4.2 Kinematics and dynamics of particles and rigid bodies

Kinematics and dynamics of particles and rigid bodies are key concepts in engineering physics. They help us understand how objects move and why. From a car's wheels spinning to a satellite orbiting Earth, these principles explain it all.

We'll explore motion without and with forces, using Newton's laws and energy principles. We'll also dive into linear and angular momentum, crucial for analyzing collisions and rotations. These concepts form the backbone of mechanical engineering and physics.

Particle and Rigid Body Motion

Kinematics Fundamentals

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• Kinematics describes motion of objects without considering forces causing the motion
• Position, velocity, and acceleration serve as fundamental kinematic variables for describing motion in one, two, and three dimensions
• Kinematic equations for constant acceleration motion include:
  • $s = ut + \frac{1}{2}at^2$
  • $v = u + at$
  • $v^2 = u^2 + 2as$
  • $s = \frac{u+v}{2}t$
  • Where s represents displacement, u initial velocity, v final velocity, a acceleration, and t time
• Relative motion analysis describes motion of one object with respect to another moving object using vector addition of velocities
• Curvilinear motion (projectile motion and circular motion) requires vector calculus and parametric equations for full description

Rigid Body Kinematics

• Additional kinematic variables describe rotational motion of rigid bodies
  • Angular position
  • Angular velocity
  • Angular acceleration
• Relationship between linear and angular kinematic variables governed by equations:
  • $v = r\omega$
  • $a = r\alpha$
  • Where r represents radius, ω angular velocity, and α angular acceleration
• Examples of rigid body motion:
  • Rotating wheel (car wheel spinning)
  • Pendulum swinging (clock pendulum)

Applying Newton's Laws

Newton's Laws of Motion

• Newton's First Law states objects remain at rest or in uniform motion unless acted upon by external force
• Newton's Second Law ($F = ma$) relates net force acting on object to its mass and acceleration
• Newton's Third Law states for every action, there exists an equal and opposite reaction
• Free body diagrams visualize and analyze all forces acting on particle or rigid body
• Examples of Newton's Laws in action:
  • Book resting on table (First Law)
  • Rocket propulsion (Third Law)

Extended Applications

• Rigid body dynamics extend Newton's Second Law to include rotational dynamics:
  • $\tau = I\alpha$
  • Where τ represents torque, I moment of inertia, and α angular acceleration
• Principle of transmissibility states effect of force on rigid body remains independent of point of application along its line of action
• D'Alembert's principle analyzes dynamic systems by treating inertial forces as additional external forces
• Examples of extended applications:
  • Torque wrench tightening bolt
  • Gyroscope precession

Problem Solving in Dynamics

Force Analysis

• Friction forces (static and kinetic) modeled using empirical relationships
• Gravitational forces near Earth's surface treated as constant
• Newton's law of universal gravitation for larger scales:
  • $F = G\frac{m_1m_2}{r^2}$
• Spring forces modeled using Hooke's law:
  • $F = -kx$
  • Where k represents spring constant and x displacement from equilibrium
• Tension forces in ropes or cables analyzed assuming massless and inextensible properties
• Constraint forces (normal forces and tensions in rigid connections) determined through motion constraints and equilibrium conditions
• Examples of force analysis:
  • Block sliding down inclined plane (friction and gravity)
  • Bungee jumping (tension and gravity)

Energy and Momentum Methods

• Work-energy principles provide alternative methods for solving dynamics problems
  • Work-energy theorem
  • Conservation of energy
• Impulse and momentum methods analyze collisions and sudden changes in motion
  • Impulse-momentum theorem
  • Conservation of momentum
• Examples of energy and momentum methods:
  • Roller coaster design (conservation of energy)
  • Car crash analysis (impulse-momentum)

Linear vs Angular Momentum

Linear Momentum

• Linear momentum defined as:
  • $p = mv$
  • Where m represents mass and v velocity
• Conservation of linear momentum states total linear momentum of closed system remains constant without external forces
• Impulse equals change in linear momentum, crucial for analyzing collisions and impacts
• Examples of linear momentum:
  • Billiard ball collisions
  • Rocket propulsion

Angular Momentum

• Angular momentum for particle defined as:
  • $L = r \times p$
• Angular momentum for rigid body defined as:
  • $L = I\omega$
  • Where I represents moment of inertia and ω angular velocity
• Conservation of angular momentum states total angular momentum of closed system remains constant without external torques
• Parallel axis theorem and perpendicular axis theorem calculate moments of inertia for complex shapes or different rotation axes
• Center of mass and center of gravity concepts simplify rigid body motion analysis and apply conservation laws to particle systems
• Examples of angular momentum:
  • Figure skater spin
  • Satellite orientation control