Central Force

Introduction

Central Force, in physics, refers to a force that acts on every particle of the system, with the same magnitude and direction. Central forces are encountered often in physics, particularly when studying celestial mechanics and electromagnetism, where they appear as gravity and Coulomb’s law, respectively.

In this tutorial, we will discuss what a central force is and how it works.

What is Central Force?

A central force is a force that emanates from a single source and acts on all particles in a system. This force acts as an attractor on other objects that are nearby, like gravity and electromagnetism. The term central refers to the fact that these forces can originate from the center of an object, rather than its edge or surface, just like the word center refers to the middle of something.

Example

One of the most well-known examples of a central force comes from Isaac Newton’s law of universal gravitation, which says that all objects with mass in the universe exert a gravitational force on one another that is directly proportional to their masses and inversely proportional to the square of their distances apart; these properties are common to any central force.

The concept of central forces was introduced by French mathematician Joseph-Louis Lagrange in 1788 as an extension of Newton’s laws of motion. In Lagrange’s formulation, forces are not dependent on where an object is located within a system; they only depend on its mass and position relative to other objects within that system.

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Equation of Central Force

The mathematical equation of central force is as follows:

F = F(r) r^

Where,

F = Conservative central force

r = Vector magnitude |r| is the distance to the center of force.

r^ = r/r

Here,

r^=r/r is a unit vector in the direction of r.

The central force is a conservative force, represented as follows:

F(r)=dU/drF(r)=dU/dr

Here,

F(r) is the magnitude of the central force.

U(r) is time-independent potential energy.

To result in the particle’s motion being uniformly circular under the central force, the centripetal force should be as follows:

mv2r=F(r)mv2r=F(r)

Where:

m = mass of the body

v = speed at which the centripetal force equation holds

r = Original Radius

Types of Motions in Central Force

The Central Force is comprised of two different types of motions:

• Bounded motion
• Unbounded motion.

Let’s have a look.

Bounded Motion

Bounded motion is a special property of a central force. It means that an object under its influence can only travel along a conic section. A conic section is any curve traced out by a plane slicing through a cone. The standard orbits of central forces are ellipses, parabolas, and hyperbolas. These three curves are all examples of conic sections.

Example

One example is the motion of the planets around the sun. All of these objects follow elliptical orbits because they are affected by a central force (gravity). In fact, every planet in our solar system follows an elliptical orbit around our star.

Unbounded Motion

During the initial and final phases of this motion, the distance between the two bodies or objects is infinite. The force of attraction that keeps them together is not constant; it varies with time. This type of motion was studied by Sir Isaac Newton and his contemporary Gottfried Leibniz.