Newtonian Mechanics

Newton's First Law of Motion, also known as the law of inertia, 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 an unbalanced force.

Newton's Second Law of Motion states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to the mass of the object. The equation for Newton's Second Law is F = ma, where F is the net force, m is the mass of the object, and a is the acceleration of the object.

Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. This means that when one object exerts a force on another object, the other object exerts a force of equal magnitude but opposite direction on the first object.

The force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The equation for the force of gravity is F = Gm1m2/r^2, where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

The momentum of an object is defined as the product of its mass and velocity. The equation for momentum is p = mv, where p is the momentum, m is the mass of the object, and v is the velocity of the object.

The work done by a force on an object is equal to the product of the force and the displacement of the object in the direction of the force. The equation for work is W = Fd, where W is the work done, F is the force, and d is the displacement of the object.

The kinetic energy of an object is equal to the work done to accelerate the object from rest to its current velocity. The equation for kinetic energy is K = 1/2mv^2, where K is the kinetic energy, m is the mass of the object, and v is the velocity of the object.

The potential energy of an object is equal to the work done to move the object from its current position to a reference position. The equation for potential energy is U = mgh, where U is the potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the reference position.

The total mechanical energy of an object is the sum of its kinetic energy and potential energy. The equation for total mechanical energy is E = K + U, where E is the total mechanical energy, K is the kinetic energy, and U is the potential energy.

The law of conservation of energy states that the total energy of a closed system remains constant. This means that energy can be transferred from one form to another, but it cannot be created or destroyed.

The period of a pendulum is the time it takes for the pendulum to make one complete swing. The equation for the period of a pendulum is T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

The potential energy stored in a spring is given by the equation U = 1/2kx^2, where U is the potential energy, k is the spring constant, and x is the displacement of the spring from its equilibrium position.

The natural frequency of a mass-spring system is the frequency at which the system will oscillate when it is disturbed from its equilibrium position. The equation for the natural frequency of a mass-spring system is f = 1/2π√(k/m), where f is the natural frequency, k is the spring constant, and m is the mass of the object.

The equation for the displacement of a simple harmonic oscillator is x = Acos(ωt + φ), where x is the displacement, A is the amplitude of the oscillation, ω is the angular frequency, t is the time, and φ is the phase angle.

The equation for the displacement of a damped harmonic oscillator is x = Ae^(-γt)cos(ωt + φ), where x is the displacement, A is the amplitude of the oscillation, γ is the damping coefficient, ω is the angular frequency, t is the time, and φ is the phase angle.