Classical mechanics is a branch of physics that deals with the motion of objects under the influence of forces. It provides a framework to understand and predict the behavior of macroscopic objects, such as planets, cars, and projectiles. The fundamental principles of classical mechanics were developed by Sir Isaac Newton in the 17th century and laid the foundation for much of physics.
Newton's Laws of Motion:
- First Law (Law of Inertia): An object at rest tends to stay at rest, and an object in motion tends to stay in motion with the same velocity, unless acted upon by an external force.
- Second Law (Law of Acceleration): The rate of change of momentum of an object is directly proportional to the force applied to it. F = ma, where F is the force, m is the mass of the object, and a is the acceleration.
- Third Law (Law of Action and Reaction): For every action, there is an equal and opposite reaction.
Principle of Conservation of Energy: The total energy of a system remains constant if no external forces are acting on it. The energy can exist in various forms, such as kinetic energy (energy due to motion) and potential energy (energy stored in the position of an object).
Principle of Conservation of Momentum: The total momentum of a system remains constant if no external forces are acting on it. Momentum is the product of an object's mass and velocity.
Work-Energy Theorem: The work done on an object is equal to the change in its kinetic energy. Mathematically, W = ΔKE, where W is the work done and ΔKE is the change in kinetic energy.
Principle of Least Action (Hamilton's Principle): The path taken by a system between two points in space and time is the one that minimizes the action integral, which is the integral of the Lagrangian function over time.
Conservation of Angular Momentum: The total angular momentum of a system remains constant if no external torques are acting on it. Angular momentum is the product of an object's moment of inertia and angular velocity.
Lagrange's Equations: A set of equations that describe the motion of a system in terms of generalized coordinates and their time derivatives. These equations are derived from the principle of least action and provide an alternative formulation to Newton's laws.
Kepler's Laws of Planetary Motion: A set of three empirical laws derived by Johannes Kepler that describe the motion of planets around the Sun. These laws are based on observations and are foundational for celestial mechanics.
Conservation of Linear Momentum: The total linear momentum of a system remains constant if no external forces are acting on it. Linear momentum is the product of an object's mass and velocity.
Hooke's Law: The force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. F = -kx, where F is the force, k is the spring constant, and x is the displacement.
Conservation of Angular Momentum (for a rotating object): The angular momentum of an object rotating about a fixed axis remains constant if no external torques are acting on it. The angular momentum is the product of the moment of inertia and angular velocity.
Torque and Moment of Inertia: Torque is the rotational equivalent of force and is defined as the product of force and the perpendicular distance from the axis of rotation. The moment of inertia represents an object's resistance to rotational motion and depends on its mass distribution.
Centripetal Force: The force directed towards the center of a circular path that keeps an object moving in a curved trajectory. It is given by F = mv²/r, where F is the centripetal force, m is the mass of the object, v is its velocity, and r is the radius of the circular path.
D'Alembert's Principle: The principle states that in a system in equilibrium or in uniform motion, the sum of the applied forces and the inertial forces (due to acceleration) is zero.
Principle of Virtual Work: It states that the work done by all forces acting on a system is zero for any virtual displacement consistent with the constraints of the system.
Principle of Moments: The principle states that for an object in rotational equilibrium, the sum of the clockwise moments about any point is equal to the sum of the counterclockwise moments about the same point.
Conservation of Energy (for conservative forces): For a conservative force, the total mechanical energy of a system, which is the sum of the potential energy and kinetic energy, remains constant.
Rigid Body Dynamics: The study of the motion of rigid bodies (objects that do not deform under applied forces) and the forces acting on them.
Simple Harmonic Motion: The repetitive motion exhibited by a system when the restoring force is directly proportional to the displacement from the equilibrium position. It follows a sinusoidal pattern and is characterized by properties like amplitude, frequency, and period.
Equations of Motion: These are the differential equations that describe the motion of objects under the influence of forces. They can be derived from Newton's laws and are often used to solve specific problems in classical mechanics.
Conservation of Mechanical Energy (for non-conservative forces): In situations where non-conservative forces, such as friction or air resistance, are present, the total mechanical energy of a system is not conserved. The work done by non-conservative forces results in a change in mechanical energy.
Principle of Work and Energy: This principle states that the work done on an object is equal to the change in its total mechanical energy. It combines the concepts of work and energy to analyze the motion of objects.
Moment of Inertia: The moment of inertia is a measure of an object's resistance to changes in rotational motion. It depends on the mass distribution and the axis of rotation. For a point mass, the moment of inertia is given by I = mr², where m is the mass and r is the distance from the axis of rotation.
Rotational Kinetic Energy: The kinetic energy associated with the rotational motion of an object is given by K = (1/2)Iω², where I is the moment of inertia and ω is the angular velocity.
Conservation of Angular Momentum (for a system of particles): The total angular momentum of a system of particles remains constant if no external torques are acting on the system. It is the sum of the individual angular momenta of the particles.
Equilibrium Conditions: In a system in equilibrium, the net force and net torque acting on the system are both zero. This leads to conditions such as static equilibrium (no translational motion) and rotational equilibrium (no rotational motion).
Principle of Impulse and Momentum: The change in momentum of an object is equal to the impulse exerted on it. Impulse is defined as the product of the force applied to an object and the time interval over which the force acts.
Collisions: The study of collisions involves the analysis of the motion of objects before and after they come into contact. Different types of collisions, such as elastic collisions (where kinetic energy is conserved) and inelastic collisions (where kinetic energy is not conserved), are examined.
Torque and Angular Acceleration: Torque is the rate of change of angular momentum, and it is given by the product of the moment of inertia and the angular acceleration. It is analogous to force in linear motion.
Rigid Body Rotation: The study of the motion of rigid bodies about a fixed axis of rotation. Concepts such as angular displacement, angular velocity, and angular acceleration are used to describe rotational motion.
These additional theories and theorems in classical mechanics further expand our understanding of the principles governing the motion of objects and systems. Classical mechanics provides a robust framework for analyzing a wide range of physical phenomena, from simple particle motion to complex systems of interacting bodies.
Here are some of the fundamental formulas in classical mechanics:
Newton's Second Law of Motion:
F = ma
Where F is the net force applied to an object, m is its mass, and a is the resulting acceleration.
Kinematic Equations:
v = u + at
This equation relates initial velocity (u), final velocity (v), acceleration (a), and time (t).
s = ut + (1/2)at^2
This equation relates displacement (s), initial velocity (u), acceleration (a), and time (t).
v^2 = u^2 + 2as
This equation relates final velocity (v), initial velocity (u), acceleration (a), and displacement (s).
Work-Energy Theorem:
W = ΔKE
Where W is the work done on an object and ΔKE is the change in its kinetic energy.
Conservation of Mechanical Energy:
E = KE + PE
Where E is the total mechanical energy of an object, KE is its kinetic energy, and PE is its potential energy.
Law of Universal Gravitation:
F = G * (m1 * m2) / r^2
Where F is the gravitational force between two objects, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
Circular Motion:
Centripetal Force:
F = m * (v^2 / r)
This equation relates the centripetal force (F) acting on an object moving in a circle of radius (r), mass (m), and velocity (v).
Centripetal Acceleration:
a = (v^2 / r)
This equation relates the centripetal acceleration (a) of an object moving in a circle of radius (r) and velocity (v).
Linear Momentum:
p = mv
Where p is the momentum of an object, m is its mass, and v is its velocity.
Impulse-Momentum Theorem:
J = Δp
Where J is the impulse applied to an object and Δp is the change in its momentum.
Hooke's Law (Force exerted by a spring):
F = -kx
Where F is the force exerted by a spring, k is the spring constant, and x is the displacement from the equilibrium position.
Torque (Moment of force):
τ = r × F
Where τ is the torque, r is the position vector, and F is the force applied.
Angular Momentum:
L = Iω
Where L is the angular momentum of an object, I is its moment of inertia, and ω is its angular velocity.
Rotational Kinematics:
ω = ω₀ + αt
This equation relates initial angular velocity (ω₀), final angular velocity (ω), angular acceleration (α), and time (t).
θ = ω₀t + (1/2)αt²
This equation relates angular displacement (θ), initial angular velocity (ω₀), angular acceleration (α), and time (t).
ω² = ω₀² + 2αθ
This equation relates final angular velocity (ω), initial angular velocity (ω₀), angular acceleration (α), and angular displacement (θ).
Moment of Inertia:
For a point mass:
I = mr²
For a rigid body rotating about an axis:
I = ∫r² dm Where I is the moment of inertia, m is the mass, r is the distance from the axis of rotation, and dm is the differential mass element.
Conservation of Angular Momentum:
L₁ = L₂
Where L₁ is the initial angular momentum and L₂ is the final angular momentum of a system.
Elastic Collisions (Conservation of Momentum):
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Where m₁ and m₂ are the masses of the colliding objects, u₁ and u₂ are their initial velocities, and v₁ and v₂ are their final velocities.
Inelastic Collisions:
Conservation of Momentum:
m₁u₁ + m₂u₂ = (m₁ + m₂)v
Where m₁ and m₂ are the masses of the colliding objects, u₁ and u₂ are their initial velocities, and v is their final common velocity.
Coefficient of Restitution:
e = (v₂ - v₁) / (u₁ - u₂)
Where e is the coefficient of restitution, v₁ and v₂ are the final velocities of the objects, and u₁ and u₂ are their initial velocities.
These formulas should provide you with a broader understanding of classical mechanics. Remember to apply them in the appropriate context and adapt them to the specific problems you are solving.