:

    • Friction (multiple forms)

Historical Context

Sir Isaac Newton

  • Lived from 1642 to 1727.

  • Attended Cambridge as a student and later became a professor.

  • In 1687, he published the book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), containing 20+ years of mathematical and physical work, considered one of the most important texts in science.

Impact of Newton's Work

Newton's Principia Mathematica

  • Defines versions of kinematic and kinetic quantities.

  • States three laws of motion for linear motion:

    1. Law of Inertia

    2. Law of Acceleration

    3. Law of Reaction

  • Introduces the Universal Law of Gravitation and develops differential calculus to describe physical phenomena.

Contributions from Newton's Contemporaries

  • Various natural philosophers contributed significantly during Newton's time.

    • Descartes: Developed mechanical philosophy, momentum, coordinate systems, and graphs.

    • Kepler, Hooke, Halley: Investigated the mathematics of planetary orbits.

    • Huygens: Worked on kinematics and wave theory of light.

    • Leibniz: Explored kinetic energy (vis viva) and advanced calculus notation.

Energy Fundamentals

Definition of Energy

  • Energy (U): Defined as the capacity to do work, which is fundamental to the universe.

  • First Law of Thermodynamics: Energy cannot be created or destroyed.

Forms of Energy

  • Mechanical Forms:

    • Kinetic Energy (UK): Stored due to the inertia of a moving body.

    • Elastic Potential Energy (UE): Stored by the elasticity of a deformed body.

    • Gravitational Potential Energy (Ug): Stored by the mass of a body raised above the Earth.

  • SI Unit of Measure: Joules (J).

Work, Force, and Power

Definitions

  • Work (W): The energy transfer between bodies.

    • SI Unit: Joules (J).

  • Force (F): The agent responsible for energy transfer between bodies; it is a vector quantity that includes both magnitude and direction.

    • SI Unit: Newtons (N).

  • Power (P): The rate of doing work.

    • This is a scalar quantity (magnitude only).

    • SI Unit: Watts (W).

Mathematical Relationships

  • Work is calculated by
    W = F imes d
    where $d$ is the distance moved in the direction of the force.

  • Power is defined as P = rac{W}{ riangle t}

    • It can also be expressed as
      P = F imes v

    • Where $v$ is the velocity of the object.

    • Conversion:

    • 1 J = 1 N·m

    • 1 W = 1 J/s.

The Concept of Gravity

Newton's Law of Universal Gravitation

  • Introduced by Newton in his Principia, focusing on gravity at Earth's surface.

  • The gravitational force is directed downward toward the center of the Earth and is proportional to mass. If unbalanced, it causes a constant downward acceleration denoted as g = 9.81 ext{ m/s}^2

  • Weight: The force of gravity acting on an object, mathematically expressed as
    F_g = m imes g
    where $m$ is the object's mass.

Inertia and Motion

Definition of Inertia

  • Described as the resistance to acceleration and is a fundamental property of matter (mass).

  • It is the basis for Newton's first two laws of motion.

Momentum and Kinetic Energy

  • Momentum (L): Defined as
    L = m imes v
    where $m$ is mass and $v$ is velocity.

  • Kinetic Energy (UK): Defined as U_k = rac{1}{2} m imes v^2

    • Both momentum and kinetic energy are conserved quantities, especially in closed systems.

Newton's Laws of Motion

First Law - Law of Inertia

  • An inertial body retains a constant linear velocity unless acted upon by an unbalanced external force, illustrating that inertia resists acceleration.

Second Law - Law of Acceleration

  • A net (unbalanced) force on a body causes it to accelerate in the direction of the force, in proportion to the net force and inversely proportional to its inertia, mathematically expressed as
    extstyle egin{align} ext{Forces: } extstyle extstyle extsum F = m imes a ext{ (where } a ext{ is acceleration)} \ ext{(Total Force = mass times acceleration)} ext{ in a system.} \ ext{This principle applies in linear motion, and all bodies undergo this law.} \ ext{(Newton-Euler Equations for fixed inertia systems: } extsum Mp = Ip imes ext{α)} ext{ (where } ext{α is angular acceleration)} \end{align}

Third Law - Law of Reaction

  • Whenever one material body exerts a force on another material body, the second body exerts a force of equal magnitude and opposite direction back on the first body. This principle underscores interactions whenever objects collide or come into contact.

Contact and Interaction in Kinematics

Contact Forces

  • Contact defined by the proximity of bodies causing interactions that lead to acceleration and/or deformation.

  • In rigid body mechanics: Examines how loads cause acceleration.

  • In deformable body mechanics: Studies how loads cause deformation.

Adhesion

  • Adhesion causes contact objects to resist being pulled apart by tensile loads

    • Examples include how muscles pull on tendons and how tendons pull on bones.

    • Most injuries in muscles and tendons are failures of the adhesives rather than of the muscles or tendons themselves.

    • Relevant structures:

    • Entheses: Connections of tendons to bones.

    • Myotendon: Connections of muscles to tendons.

Impulse and Collisions

Impulse Definition

  • Impulse: Defined as the product of applied force and the duration of the force, denoted mathematically as:
    extstyle J = F imes riangle t

  • This principle leads to changes in momentum during collisions, where:
    extstyle riangle L = J

  • For collision dynamics, changes in momentum for each body involved are equal in magnitude and opposite in direction.

Friction and Its Types

Friction Defined

  • Friction is a resistance to motion that dissipates energy, converting it to heat, sound, etc.

  • Types of friction include:

    • External Friction: Occurs between objects (solids, fluids, gases) in contact, including:

    • Sliding Friction: Between sliding objects.

    • Rolling Resistance: For rolling objects.

    • Fluid Drag: For objects moving through fluids such as water and air.

    • Internal Friction: Described as viscosity, and occurs during deformation.

Angular Kinetics

Overview

  • Similar to linear kinetics, but inertia and the effect of forces differ significantly.

    • Newton did not fully address this in Principia, while Leonhard Euler later developed the needed concepts.

Moment of Inertia (I)

  • Defined as a body's resistance to angular acceleration, determined by its mass and spatial distribution relative to an axis of rotation.

  • A mass further from the axis must accelerate more to achieve the same angular velocity.

  • Units of Measure: kg·m².

Rotation Dynamics

Free Body Rotation

  • A linear force can induce a free body's rotation about its center of mass (CoM) only if the force is not aligned with the CoM.

Fixed Centre-of-Rotation

  • A linear force can also cause rotation about a fixed point if it is not in line with that point.

Moment and Torque

Definition

  • Moment (Torque, T): The measure of a force's tendency to cause rotation about a point or axis, expressed as: Mp = F imes dot where:

    • $d_ot$ is the perpendicular distance from the line of action to the center of rotation.

Example Calculation ( m_1 = 80 kg )

  • For a force ( F1 = 784 N ) at a perpendicular distance ( d{ot 1} = 0.5 m ):

    • Moment ( M_1 = 392 N·m ) counter-clockwise.

  • For ( m2 = 20 kg ), and ( F2 = 196 N ) at ( d_{ot 2} = 2 m ):

    • Moment ( M_2 = 392 N·m ) clockwise.

Muscle Moment Arms

Leverage of Muscles

  • Bones function as levers fixed at a fulcrum (joint), where the leverage of a muscle force depends on its moment arm around the joint it spans.

  • The moment arm is the perpendicular distance from the muscle's line of action to the center of rotation (CoR) of the joint, varying with joint position.

Importance of Understanding Kinetics

Why Study This Material?

  • Understanding kinetic variables can impact muscle strength (depends on moment arms), the effect of gravity (also influenced by moment arms), and overall biomechanics determined by 3-D anatomical positioning.

Summary of Kinetic Variables

  • Motion Types:

    • Linear Motion:

    • Inertia (m): Mass in kilograms (kg)

    • Load (F): Force in Newtons (N)

    • Work (W): Calculated as
      W = F imes d
      SI Unit: Joules (J)

    • Power (P): Given by
      P = rac{W}{t}
      SI Unit: Watts (W)

    • Angular Motion:

    • Moment of Inertia (Ip): in kg·m²

    • Moment of Force (Mp): in N·m

    • Work is calculated as:
      W = M_p imes riangle heta
      where ( riangle heta ) is angular displacement.

Restating Newton's Laws

Comprehensive Summary

  • Law of Inertia: Momentum remains constant unless affected by an unbalanced load.

  • Law of Acceleration: Rate of change of momentum relates to unbalanced load and is inversely proportional to inertia:
    extsum F = m imes a
    extsum Mp = Ip imes ext{α}

  • Law of Reaction: Every load interaction maintains equality in magnitude and opposite direction forces.

Conclusion

  • The study of kinetics encompasses understanding forces, motion responses, energy dynamics, and their various interrelationships, critical for fields like kinesiology and physical education.