Classical Mechanics: Kinematics and Dynamics

• Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, and stars.

Core Concepts

• Kinematics: Describes motion without considering its causes, focusing on displacement, velocity, and acceleration.
• Displacement is the change in position of an object.
• Velocity is the rate of change of displacement.
• Acceleration is the rate of change of velocity.
• Dynamics: Relates motion to its causes, mainly forces and torques.
• Force is an interaction that, when unopposed, will change the motion of an object.
• Torque is a twisting force that tends to cause rotation.
• Newton's Laws of Motion: Three fundamental laws that form the basis of classical mechanics.
  • First Law (Law of Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by a force.
  • Second Law: The acceleration of an object is directly proportional to the net force acting on the object, is in the same direction as the net force, and is inversely proportional to the mass of the object (F = ma).
  • Third Law: For every action, there is an equal and opposite reaction.
• Work and Energy: Concepts related to forces acting over a distance.
• Work is the energy transferred to or from an object by a force acting on the object.
• Energy is the capacity to do work.
• Kinetic energy is the energy of motion.
• Potential energy is the energy an object has due to its position or configuration.
• Conservation Laws: Fundamental principles stating that certain physical quantities remain constant over time.
  • Conservation of Energy: The total energy of an isolated system remains constant.
  • Conservation of Momentum: The total momentum of an isolated system remains constant.
  • Conservation of Angular Momentum: The total angular momentum of an isolated system remains constant.
• Reference Frames: Coordinate systems used to describe the position and motion of objects.
• Inertial reference frames are those in which Newton's first law holds true.
• Non-inertial reference frames are accelerating frames in which fictitious forces appear.

Key Equations

• Constant Acceleration Kinematics:
  • v = v₀ + at (velocity as a function of time)
  • Δx = v₀t + (1/2)at² (displacement as a function of time)
  • v² = v₀² + 2aΔx (velocity as a function of displacement)
• Newton's Second Law: F = ma (force equals mass times acceleration)
• Work: W = Fdcosθ (work done by a constant force)
• Kinetic Energy: KE = (1/2)mv²
• Potential Energy:
  • Gravitational PE: PE = mgh (near the Earth's surface)
  • Elastic PE: PE = (1/2)kx² (for a spring)
• Momentum: p = mv (momentum equals mass times velocity)
• Impulse: J = Δp = FΔt (impulse equals change in momentum)

Types of Forces

• Gravitational Force: The force of attraction between objects with mass.
  • F = Gm₁m₂/r² (Newton's Law of Universal Gravitation)
• Normal Force: The force exerted by a surface that supports the weight of an object.
• Friction: A force that opposes motion between surfaces in contact.
  • Static friction: Prevents motion from starting.
  • Kinetic friction: Opposes motion that is already occurring.
• Tension: The force transmitted through a string, rope, cable, or wire when it is pulled tight by forces acting from opposite ends.
• Spring Force: The force exerted by a spring when it is stretched or compressed.
  • F = -kx (Hooke's Law)

Work and Energy

• Work-Energy Theorem: The net work done on an object equals the change in its kinetic energy: W_net = ΔKE
• Power: The rate at which work is done: P = W/t = Fvcosθ
• Conservative Forces: Forces for which the work done is independent of the path taken. Examples include gravity and spring force.
• Non-Conservative Forces: Forces for which the work done depends on the path taken. Examples include friction and air resistance.

Momentum and Collisions

• Conservation of Momentum: In a closed system (no external forces), the total momentum before a collision equals the total momentum after the collision.
• Elastic Collisions: Collisions in which kinetic energy is conserved.
• Inelastic Collisions: Collisions in which kinetic energy is not conserved (some is converted to other forms of energy).
• Perfectly Inelastic Collisions: Collisions in which the objects stick together after the collision.

Rotational Motion

• Angular Displacement: The angle through which an object rotates.
• Angular Velocity: The rate of change of angular displacement.
• Angular Acceleration: The rate of change of angular velocity.
• Torque: A twisting force that causes rotation.
• Moment of Inertia: A measure of an object's resistance to rotational motion.
• Angular Momentum: A measure of an object's rotational momentum.
• Rotational Kinetic Energy: The kinetic energy of an object due to its rotation.
• Equations:
  • τ = rFsinθ (torque)
  • I = Σmr² (moment of inertia for a system of particles)
  • L = Iω (angular momentum)
  • KE_rotational = (1/2)Iω² (rotational kinetic energy)

Simple Harmonic Motion (SHM)

• Periodic motion in which the restoring force is proportional to the displacement.
• Examples: Mass-spring system, simple pendulum.
• Key Quantities:
  • Amplitude: The maximum displacement from equilibrium.
  • Period: The time for one complete cycle.
  • Frequency: The number of cycles per unit time.
• Equations:
  • x(t) = Acos(ωt + φ) (displacement as a function of time)
  • ω = √(k/m) (angular frequency for a mass-spring system)
  • ω = √(g/L) (angular frequency for a simple pendulum)
  • T = 2π/ω = 2π√(m/k) (period for a mass-spring system)
  • T = 2π/ω = 2π√(L/g) (period for a simple pendulum)

Waves

• A disturbance that transfers energy through a medium.
• Types: Transverse waves (displacement perpendicular to wave direction), longitudinal waves (displacement parallel to wave direction).
• Key Quantities:
  • Wavelength: The distance between two successive crests or troughs.
  • Frequency: The number of waves passing a point per unit time.
  • Amplitude: The maximum displacement of a wave from its equilibrium position.
  • Speed: The speed at which the wave propagates through the medium.
• Equation: v = fλ (wave speed equals frequency times wavelength)

Fluids

• Pressure: Force per unit area exerted by a fluid: P = F/A
• Density: Mass per unit volume: ρ = m/V
• Buoyancy: The upward force exerted by a fluid on an object immersed in it.
• Archimedes' Principle: The buoyant force on an object is equal to the weight of the fluid displaced by the object.
• Fluid Dynamics: The study of fluids in motion.
• Viscosity: A measure of a fluid's resistance to flow.
• Bernoulli's Equation: Relates pressure, velocity, and height for a fluid in motion: P + (1/2)ρv² + ρgh = constant

Thermodynamics

• Temperature: A measure of the average kinetic energy of the particles in a substance.
• Heat: The transfer of energy between objects due to a temperature difference.
• Specific Heat: The amount of heat required to raise the temperature of 1 kg of a substance by 1 degree Celsius.
• Laws of Thermodynamics:
  • Zeroth Law: If two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.
  • First Law: The change in internal energy of a system is equal to the heat added to the system minus the work done by the system: ΔU = Q - W
  • Second Law: The entropy of an isolated system always increases or remains constant.
  • Third Law: The entropy of a system approaches a minimum value as the temperature approaches absolute zero.

Important Considerations

• Units: Ensure consistency in units when applying equations. Use the SI system (meters, kilograms, seconds).
• Free-Body Diagrams: Draw free-body diagrams to visualize forces acting on an object.
• Assumptions: Be aware of the assumptions made when applying specific formulas or models (e.g., neglecting air resistance).
• Problem-Solving Strategies:
  • Read the problem carefully and identify what is being asked.
  • Draw a diagram if appropriate.
  • Identify relevant principles and equations.
  • Solve the equations algebraically before plugging in numbers.
  • Check your answer for reasonableness and correct units.