Classical Mechanics: Motion, Forces, and Newton's Laws

• Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, and stars.
• It explains and predicts a wide range of phenomena accurately.

Core concepts

• Space, time, mass, and force are fundamental concepts.
• Space and time are absolute and independent of the observer in Newtonian mechanics.
• Mass is the measure of an object's resistance to acceleration and is a scalar quantity.
• Force is an interaction that, when unopposed, will change the motion of an object and is a vector quantity.

Newton's Laws of Motion

• First law: 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. This is also known as the law of inertia.
• 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.

Kinematics

• Kinematics describes the motion of objects without considering the forces that cause the motion.
• Displacement is the change in position of an object.
• Velocity is the rate of change of displacement with respect to time.
• Acceleration is the rate of change of velocity with respect to time.
• Equations of motion under constant acceleration relate displacement, initial velocity, final velocity, acceleration, and time.

Work and Energy

• Work is done when a force causes a displacement of an object.
• Energy is the capacity to do work.
• Kinetic energy (KE) is the energy of motion (KE = 1/2 mv^2).
• Potential energy (PE) is stored energy due to position or configuration.
• Gravitational potential energy (GPE) is the energy stored when an object is raised against gravity (GPE = mgh).
• Elastic potential energy is the energy stored in a deformed elastic object, such as a spring.
• The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.

Conservation of Energy

• The total energy of an isolated system remains constant.
• Energy can be transformed from one form to another (e.g., potential to kinetic), but it cannot be created or destroyed.
• Conservative forces (e.g., gravity, spring force) are those for which the work done is independent of the path taken.
• Non-conservative forces (e.g., friction) are those for which the work done depends on the path taken, and mechanical energy is not conserved.

Momentum and Collisions

• Momentum is the product of an object's mass and velocity (p = mv).
• Impulse is the change in momentum of an object.
• The law of conservation of momentum states that the total momentum of an isolated system remains constant.
• Collisions can be elastic (kinetic energy is conserved) or inelastic (kinetic energy is not conserved).
• In a perfectly inelastic collision, objects stick together after colliding.

Rotational Motion

• Angular displacement is the change in angular position of a rotating object.
• Angular velocity is the rate of change of angular displacement with respect to time.
• Angular acceleration is the rate of change of angular velocity with respect to time.
• Torque is a twisting force that causes rotation.
• Moment of inertia is a measure of an object's resistance to rotational motion.
• Rotational kinetic energy is the energy of rotational motion (KE = 1/2 Iω^2).
• Angular momentum is the product of an object's moment of inertia and angular velocity.
• The law of conservation of angular momentum states that the total angular momentum of an isolated system remains constant.

Gravitation

• Newton's law of universal gravitation states that 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 them.
• Gravitational potential energy is the energy stored when an object is separated from another massive object, like a planet.
• Kepler's laws of planetary motion describe the motion of planets around the Sun.

Oscillations

• Simple harmonic motion (SHM) is a type of periodic motion in which the restoring force is proportional to the displacement from equilibrium.
• Examples of SHM include a mass-spring system and a simple pendulum.
• The period of oscillation is the time it takes for one complete cycle.
• The frequency of oscillation is the number of cycles per unit time.
• Damped oscillations are oscillations that decrease in amplitude over time due to energy dissipation.
• Forced oscillations occur when an external force is applied to an oscillating system.
• Resonance occurs when the frequency of the driving force is equal to the natural frequency of the system, resulting in a large amplitude of oscillation.

Waves

• A wave is a disturbance that propagates through space and time, transferring energy.
• Transverse waves are waves in which the displacement is perpendicular to the direction of propagation (e.g., light waves).
• Longitudinal waves are waves in which the displacement is parallel to the direction of propagation (e.g., sound waves).
• Wavelength is the distance between two successive crests or troughs of a wave.
• Frequency is the number of waves that pass a point per unit time.
• The speed of a wave is the product of its wavelength and frequency.
• Superposition is the phenomenon that occurs when two or more waves overlap.
• Interference is the result of the superposition of waves, which can be constructive or destructive.
• Diffraction is the bending of waves around obstacles or through openings.
• The Doppler effect is the change in frequency of a wave due to the motion of the source or the observer.