Classical Mechanics: Motion and Forces

• Classical mechanics describes the motion of macroscopic objects such as projectiles, machinery parts, spacecraft, planets, and stars.
• It relies on Newton's laws of motion.
• Lagrangian and Hamiltonian formalisms are included.
• It studies object motion under the influence of forces
• Space and time are considered absolute, with an infinite speed of light.

Core Concepts

• Kinematics describes motion without regard to its causes
  • Displacement, velocity, and acceleration are essential kinematic quantities.
  • Equations of motion show relations under constant acceleration.
• Dynamics connects motion to forces involved.
  • Newton's laws of motion are the base of dynamics.
    • First law: Objects remain at rest or in uniform motion unless a force acts upon them.
    • Second law: Net force equals mass times acceleration (F = ma).
    • Third law: Every action has an equal, opposite reaction.
• Work and Energy: Work is energy transfer.
  • Kinetic energy: Energy of motion (KE = 1/2 mv^2).
  • Potential energy: Stored energy based on position or configuration.
    • Gravitational potential energy: PE = mgh.
    • Elastic potential energy: PE = 1/2 kx^2.
  • Work-energy theorem: Work done equals change in kinetic energy.
  • Conservative forces: Work is path-independent (e.g., gravity, spring force).
  • Non-conservative forces: Work is path-dependent (e.g., friction).
• Momentum measures mass in motion (p = mv).
  • Conservation of momentum: Total momentum in a closed system stays constant.
  • Impulse is an object's change in momentum.
  • Collisions can be elastic (kinetic energy conserved) or inelastic (kinetic energy not conserved).
• Rotational Motion describes objects rotating around an axis.
  • Angular displacement, velocity, and acceleration are key.
  • Torque is the rotational equivalent of force.
  • Moment of inertia resists changes in rotational motion.
  • Angular momentum measures rotational inertia in motion.
  • Conservation of angular momentum: Total angular momentum in a closed system remains constant.

Newton's Laws

• First Law (Law of Inertia):
  • Objects at rest stay at rest; objects in motion stay in motion with the same speed and direction unless acted on by a net force.
  • Inertia is an object's resistance to changes in motion; mass measures inertia.
• Second Law (F = ma):
  • Acceleration is proportional to net force, in the same direction, and inversely proportional to mass.
  • F = net force.
  • m = mass.
  • a = acceleration.
• Third Law (Action-Reaction):
  • Every action has an equal, opposite reaction.
  • If object A exerts force on B, B exerts an equal, opposite force on A.
  • Forces act on different objects.

Work and Energy

• Work: Force multiplied by displacement in the direction of the force.
  • W = Fd cos(θ), where θ is the angle between force and displacement.
  • Work is a scalar, measured in joules (J).
• Kinetic Energy (KE): Energy from an object's motion.
  • KE = 1/2 mv^2, where m = mass, v = speed.
• Potential Energy (PE): Stored energy based on position/configuration.
  • Gravitational PE: PE = mgh (h = height above reference).
  • Elastic PE: PE = 1/2 kx^2 (k = spring constant, x = displacement from equilibrium).
• Conservation of Energy:
  • Total mechanical energy (KE + PE) stays constant in a closed system with only conservative forces.
  • Non-conservative forces convert mechanical energy into other forms (e.g., heat from friction).
• Power:
  • The rate of work done or energy transferred.
  • P = W/t (t = time interval).
  • Measured in watts (W).

Momentum and Collisions

• Momentum (p):
  • Mass in motion.
  • p = mv (mass x velocity).
  • Momentum is a vector.
• Impulse (J):
  • Change in momentum.
  • J = Δp = FΔt (force x time interval).
• Conservation of Momentum:
  • Total momentum remains constant in a closed system (no external forces).
  • m1v1i + m2v2i = m1v1f + m2v2f (i = initial, f = final).
• Collisions:
  • Elastic: Both momentum and kinetic energy are conserved.
  • Inelastic: Momentum is conserved, kinetic energy is not.
  • Perfectly inelastic: Objects stick together after the collision.

Rotational Motion

• Angular Quantities:
  • Angular displacement (θ): Angle of rotation.
  • Angular velocity (ω): Rate of change of angular displacement (ω = dθ/dt).
  • Angular acceleration (α): Rate of change of angular velocity (α = dω/dt).
  • Relationships are analogous to linear motion equations.
• Torque (τ):
  • Rotational equivalent of force, causing rotation.
  • τ = rFsin(θ), where r = distance from axis to force point, θ = angle between force and lever arm.
• Moment of Inertia (I):
  • Resistance to changes in rotational motion.
  • Depends on mass and its distribution relative to the rotation axis.
• Rotational Kinetic Energy:
  • KE_rot = 1/2 Iω^2.
• Angular Momentum (L):
  • Rotational inertia in motion.
  • L = Iω.
• Conservation of Angular Momentum:
  • Total angular momentum stays constant in a closed system.
• Newton's Second Law for Rotation:
  • τ_net = Iα (analogous to F = ma).