Centripetal force Totally Explained

Everything about The Centripetal totally explained

The centripetal force is the external force required to make a body follow a curved path. Hence centripetal force is a force requirement, not a particular kind of force, like gravity or electromagnetic force. Any force (gravitational, electromagnetic, etc.) (or combination of forces) can act to provide a centripetal force. An example for the case of uniform circular motion is shown in Figure 1. Centripetal force is directed inward, toward the center of curvature of the path. The term centripetal force comes from the Latin words centrum ("center") and petere ("tend towards", "aim at."), and can also be derived from Isaac Newton's original definitions described in Philosophiae Naturalis Principia Mathematica.
Centripetal force shouldn't be confused with centrifugal force (see the Common misunderstandings section below).

Simple example: uniform circular motion

The velocity vector is defined by the speed and also by the direction of motion. Objects experiencing no net force don't accelerate and, hence, move in a straight line with constant speed: they've a constant velocity. However, even an object moving in a circle at constant speed has a changing direction of motion. The rate of change of the object's velocity vector in this case is the centripetal acceleration. See Figure 1.
The centripetal acceleration varies with the radius R of the path and speed v of the object, becoming larger for greater speed (at constant radius) and smaller radius (at constant speed). If an object is traveling in a circle with a varying speed, its acceleration can be divided into two components, a radial acceleration (the centripetal acceleration that changes the direction of the velocity) and a tangential acceleration that changes the magnitude of the velocity. Below a number of examples of increasing complexity are discussed, and formulas for the motion are derived.

How is centripetal force provided?

Centripetal force is inferred from the trajectory of the object, without regard for how the path was arrived at (regardless of the origin of the forces involved). The studies of trajectories and the forces they imply is kinematics, while the study of which motions result from given physical forces is kinetics, the other branch of dynamics.
Supposing the analysis of a trajectory (kinematics) has concluded that for an object to follow the observed path a centripetal force is required, one might reasonably ask the (kinetic) question: "Where is the centripetal force coming from?"
For a satellite in orbit around a planet, the centripetal force is supplied by the gravitational attraction between the satellite and the planet, and acts toward the center of mass of the two objects. For an object at the end of a rope rotating about a vertical axis, the centripetal force is the horizontal component of the tension of the rope, which acts towards the center of mass between the axis of rotation and the rotating object. For a spinning object, internal tensile stress is the centripetal force that holds the object together in one piece.

Common misunderstandings

Centripetal force shouldn't be confused with centrifugal force. Centripetal force is a force requirement deduced from an observed trajectory, not a kinetic force like gravity or electrical forces. Centripetal force requirements may be deduced from a trajectory in any frame of reference (although the trajectory of an object and the deduced centripetal force will vary from one frame to another). Because centripetal force is inferred from an established trajectory, and isn't used to deduce a trajectory from a physical situation, centripetal force is not included in the inventory of forces that are used in applying Newton's laws F = m a to calculate a trajectory.
   Centrifugal force, on the other hand, is a fictitious force that arises only when motion is described or experienced in a rotating reference frame, and it doesn't exist in an inertial frame of reference. In both inertial frames and rotating frames of reference one uses Newton's laws of motion, such as F = ma, but inertial frames never use fictitious forces, while rotating frames must include fictitious forces that express the effects of rotation, in particular, the centrifugal force and the Coriolis force. Examples are provided in the article on centrifugal force.
   Centripetal force shouldn't be confused with central force, either. To reiterate, centripetal force is a force requirement necessary for a curved trajectory to be possible: it isn't a type of force, such as a nuclear or gravitational force. In contrast, central forces does refer to a type of force, more exactly to a class of physical forces between two objects that meet two conditions: (1) their magnitude depends only on the distance between the two objects and (2) their direction points along the line connecting the centers of these two objects. Examples of central forces include the gravitational force between two masses and the electrostatic force between two charges.
   As an example relating these terms, the centripetal force implied by the circular motion of an object often is provided by a central force.

Analysis of several cases

Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration.

Uniform circular motion

Uniform circular motion refers to the case of constant rate of rotation. Here are two approaches to describing this case.

Geometric derivation

The circle on the left in Figure 2 shows an object moving on a circle at constant speed at two different times in its orbit. Its position is given by R and its velocity is v.
The velocity vector v is always perpendicular to the position vector (since the velocity vector is always tangent to the R circle); thus, since R moves in a circle, so does v. The circular motion of the velocity is shown in the circle on the right of Figure 2, along with its acceleration a. Just as velocity is the rate of change of position, acceleration is the rate of change of velocity.
Since the position and velocity vectors move in tandem, they go around their circles in the same time T. That time equals the distance traveled divided by the velocity ť

T = frac
ight]   .

These results agree with those above for nonuniform circular motion. See also the article on non-uniform circular motion. If this acceleration is multiplied by the particle mass, the leading term is the centripetal force and the negative of the second term related to angular acceleration is sometimes called the Euler force.

Notes and references

Further Information

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