Classical Mechanics: Core Concepts

• Classical mechanics is a branch of physics that describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, and stars.
• It describes the motion of these objects accurately, as long as two conditions are met: the object must be large enough to be visible with the naked eye, and its speed must be much slower than the speed of light.

Core Concepts

• Kinematics: Describes the motion of objects without considering the forces that cause the motion. It involves displacement, velocity, and acceleration.
• Dynamics: Studies the forces that cause motion and how they affect the motion of objects. It involves Newton's laws of motion.
• Statics: Deals with objects at rest, where the net force and net torque are zero.

Newtonian Mechanics

• Newtonian mechanics is formulated in terms of three laws of 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 a force.
• Second Law: The acceleration of an object is directly proportional to the net force acting on it, is in the same direction as the net force, and is inversely proportional to the mass of the object. F = ma, where F is the net force, m is the mass, and a is the acceleration.
• Third Law: For every action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B exerts a force of equal magnitude and opposite direction on object A.
• Mass is the measure of an object's inertia, or its resistance to acceleration.
• Force is an interaction that, when unopposed, will change the motion of an object.

Work and Energy

• Work: The work done by a force on an object is the product of the force and the displacement in the direction of the force. W = Fd cosθ, where W is work, F is the magnitude of the force, d is the magnitude of the displacement, and θ is the angle between the force and the displacement.
• Kinetic Energy: The energy an object possesses due to its motion. KE = (1/2)mv², where KE is kinetic energy, m is mass, and v is speed.
• Potential Energy: The energy an object possesses due to its position or configuration. Examples include gravitational potential energy (PE = mgh) and elastic potential energy (PE = (1/2)kx²).
• Conservation of Energy: In a closed system, the total energy remains constant. Energy can be transformed from one form to another, but it cannot be created or destroyed.

Momentum and Impulse

• Momentum: A measure of an object's mass in motion. p = mv, where p is momentum, m is mass, and v is velocity.
• Impulse: The change in momentum of an object. J = FΔt, where J is impulse, F is the average force, and Δt is the time interval over which the force acts. Impulse is also equal to the change in momentum: J = Δp.
• Conservation of Momentum: In a closed system, the total momentum remains constant. This is particularly useful in analyzing collisions.

Rotational Motion

• Angular Displacement: The angle through which an object rotates. Measured in radians.
• Angular Velocity: The rate of change of angular displacement. ω = dθ/dt, where ω is angular velocity, θ is angular displacement, and t is time.
• Angular Acceleration: The rate of change of angular velocity. α = dω/dt, where α is angular acceleration, ω is angular velocity, and t is time.
• Torque: A twisting force that causes rotation. τ = rFsinθ, where τ is torque, 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: A measure of an object's resistance to rotational acceleration. I = Σmr², where I is the moment of inertia, m is the mass of each particle, and r is the distance of each particle from the axis of rotation.
• Angular Momentum: A measure of an object's rotational momentum. L = Iω, where L is angular momentum, I is the moment of inertia, and ω is angular velocity.
• Conservation of Angular Momentum: In a closed system, the total angular momentum remains constant.

Gravitation

• Newton's Law of Universal Gravitation: The gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. F = Gm₁m₂/r², where F is the gravitational force, G is the gravitational constant, m₁ and m₂ are the masses of the two objects, and r is the distance between their centers.
• Gravitational Potential Energy: The potential energy of an object due to its position in a gravitational field. PE = -Gm₁m₂/r.

Oscillations and Waves

• Simple Harmonic Motion (SHM): A type of periodic motion where the restoring force is directly proportional to the displacement. Examples include a mass-spring system and a simple pendulum.
• Frequency: The number of oscillations per unit time. f = 1/T, where f is frequency and T is the period.
• Period: The time it takes for one complete oscillation.
• Amplitude: The maximum displacement from the equilibrium position.
• Waves: Disturbances that propagate through a medium or space, transferring energy without transferring matter.
• Transverse waves: Oscillations are perpendicular to the direction of wave propagation (e.g., light waves).
• Longitudinal waves: Oscillations are parallel to the direction of wave propagation (e.g., sound waves).
• Wavelength: The distance between two successive crests or troughs of a wave.
• Wave Speed: The speed at which a wave propagates. v = fλ, where v is wave speed, f is frequency, and λ is wavelength.

Fluid Mechanics

• Pressure: Force per unit area. P = F/A, where P is pressure, F is force, and A is area.
• Density: Mass per unit volume. ρ = m/V, where ρ is density, m is mass, and V is volume.
• Buoyancy: The upward force exerted by a fluid on an object immersed in it. Equal to the weight of the fluid displaced by the object (Archimedes' principle).
• Fluid Dynamics: Deals with fluids in motion.
• Viscosity: A measure of a fluid's resistance to flow.