Motion, Kinematics and Dynamics

• Mechanics is the branch of physics concerned with the motion of bodies under the action of forces, including the special case in which a body remains at rest.

Kinematics

• Kinematics describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move.
• Displacement refers to 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.
• Uniform motion implies constant velocity (zero acceleration).
• Non-uniform motion involves changing velocity (non-zero acceleration).

Dynamics

• Dynamics is the study of the forces that cause motion and changes in motion.
• Force is an interaction that, when unopposed, will change the motion of an object.
• Newton's first law states that 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.
• Newton's second law states that 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).
• Newton's third law states that when one object exerts a force on a second object, the second object simultaneously exerts a force equal in magnitude and opposite in direction on the first object.
• Mass is a measure of an object's resistance to acceleration.
• Weight is the force exerted on an object due to gravity (W = mg).

Work and Energy

• Work is done when a force causes a displacement.
• The work done by a constant force is defined as W = Fd cosθ, where F is the magnitude of the force, d is the magnitude of the displacement, and θ is the angle between the force and displacement vectors.
• Energy is the capacity to do work.
• 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 associated with the position or configuration of an object.
• Gravitational potential energy is PE = mgh, where h is the height above a reference point.
• Elastic potential energy is PE = 1/2 kx^2, where k is the spring constant and x is the displacement from the equilibrium position.
• The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.
• Power is the rate at which work is done or energy is transferred, P = W/t.

Momentum and Collisions

• Momentum is the product of mass and velocity, p = mv.
• Impulse is the change in momentum of an object, J = Δp = FΔt.
• The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it.
• In an elastic collision, both momentum and kinetic energy are conserved.
• In an inelastic collision, momentum is conserved, but kinetic energy is not.
• A perfectly inelastic collision is one in which the objects stick together after the collision.

Rotational Motion

• Angular displacement is the angle through which an object rotates.
• 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, τ = rFsinθ, 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 is a measure of an object's resistance to rotational acceleration, I = Σmr^2 for discrete particles or I = ∫r^2dm for continuous objects.
• Newton's second law for rotation is τ = Iα, where α is the angular acceleration.
• Rotational kinetic energy is KE = 1/2 Iω^2, where ω is the angular velocity.
• Angular momentum is the product of the moment of inertia and the angular velocity, L = Iω.
• The law of conservation of angular momentum states that the total angular momentum of a closed system remains constant if no external torques act on it.

Simple Harmonic Motion

• Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.
• 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.
• The amplitude of SHM is the maximum displacement from the equilibrium position.
• For a mass-spring system, the period is T = 2π√(m/k), where m is the mass and k is the spring constant.
• For a simple pendulum, the period is T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity.

Fluids

• Pressure is force per unit area, P = F/A.
• Density is mass per unit volume, ρ = m/V.
• Pascal's principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
• Archimedes' principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
• Buoyant force is the upward force exerted by a fluid on an object submerged in it.
• Fluid dynamics is the study of fluids in motion.
• Viscosity is a measure of a fluid's resistance to flow.
• Bernoulli's equation relates pressure, velocity, and height for an ideal fluid in steady flow.

Thermodynamics

• Thermodynamics is the study of energy and its transformations.
• Temperature is a measure of the average kinetic energy of the particles in a system.
• Heat is the transfer of energy due to a temperature difference.
• 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 entropy of an isolated system tends to increase.
• Entropy is a measure of the disorder or randomness of a system.
• Heat transfer can occur through conduction, convection, and radiation.

Waves

• A wave is a disturbance that transfers energy through a medium.
• Transverse waves are waves in which the displacement of the medium is perpendicular to the direction of propagation of the wave.
• Longitudinal waves are waves in which the displacement of the medium is parallel to the direction of propagation of the wave.
• Wavelength is the distance between two consecutive crests or troughs of a wave.
• Frequency is the number of waves that pass a given point per unit time.
• Amplitude is the maximum displacement of a wave from its equilibrium position.
• The speed of a wave is the product of its wavelength and frequency, v = λf.
• Interference occurs when two or more waves overlap.
• Diffraction is the bending of waves around obstacles or through openings.

Sound

• Sound is a longitudinal wave that travels through a medium.
• The speed of sound depends on the properties of the medium.
• Intensity is the power of a sound wave per unit area.
• The decibel scale is used to measure sound intensity level.
• The Doppler effect is the change in frequency of a sound wave due to the motion of the source or the observer.