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Semester 3: GENERAL MECHANICS AND CLASSICAL MECHANICS
Laws of Motion - Newton's Laws, Forces, Equations of Motion, Gravitation, Kepler's Laws
Laws of Motion
Newton's Laws of Motion
Newton's Laws describe the relationship between the motion of an object and the forces acting on it. The three laws are: 1) An object at rest stays at rest and an object in motion continues in motion unless acted upon by a net external force. 2) The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. 3) For every action, there is an equal and opposite reaction.
Forces
A force is a vector quantity that causes an object to undergo a change in speed, direction, or shape. Common types of forces include gravitational force, frictional force, tension, and normal force. Forces are measured in newtons and can be represented graphically using free-body diagrams.
Equations of Motion
The equations of motion describe the relationship between velocity, acceleration, time, and displacement for an object under uniform acceleration. The three key equations are: 1) v = u + at, 2) s = ut + 0.5at², 3) v² = u² + 2as, where u is the initial velocity, v is the final velocity, a is acceleration, s is displacement, and t is time.
Gravitation
Gravitation is a universal force that attracts two bodies towards each other. According to Newton's law of universal gravitation, the force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Kepler's Laws
Kepler's Laws describe the motion of planets around the sun. They include: 1) The orbit of a planet is an ellipse with the sun at one of the foci. 2) A line segment joining a planet and the sun sweeps out equal areas during equal intervals of time. 3) The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit.
Conservation Laws of Linear and Angular Momentum, Internal Forces and Momentum Conservation
Conservation Laws of Linear and Angular Momentum, Internal Forces and Momentum Conservation
Introduction to Conservation Laws
Conservation laws are fundamental principles in physics that describe the constancy of certain quantities in isolated systems. These include linear momentum, angular momentum, and energy.
Linear Momentum Conservation
The linear momentum of a system remains constant if no external forces act on it. This implies that the total momentum before an event is equal to the total momentum after the event, expressed mathematically as p_initial = p_final.
Angular Momentum Conservation
Angular momentum is conserved in a closed system where there are no external torques. The principle states that the total angular momentum L_initial = L_final, where angular momentum L is defined as the product of the moment of inertia and angular velocity.
Internal Forces and Momentum Conservation
Internal forces are forces that act between the components of a system. They do not affect the total momentum of the system, as action-reaction pairs exert force and do not change the overall momentum.
Applications of Momentum Conservation
Momentum conservation principles are applicable in various physical systems, including collisions in particle physics, movement of celestial bodies, and the behavior of fluids.
Examples and Problem Solving
To effectively apply conservation laws, one must analyze the scenarios such as elastic and inelastic collisions, where kinetic energy conservation can also be assessed along with momentum.
Conservation Laws of Energy - Work, Power, Energy, Conservative and Non-Conservative Forces
Conservation Laws of Energy
Conservation laws state that energy cannot be created or destroyed, only transformed from one form to another. This principle is fundamental in understanding mechanics and energy interactions within physical systems.
Work
Work is defined as the transfer of energy that occurs when a force is applied to an object and it moves a distance. Mathematically, work is expressed as W = F * d * cos(θ), where F is the force applied, d is the distance moved, and θ is the angle between the force and the direction of motion.
Power
Power is the rate at which work is done or energy is transferred over time. It is calculated as P = W / t, where P is power, W is work done, and t is the time taken. The SI unit of power is the watt (W).
Energy
Energy is the capacity to do work or produce change. It exists in various forms such as kinetic energy, potential energy, thermal energy, etc. The total energy in an isolated system remains constant, illustrating the conservation of energy principle.
Conservative Forces
Conservative forces are those that do not dissipate energy. The work done by these forces depends only on the initial and final positions, not the path taken. Examples include gravitational force and spring force. The potential energy associated with conservative forces can be easily defined.
Non-Conservative Forces
Non-conservative forces, such as friction and air resistance, do dissipate energy as heat or sound. The work done against non-conservative forces is path-dependent, meaning the energy lost depends on the route taken during the movement.
Rigid Body Dynamics - Translational and Rotational Motion, Angular Momentum, Moment of Inertia, Rotation About Fixed Axis
Rigid Body Dynamics
Translational Motion
Translational motion refers to the movement of a rigid body where all points of the body move in parallel paths. It is characterized by the body's center of mass, and the translational motion can be described using Newton's laws. In this context, the net force acting on the body results in acceleration, and the equations of motion can be applied.
Rotational Motion
Rotational motion involves the movement of a rigid body about an axis. It is described using angular displacement, angular velocity, and angular acceleration. The relationship between linear and angular motion is described by the equations that connect linear quantities with their rotational counterparts.
Angular Momentum
Angular momentum is a measure of the rotational motion of a body and is defined as the product of the moment of inertia and the angular velocity. It is a vector quantity and follows the principle of conservation, meaning the total angular momentum remains constant in an isolated system unless acted upon by an external torque.
Moment of Inertia
Moment of inertia quantifies a body's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation and is calculated using integrals for irregular shapes or by summing values for discrete particles. The greater the moment of inertia, the harder it is to change the body's speed or direction of rotation.
Rotation About Fixed Axis
When a rigid body rotates about a fixed axis, every point in the body moves in a circular path around that axis. The dynamics of such motion can be analyzed using torques and angular acceleration. The equations governing this motion reflect the relationships between torque, moment of inertia, and angular acceleration.
Lagrangian Mechanics - Generalized Coordinates, Principle of Virtual Work, Lagrange's Equation, Simple Pendulum
Lagrangian Mechanics
Generalized Coordinates
Generalized coordinates are a set of variables that uniquely define the configuration of a system relative to some reference configuration. They simplify the description of a system with constraints, allowing more flexibility than traditional Cartesian coordinates. Each generalized coordinate may encompass several physical dimensions or constraints.
Principle of Virtual Work
The Principle of Virtual Work states that for a system in equilibrium, the total virtual work done by the applied forces during any virtual displacement is zero. This principle provides a foundation for deriving equations of motion in a systematic way, especially when considering systems with constraints.
Lagrange's Equation
Lagrange's Equation is derived from the principle of least action and describes the dynamics of a system in terms of its generalized coordinates. The equation is expressed as d/dt(dL/d(q_dot)) - dL/dq = 0, where L is the Lagrangian function defined as the difference between kinetic and potential energy.
Simple Pendulum
A simple pendulum consists of a mass attached to a string of fixed length that swings back and forth under the influence of gravity. By applying Lagrange's equation, one can analyze the motion of the pendulum. The Lagrangian for the pendulum is L = T - V, where T is kinetic energy and V is potential energy. The resulting equation describes simple harmonic motion.
