conservative force

Examples of conservative force in the following topics:

• Fundamental Theorem for Line Integrals

  • In physics this theorem is one of the ways of defining a "conservative force."
  • By placing $\varphi$ as potential, $\nabla$ is a conservative field.
  • Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows.
  • The gradient theorem also has an interesting converse: any conservative vector field can be expressed as the gradient of a scalar field.
  • Therefore, electric force is a conservative force.

• Vector Fields

  • Vector fields are often used to model the speed and direction of a moving fluid throughout space, for example, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.
  • When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and, under this interpretation, conservation of energy is exhibited as a special case of the fundamental theorem of calculus.
  • A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that:

• Conservative Vector Fields

  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • Conversely, path independence is equivalent to the vector field's being conservative.
  • Conservative vector fields are also irrotational, meaning that (in three dimensions) they have vanishing curl.
  • Therefore, the curl of a conservative vector field $\mathbf{v}$ is always $0$.

• Physics and Engineering: Fluid Pressure and Force

  • Pressure ($p$) is force per unit area applied in a direction perpendicular to the surface of an object.
  • It relates the vector surface element (a vector normal to the surface) with the normal force acting on it.
  • The total force normal to the contact surface would be:
  • Using this expression, we can calculate the total force that the fluid pressure gives rise to:
  • This equation, for example, can be used to calculate the total force on a submarine submerged in the sea.

• Work

  • Forces may do work on a system.
  • For moving objects, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector) and the velocity vector of the point of application.
  • This scalar product of force and velocity is classified as instantaneous power delivered by the force.
  • Work is the result of a force on a point that moves through a distance.
  • If the force is always directed along this line, and the magnitude of the force is $F$, then this integral simplifies to:

• Stokes' Theorem

  • Electric field is a conservative vector field.

• Applications of Second-Order Differential Equations

  • In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, $F$, proportional to the displacement, $x$: $\vec F = -k \vec x \,$, where $k$ is a positive constant.
  • If $F$ is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency.
  • In many vibrating systems the frictional force $Ff$ can be modeled as being proportional to the velocity v of the object: $Ff = −cv$, where $c$ is called the viscous damping coefficient.
  • Driven harmonic oscillator: Driven harmonic oscillators are damped oscillators further affected by an externally applied force $F(t)$.

• Planetary Motion According to Kepler and Newton

  • From this, Newton defined the force acting on a planet as the product of its mass and acceleration.
  • Therefore, by Newton's law, every planet is attracted to the Sun, and the force acting on a planet is directly proportional to the mass and inversely proportional to the square of its distance from the Sun.
  • The green arrow represents the planet's velocity, and the purple arrows represent the force on the planet.

• Vectors in Three Dimensions

  • Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors.

• Differentiation and Rates of Change in the Natural and Social Sciences

  • Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body.