Introduction to Mechanics

Mechanics

• Mechanics is the study of the motion of objects and the forces that cause that motion
• Kinematics describes motion without considering its causes, focusing on displacement, velocity, and acceleration
• Dynamics studies the forces that cause changes in motion, using Newton's laws of motion
• Newton's first law states that an object at rest stays at rest, and an object in motion stays in motion with the same velocity unless acted upon by a net force
• Newton's second law states that the net force on an object is equal to the mass of the object times its acceleration (F = ma)
• Newton's third law states that for every action, there is an equal and opposite reaction
• Work is the energy transferred to or from an object by a force acting on it, calculated as the force times the distance moved in the direction of the force (W = Fd cosθ)
• Energy is the capacity to do work, and exists in various forms, including kinetic energy (energy of motion) and potential energy (stored energy)
• Kinetic energy is given by KE = (1/2)mv^2, where m is mass and v is velocity
• Potential energy can be gravitational (PE = mgh, where m is mass, g is gravitational acceleration, and h is height) or elastic (PE = (1/2)kx^2, where k is the spring constant and x is the displacement from equilibrium)
• The law of conservation of energy states that energy cannot be created or destroyed, but can be transformed from one form to another
• Power is the rate at which work is done, or energy is transferred (P = W/t)
• Momentum is the product of mass and velocity (p = mv)
• The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it
• Impulse is the change in momentum of an object, equal to the force applied multiplied by the time interval over which it acts (J = FΔt = Δp)
• Rotational motion describes the motion of objects around an axis
• Angular displacement is the angle through which an object rotates
• Angular velocity is the rate of change of angular displacement
• Angular acceleration is the rate of change of angular velocity
• Torque is a twisting force that tends to cause rotation (τ = rFsinθ, where r is the distance from the axis of rotation to the point where the force is applied, F is the force, and θ is the angle between r and F)
• Moment of inertia is a measure of an object's resistance to changes in its rotational motion (I = Σmr^2 for discrete particles)
• Angular momentum is the product of moment of inertia and angular velocity (L = Iω)
• Simple harmonic motion (SHM) is a type of periodic motion where 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 SHM is the time it takes for one complete oscillation
• The frequency of SHM is the number of oscillations per unit time

Thermodynamics

• Thermodynamics is the study of heat and its relation to other forms of energy
• Temperature is a measure of the average kinetic energy of the particles in a substance
• Heat is the transfer of thermal energy between objects due to a temperature difference
• The zeroth law of thermodynamics states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other
• The first law of thermodynamics states that 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)
• The second law of thermodynamics states that the total entropy of an isolated system can only increase over time
• Entropy is a measure of the disorder or randomness of a system
• Heat engines convert thermal energy into mechanical work
• Refrigerators transfer heat from a cold reservoir to a hot reservoir, requiring work input
• Heat transfer can occur through conduction, convection, and radiation
• Conduction is the transfer of heat through a material by direct contact
• Convection is the transfer of heat by the movement of fluids (liquids or gases)
• Radiation is the transfer of heat by electromagnetic waves

Waves and Optics

• A wave is a disturbance that transfers energy through a medium or space
• Transverse waves oscillate perpendicular to the direction of propagation (e.g., light waves)
• Longitudinal waves oscillate 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
• Amplitude is the maximum displacement of a wave from its equilibrium position
• The speed of a wave is related to its wavelength and frequency by the equation v = fλ
• Superposition is the combination of two or more waves at the same point
• Interference is the phenomenon that occurs when waves superpose, resulting in constructive or destructive interference
• Diffraction is the bending of waves around obstacles or through openings
• Reflection is the bouncing back of a wave when it strikes a boundary
• Refraction is the change in direction of a wave as it passes from one medium to another
• Optics is the study of light and its behavior
• Reflection of light follows the law of reflection, which states that the angle of incidence equals the angle of reflection
• Refraction of light follows Snell's law, which relates the angles of incidence and refraction to the indices of refraction of the two media (n1sinθ1 = n2sinθ2)
• Lenses are used to focus or diverge light, forming images
• Converging lenses focus light to a point, while diverging lenses spread light out
• The focal length of a lens is the distance from the lens to the point where parallel rays of light converge (or appear to diverge from)
• The lens equation relates the object distance, image distance, and focal length of a lens (1/f = 1/do + 1/di)
• Interference and diffraction effects are prominent in the study of light, leading to phenomena like interference patterns and diffraction gratings

Electricity and Magnetism

• Electric charge is a fundamental property of matter that causes it to experience a force in an electromagnetic field
• Coulomb's law describes the force between two point charges (F = k(q1q2)/r^2, where k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between them)
• Electric field is the force per unit charge at a point in space
• Electric potential is the electric potential energy per unit charge at a point in space
• Capacitance is the ability of a system to store electric charge (C = Q/V, where Q is the charge and V is the voltage)
• Current is the rate of flow of electric charge (I = ΔQ/Δt)
• Resistance is the opposition to the flow of electric current (R = V/I, Ohm's Law)
• Ohm's law relates voltage, current, and resistance in a circuit (V = IR)
• Electric power is the rate at which electrical energy is converted into other forms of energy (P = VI = I^2R = V^2/R)
• Magnetism is a force caused by moving electric charges
• Magnetic field is the region around a magnet or current-carrying wire where a magnetic force is exerted
• The magnetic force on a moving charge is given by F = qvBsinθ, where q is the charge, v is its velocity, B is the magnetic field, and θ is the angle between v and B
• Electromagnetic induction is the production of an electromotive force (EMF) in a circuit due to a changing magnetic field
• Faraday's law of induction states that the EMF induced in a circuit is proportional to the rate of change of magnetic flux through the circuit (EMF = -N(dΦ/dt), where N is the number of turns in the coil and Φ is the magnetic flux)
• Lenz's law states that the direction of the induced current is such that it opposes the change in magnetic flux that produces it
• Electromagnetic waves are disturbances that propagate through space by the interplay of electric and magnetic fields
• Electromagnetic waves include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays

Modern Physics

• Modern physics encompasses the major breakthroughs in physics from the early 20th century onwards, including relativity and quantum mechanics
• Special relativity deals with the relationship between space and time
• One of the main postulates is that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source
• Time dilation and length contraction are consequences of special relativity
• Mass-energy equivalence is expressed by the famous equation E = mc^2, where E is energy, m is mass, and c is the speed of light
• General relativity extends special relativity to include gravity, describing it as the curvature of spacetime caused by mass and energy
• Quantum mechanics deals with the behavior of matter and energy at the atomic and subatomic levels
• Key concepts include quantization of energy, wave-particle duality, and the uncertainty principle
• The Heisenberg uncertainty principle states that it is impossible to simultaneously know both the position and momentum of a particle with perfect accuracy
• Atomic physics deals with the structure and properties of atoms
• Nuclear physics deals with the structure and properties of atomic nuclei
• Particle physics studies the fundamental particles and forces that make up the universe
• The Standard Model is a theory that describes the fundamental particles and forces of nature