Classical Mechanics: Core Concepts

• Mechanics is the branch of physics concerned with the motion of bodies under the influence of forces, including the special case in which a body remains at rest.
• Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies.
• If the objects being examined are extremely small, it is necessary to introduce the other major sub-field of mechanics, quantum mechanics.
• A physicist uses mechanics to predict what will happen in a range of circumstances.

Core Concepts

• Space, time, mass, and force are core concepts in Classical Mechanics.
• Space and time are absolute: Space is conceived of as a three-dimensional Euclidean space, and time is independent of space.
• Mass is a measure of the quantity of matter in a body and is assumed to be an invariable constant.
• Force is the external agent that causes a body to accelerate.
• Particle: An object with negligible extent.
• External force: An influence on a mechanical system from an agent outside the system.
• Internal force: A force between two parts of the system.

Newton's Laws of Motion

• Newton's laws of motion are fundamental principles that relate the forces acting on an object to its motion.
• 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 an external 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.

Key Principles and Equations

• Momentum (p) is the product of an object's mass (m) and its velocity (v): p = mv.
• Conservation of Momentum: In a closed system, the total momentum remains constant if no external forces act on the system.
• Work (W) is the energy transferred to or from an object by a force acting on the object: W = Fd cos(θ), where F is the force, d is the displacement, and θ is the angle between the force and displacement vectors.
• Kinetic Energy (KE) is the energy possessed by an object due to its motion: KE = (1/2)mv^2.
• Potential Energy (PE) is the energy stored in an object due to its position or configuration:
• Gravitational PE: PE = mgh (where g is the acceleration due to gravity and h is the height).
• Elastic PE: PE = (1/2)kx^2 (where k is the spring constant and x is the displacement from equilibrium).
• Conservation of Energy: In a closed system, the total energy remains constant if no external forces do work on the system.
• Power (P) is the rate at which work is done or energy is transferred: P = W/t.

Types of Forces

• Gravitational Force: The attractive force between objects with mass.
• Electromagnetic Force: The force between objects with electric charge.
• Normal Force: The force exerted by a surface on an object in contact with the surface, perpendicular to the surface.
• Frictional Force: The force that opposes motion between surfaces in contact.
• Static friction: Prevents the start of motion.
• Kinetic friction: Opposes motion when objects are sliding.
• Tension Force: 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.

Work and Energy

• Work is the energy transferred to or from an object by a force acting on the object.
• The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.
• Energy can be transformed from one form to another, but the total energy in a closed system remains constant.
• Power is the rate at which work is done or energy is transferred.

Rotational Motion

• Angular Displacement (θ) is the angle through which an object rotates.
• Angular Velocity (ω) is the rate of change of angular displacement: ω = dθ/dt.
• Angular Acceleration (α) is the rate of change of angular velocity: α = dω/dt.
• Torque (τ) is the rotational equivalent of force: τ = rFsin(θ), where r is the distance from the axis of rotation to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force and the lever arm.
• Moment of Inertia (I) is a measure of an object's resistance to changes in its rotational motion.
• Rotational Kinetic Energy (KE_rot) is the energy possessed by an object due to its rotational motion: KE_rot = (1/2)Iω^2.
• Angular Momentum (L) is the rotational equivalent of linear momentum: L = Iω.
• Conservation of Angular Momentum: In a closed system, the total angular momentum remains constant if no external torques act on the system.

Oscillations and Waves

• Simple Harmonic Motion (SHM) is a type of periodic motion in which the restoring force is proportional to the displacement from equilibrium.
• Period (T) is the time it takes for one complete oscillation.
• Frequency (f) is the number of oscillations per unit time: f = 1/T.
• Amplitude (A) is the maximum displacement from equilibrium.
• Waves are disturbances that propagate through a medium or space, transferring energy without transferring mass.
• Wavelength (λ) is the distance between two consecutive points in a wave that are in phase.
• Wave Speed (v) is the speed at which a wave propagates: v = fλ.
• Transverse Waves: Waves in which the displacement is perpendicular to the direction of propagation.
• Longitudinal Waves: Waves in which the displacement is parallel to the direction of propagation.
• Superposition: When two or more waves overlap, the resulting displacement is the sum of the individual displacements.
• Interference: The phenomenon that occurs when two or more waves overlap.
• Diffraction: The spreading of waves as they pass through an opening or around an obstacle.
• Doppler Effect: The change in frequency of a wave due to the motion of the source or the observer.

Limitations of Classical Mechanics

• Classical mechanics is an excellent approximation for describing the motion of macroscopic objects at speeds much lower than the speed of light.
• Fails when dealing with:
• Extremely small objects (quantum mechanics is required).
• Objects moving at speeds approaching the speed of light (relativity is required).
• Very strong gravitational fields (general relativity is required).

Coordinate Systems

• Cartesian Coordinates (x, y, z): Use three mutually perpendicular axes to specify the position of a point in space.
• Polar Coordinates (r, θ): Use a radial distance (r) and an angle (θ) to specify the position of a point in a plane.
• Cylindrical Coordinates (r, θ, z): Extend polar coordinates to three dimensions by adding a z-coordinate.
• Spherical Coordinates (ρ, θ, φ): Use a radial distance (ρ), an azimuthal angle (θ), and a polar angle (φ) to specify the position of a point in space.

Frames of Reference

• Inertial Frame of Reference: A frame of reference in which an object at rest remains at rest and an object in motion continues to move with constant velocity unless acted upon by a force.
• Non-inertial Frame of Reference: A frame of reference that is accelerating or rotating with respect to an inertial frame. Fictitious forces (e.g., centrifugal force, Coriolis force) appear in non-inertial frames.