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are game-changers in physics. They're like reliable friends who always give you the same amount of , no matter which path you take. Gravitational and spring forces are prime examples.

These forces are tied to , making calculations easier. The work they do equals the negative change in potential energy. This relationship helps us understand in systems, a crucial concept in mechanics.

Conservative Forces

Characteristics of conservative forces

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  • Work done by conservative forces is independent of the path taken between initial and final positions
  • Work done by conservative forces on a closed path (starting and ending at the same point) is always zero
  • Examples of conservative forces include (Earth's gravity), (), and ()
  • Work done by a conservative force can be expressed as the negative change in potential energy W=−ΔUW = -\Delta U, where WW is work done and ΔU\Delta U is change in potential energy
  • Conservative forces are not path-dependent, unlike

Mathematical conditions for conservative forces

  • A force F⃗\vec{F} is conservative if it satisfies the following conditions:
    • of the force is zero ∇×F⃗=0\nabla \times \vec{F} = 0
    • Force can be expressed as the negative of a energy function UU: F⃗=−∇U\vec{F} = -\nabla U
  • These conditions ensure work done by the force is independent of the path taken and the force is conservative

Conservative forces and potential energy

  • Conservative forces are associated with potential energy in a system
  • Work done by a conservative force results in a change in the potential energy of the system
  • Potential energy function UU is related to the conservative force F⃗\vec{F} by F⃗=−∇U\vec{F} = -\nabla U
  • Change in potential energy between two points is equal to the negative of the work done by the conservative force ΔU=−W\Delta U = -W

Non-Conservative Forces

Conservative vs non-conservative forces

  • do not satisfy the conditions for conservative forces
    • Work done by non-conservative forces depends on the path taken between initial and final positions
    • Work done by non-conservative forces on a closed path is not always zero
  • Examples of non-conservative forces include:
    • - work done depends on the path taken and is always negative (dissipative)
    • Air resistance - dissipative force that depends on the path taken
  • In contrast, conservative forces (gravitational, spring) do not depend on the path taken and are not dissipative

Work done by conservative forces

  • When a conservative force acts on an object, work done can be calculated using the change in potential energy:
    1. For a gravitational force W=−ΔU=−mgΔhW = -\Delta U = -mg\Delta h, where mm is mass, gg is acceleration due to gravity, and Δh\Delta h is change in height
    2. For a spring force W=−ΔU=−12k(Δx)2W = -\Delta U = -\frac{1}{2}k(\Delta x)^2, where kk is spring constant and Δx\Delta x is change in spring's displacement
  • Total (kinetic + potential) remains constant in a system with only conservative forces
    • Known as ΔK+ΔU=0\Delta K + \Delta U = 0, where ΔK\Delta K is change in

Energy Conservation and Closed Loops

  • is a fundamental principle in systems with only conservative forces
  • In a , the done by conservative forces is zero
  • , such as friction, do not conserve energy in a system

Key Terms to Review (34)

Closed Loop: A closed loop refers to a system where the output is continuously fed back into the input, creating a circular or cyclic path of information and control. This concept is crucial in understanding the behavior of conservative and non-conservative forces in physics.
Conservation of Mechanical Energy: The conservation of mechanical energy is a fundamental principle in physics that states the total mechanical energy of an isolated system remains constant, it is said to be conserved. Mechanical energy is the sum of an object's potential energy and kinetic energy, and this total energy is maintained unless an external non-conservative force acts on the system.
Conservative Forces: Conservative forces are a type of force that does not depend on the path taken by an object, but only on its initial and final positions. These forces have the property that the work done by them on an object moving between two points is independent of the path taken by the object.
Coulomb's Law: Coulomb's law is a fundamental principle in electromagnetism that describes the force of interaction between two stationary electric charges. It states that the force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
Curl: Curl is a vector calculus operator that describes the infinitesimal rotation of a vector field around a given point. It measures the amount of twisting or spinning of a vector field, and is a fundamental concept in the study of electromagnetism and fluid dynamics.
Dissipative Forces: Dissipative forces are a type of non-conservative force that cause energy to be lost or dissipated from a system over time. These forces act to remove or diminish the mechanical energy of a system, often in the form of heat or sound energy.
Electrostatic Force: Electrostatic force is the attractive or repulsive force that exists between stationary electric charges. It is a fundamental force in nature that governs the interactions between charged particles and plays a crucial role in various physical phenomena.
Energy conservation: Energy conservation is the principle stating that the total energy in an isolated system remains constant over time. Energy can neither be created nor destroyed, only transformed from one form to another.
Energy Conservation: Energy conservation is the fundamental principle that states the total energy of an isolated system is constant; it is said to be conserved over time. This means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another.
Exact differential: An exact differential is a differential form that can be expressed as the gradient of some scalar function. It indicates a path-independent process in thermodynamics and mechanics.
F = -∇U: The equation F = -∇U represents the relationship between the force (F) acting on an object and the gradient of the potential energy (U) of the system. This equation is a fundamental principle in classical mechanics and is particularly relevant in the context of conservative and non-conservative forces.
Friction: Friction is a force that opposes the relative motion between two surfaces in contact. It arises due to the microscopic irregularities on the surfaces, which create resistance to sliding or rolling. Friction is a fundamental concept in physics that plays a crucial role in various topics, including solving problems, understanding forces, and analyzing energy transformations.
Gradient: The gradient of a quantity is a vector field that points in the direction of the greatest rate of increase of that quantity, and whose magnitude is the rate of change in that direction. It is a fundamental concept in physics, particularly in the study of conservative and non-conservative forces.
Gravitational Force: Gravitational force is the attractive force that exists between any two objects with mass. It is the force that causes objects to be pulled towards each other, and is the fundamental force responsible for the motion of celestial bodies and the behavior of objects on Earth.
Hooke's Law: Hooke's law is a fundamental principle in physics that describes the linear relationship between the force applied to an elastic object and the resulting deformation or displacement of that object. It is a crucial concept that underpins the understanding of various physical phenomena, including work, conservative and non-conservative forces, potential energy diagrams and stability, stress, strain, and elasticity, as well as simple harmonic motion.
Joule: A joule is the SI unit of work or energy, equivalent to one newton-meter. It represents the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.
Joule: The joule (J) is the standard unit of energy in the International System of Units (SI). It represents the amount of work done or energy expended when a force of one newton acts through a distance of one meter.
Kinetic energy: Kinetic energy is the energy possessed by an object due to its motion. It depends on the mass and velocity of the object.
Mechanical energy: Mechanical energy is the sum of kinetic energy and potential energy in a system. It is the energy associated with the motion and position of an object.
Mechanical Energy: Mechanical energy is the sum of the kinetic energy and potential energy possessed by an object due to its motion and position within a physical system. It represents the total energy available to do work or cause change in the system.
Net work: Net work is the total work done on an object, accounting for all forces acting on it. It determines the change in the object's kinetic energy.
Newton: Newton is the standard unit of force in the International System of Units (SI), named after the renowned English physicist and mathematician, Sir Isaac Newton. It is a fundamental unit that is essential in understanding and describing the behavior of objects under the influence of various forces, as well as in the study of mechanics, dynamics, and other related areas of physics.
Non-conservative forces: Non-conservative forces are forces where the work done depends on the path taken. Examples include friction and air resistance, which convert mechanical energy into heat or other forms of non-recoverable energy.
Non-Conservative Forces: Non-conservative forces are forces that do not satisfy the work-energy theorem, meaning the work done by these forces depends on the path taken by the object rather than just the initial and final positions. Unlike conservative forces, non-conservative forces can change the total mechanical energy of a system.
Path Dependence: Path dependence is a concept that describes how the set of decisions one faces for any given circumstance is limited by the decisions one has made in the past, even though past circumstances may no longer be relevant. It suggests that the trajectory of change in a system is heavily influenced by the system's own history.
Potential Energy: Potential energy is the stored energy possessed by an object due to its position or state, which can be converted into kinetic energy or other forms of energy when the object is moved or transformed. This term is central to understanding various physical phenomena and the conservation of energy.
Scalar Potential: Scalar potential is a scalar field that describes the potential energy per unit charge at a given point in space. It is a fundamental concept in electromagnetism and is closely related to the concepts of conservative and non-conservative forces.
Spring Force: Spring force is a type of force that arises due to the deformation of an elastic object, such as a spring. It is a conservative force that stores and releases energy as the object is stretched or compressed, and it plays a crucial role in the concepts of conservative and non-conservative forces, as well as the conservation of energy.
W = -½k(Δx)²: The term W = -½k(Δx)² represents the work done by a conservative force, specifically a spring force, when a system undergoes a displacement Δx. This expression is derived from the potential energy formula for a spring, and it describes the amount of work required to compress or extend a spring from its equilibrium position.
W = -mgΔh: W = -mgΔh is an equation that represents the work done by the force of gravity on an object as it undergoes a change in height. The term 'W' represents the work done, 'm' is the mass of the object, 'g' is the acceleration due to gravity, and 'Δh' is the change in height of the object.
W = -ΔU: The work done by a conservative force on a system is equal to the negative change in the system's potential energy. This relationship is a fundamental principle in physics that connects the concepts of work and potential energy.
Work: Work is a physical quantity that describes the energy transferred by a force acting on an object as the object is displaced. It is the product of the force applied and the displacement of the object in the direction of the force.
ΔK + ΔU = 0: The principle that the change in kinetic energy (ΔK) plus the change in potential energy (ΔU) of a system is equal to zero. This relationship is a fundamental concept in the study of conservative and non-conservative forces, as it describes the conservation of energy within a closed system.
ΔU = -W: The equation ΔU = -W represents the relationship between the change in a system's internal energy (ΔU) and the work done on or by the system (W). This equation is a fundamental principle in the study of thermodynamics and energy transformations.
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Glossary
Glossary
8.3 Conservation of Energy