What is Conservative Force?
- Conservative force formula
- What is Non-Conservative Force?
- Work Done by a Conservative Force
- Work Done by Gravitational Force
- Difference Between Conservative Force and Non-conservative Force
Conservative and Non-Conservative Forces FAQs

Conservative and Non-Conservative Forces

Q: What are conservative and non-conservative forces?

Conservative Forces : These forces perform work that is independent of the path taken, relying solely on the initial and final positions. They are associated with potential energy, and the total mechanical energy (kinetic + potential) remains conserved in their presence. Non-Conservative Forces : The work done by these forces depends on the path taken between two points. They often convert mechanical energy into other forms, such as heat or sound, leading to a loss of mechanical energy in the system.

Q: How to know if a force is conservative or non-conservative?

To determine if a force is conservative: Path Independence : If the work done by the force between any two points is independent of the path taken, the force is conservative. Closed Path Test : If the total work done by the force around any closed loop is zero, the force is conservative. Mathematically, this is expressed as: ∮ F · d r = 0, where the integral is over a closed path. Potential Energy Function : A force is conservative if it can be derived from a potential energy function, meaning F = -∇U, where U is the potential energy.

Q: What is an example of a conservative field and a non-conservative field?

Conservative Field Example : The gravitational field is conservative. The work done by gravity on an object moving between two points depends only on the vertical displacement, not the path taken. Non-Conservative Field Example : A frictional force field is non-conservative. The work done by friction depends on the path taken; longer paths result in more work done against friction, leading to energy dissipation as heat.

According to the law of conservation of energy, in a closed system isolated from its surroundings, the total energy remains constant. Conservative forces follow this principle. In this article, we will explore the concepts of conservative and non-conservative forces.

What is Conservative Force?

A conservative force is a type of force where the work done by or against the force on an object moving between two points depends only on the initial and final positions, not the path taken. This property is directly linked to the law of conservation of energy, as energy within the system remains constant.

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Examples of Conservative Forces

Gravitational Force: Work done depends only on the height difference, not the path taken.
Electrostatic Force: Work done in moving a charge between two points depends only on the potential difference.
Spring Force: Work done by a spring depends only on the displacement from its equilibrium position.

Conservative force formula

The formula for a conservative force is derived from the relationship between the force and the potential energy associated with it. A force is conservative if it can be expressed as the negative gradient of a potential energy function.

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Formula

F = -∇U

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F: Conservative force vector.
U: Potential energy function.
∇U: Gradient of the potential energy, which represents the rate of change of U with respect to position.

What is Non-Conservative Force?

A non-conservative force is a type of force where the work done by or against the force depends on the path taken, not just the initial and final positions. Non-conservative forces dissipate mechanical energy from the system, usually as heat, sound, or deformation, and thus, the total mechanical energy is not conserved.

Examples of Non-Conservative Forces

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Frictional Force: Converts kinetic energy into heat, depending on the distance over which it acts.
Air Resistance: Opposes motion and dissipates energy as heat and sound.
Tension in a Rope or Cable: When there is relative motion, energy may be lost due to stretching or internal friction.
Viscous Forces: In fluids, they dissipate energy as heat.

Work Done by a Conservative Force

The work done by a conservative force depends only on the initial and final positions of the object and is independent of the path taken. This property is a defining characteristic of conservative forces and is closely related to the concept of potential energy.

Key Features of Work Done by a Conservative Force

Path Independence:

The work done by a conservative force moving an object between two points depends solely on the start and end points. Example: For gravity, the work done to move an object to a certain height depends only on the height difference, not the path.

Potential Energy Relationship:

Work done by a conservative force is equal to the negative change in potential energy (U) of the system:

W = -ΔU

For example, in a gravitational field:

W = - (U_final - U_initial) = - m g (h_final - h_initial)

Work Along a Closed Path:

For a closed loop, the total work done by a conservative force is zero:

∮ F ⋅ dr = 0

Energy Conservation:

In a system where only conservative forces act, the total mechanical energy (kinetic + potential) remains constant:

KE_initial + PE_initial = KE_final + PE_final

Work Done by Gravitational Force

Consider moving an object of mass m from height h₁ to h₂:

W = -ΔU = - m g (h₂ - h₁)

The work depends only on h₁ and h₂, not the path.

Difference Between Conservative Force and Non-conservative Force

The difference between conservative force and non-conservative force are given below-

Aspect	Conservative Force	Non-conservative Force
Definition	A force where the work done depends only on the initial and final positions, not the path taken.	A force where the work done depends on the path taken, not just the initial and final positions.
Energy Conservation	Mechanical energy is conserved in the presence of only conservative forces.	Mechanical energy is not conserved as it is dissipated (e.g., as heat or sound).
Work Done in a Closed Path	Work done over a closed path is zero.	Work done over a closed path is not zero.
Associated Potential Energy	Can be associated with a potential energy function.	Cannot be associated with a potential energy function.
Examples	Gravitational force, electrostatic force, spring force.	Frictional force, air resistance, viscous force.
Path Dependence	Path-independent; work depends only on displacement.	Path-dependent; work depends on the entire path.

Conservative and Non-Conservative Forces FAQs

What are conservative and non-conservative forces?

Conservative Forces: These forces perform work that is independent of the path taken, relying solely on the initial and final positions. They are associated with potential energy, and the total mechanical energy (kinetic + potential) remains conserved in their presence.
Non-Conservative Forces: The work done by these forces depends on the path taken between two points. They often convert mechanical energy into other forms, such as heat or sound, leading to a loss of mechanical energy in the system.

What is a conservative force with an example?

A conservative force is one where the work done in moving an object between two points depends only on those points, not the path taken. Example: The gravitational force is conservative. When lifting an object to a certain height, the work done against gravity depends only on the height difference, not the path taken. This work is stored as gravitational potential energy, which can be fully recovered if the object returns to its original position.

How to know if a force is conservative or non-conservative?

To determine if a force is conservative:

Path Independence: If the work done by the force between any two points is independent of the path taken, the force is conservative.
Closed Path Test: If the total work done by the force around any closed loop is zero, the force is conservative. Mathematically, this is expressed as: ∮ F · dr = 0, where the integral is over a closed path.
Potential Energy Function: A force is conservative if it can be derived from a potential energy function, meaning F = -∇U, where U is the potential energy.

What is an example of a conservative field and a non-conservative field?

Conservative Field Example: The gravitational field is conservative. The work done by gravity on an object moving between two points depends only on the vertical displacement, not the path taken.
Non-Conservative Field Example: A frictional force field is non-conservative. The work done by friction depends on the path taken; longer paths result in more work done against friction, leading to energy dissipation as heat.

What is Conservative Force?