The centripetal force is the external force required to make a body follow a circular path at constant speed. The force is directed inward, toward the center of the circle. Hence it is a force requirement, not a particular kind of force. Any force (gravitational, electromagnetic, etc.) can act as a centripetal force. The term centripetal force comes from the Latin words centrum ("center") and petere ("tend towards"). Gravity is a force of attraction that acts between bodies that have mass. ... In physics, the electromagnetic force is the force that the electromagnetic field exerts on electrically charged particles. ... Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome. ...

The centripetal force always acts perpendicular to the direction of motion of the body. In the case of an object that moves along a circular arc with a changing speed, the net force on the body may be decomposed into a perpendicular component that changes the direction of motion (the centripetal force), and a parallel, or tangential component, that changes the speed. Illustration of tangential and normal components of a vector to a surface. ... Illustration of tangential and normal components of a vector to a surface. ...

Basic formula

The velocity vector is defined by the speed and also by the direction of motion. Objects experiencing no net force do not accelerate and, hence, move in a straight line with constant speed: they have a constant velocity. However, 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 is the centripetal acceleration. The velocity of an object is its speed in a particular direction. ...

The centripetal acceleration varies with the radius r of the circle and speed v of the object, becoming larger for greater speed and smaller radius. More precisely, the centripetal acceleration is given by

where ω = v / r is the angular velocity. The negative sign indicates that the direction of this acceleration is towards the center of the circle, i.e., opposite to the position vector r. (We assume that the origin of r is the center of the circle.) Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...

By Newton's second law of motion F = ma, a physical force F must be applied to a mass m to produce this acceleration. The amount of force needed to move at speed v on a circle of radius r is: Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

where the formula has been written in several equivalent ways; here, is the unit vector in the r direction and ω is the angular velocity vector. Again, the negative sign indicates that the direction of the force is inwards, towards the center of the circle and opposite to the direction of the radius vector r. If the applied force is less or more than F_c, the object will "slip outwards" or "slip inwards", moving on a larger or smaller circle, respectively. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

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. This article or section does not cite its references or sources. ...

Examples

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. An Earth observation satellite, ERS 2 For other uses, see Satellite (disambiguation). ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ... Horizontal plane is used in radio to plot a antennas relative field strength (which directly affects a stations coverage area) on a polar graph. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ... Tensile stress (or tension) is the stress state leading to expansion; that is, the length of a material tends to increase in the tensile direction. ...

Common misunderstandings

Centripetal force should not be confused with centrifugal force. The centrifugal force is a fictitious force that arises from being in a rotating reference frame. To eliminate all such fictitious forces, one needs to be in a non-accelerating reference frame, i.e., in an inertial reference frame. Only then can one safely use Newton's laws of motion, such as F = ma. Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... A fictitious force is an apparent force that acts on all masses in a non-inertial frame of reference, e. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

Centripetal force should not be confused with central force, either. Central forces are 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 centres of these two objects. Examples of central forces include the gravitational force between two masses and the electrostatic force between two charges. The centripetal force maintaining an object in circular motion is often a central force. A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector from the origin to the object. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... Gravity is a force of attraction that acts between bodies that have mass. ... Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... Electrostatics (also known as Static Electricity) is the branch of physics that deals with the forces exerted by a static (i. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ...

Geometric derivation

Figure 1: The position and velocity vectors both move in a circle.

The circle on the left in Figure 1 shows an object moving on a circle at constant speed at four different times in its orbit. Its position is given by R and its velocity is v. Image File history File links Centripetal_derivation_circles. ...

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 1, 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

and, by analogy,

Setting these two equations equal and solving for a, we get

Comparing the two circles in Figure 1 also shows that the acceleration points toward the center of the R circle. For example, in the left circle in Figure 1, the position vector R pointing at 12 o'clock has a velocity vector v pointing at 9 o'clock, which (switching to the circle on the right) has an acceleration vector a pointing at 6 o'clock. So the acceleration vector is opposite to R and toward the center of the R circle.

Derivation using calculus

Another derivation strategy is to use a polar coordinate system, assume a constant radius, and differentiate twice. A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...

Let R(t) be a vector that describes the position of a point mass as a function of time. Since we are assuming uniform circular motion, let R(t) = r·u_r, where r is a constant (the radius of the circle) and u_R is the unit vector pointing from the origin to the point mass. This direction is described by θ, the angle between the x-axis and the unit vector, measured counterclockwise from the x-axis. In terms of cartesian unit vectors in the x and y directions (i and j respectively): A point mass in physics is an idealisation of a body whose dimensions can be neglected compared to the distances of its movement. ... In physics, circular motion is rotation along a circle: a circular path or a circular orbit. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

u_R = cos(θ)i + sin(θ)j

Note: unlike cartesian unit vectors, which are constant, in polar coordinates the direction of the unit vectors depend on θ, and so in general have non-zero time derivatives. This article describes some of the common coordinate systems that appear in elementary mathematics. ...

We differentiate to find velocity:

where ω is the angular velocity dθ/dt, and u_θ is the unit vector that is perpendicular to u_R and points in the direction of increasing θ. In Cartesian terms, u_θ = −sin(θ)i + cos(θ)j.

This result for the velocity is good because it matches our expectation that the velocity should be directed around the circle, and that the magnitude of the velocity should be ωR. Differentiating again, we find that the acceleration, a is:

Thus, the radial component of the acceleration is:

a_R = −ω²r

See also

• Centrifugal force
• Circular motion
• Coriolis force

Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... In physics, circular motion is rotation along a circle: a circular path or a circular orbit. ... In the inertial frame of reference (upper part of the picture), the black object moves in a straight line. ...

References & External Links

• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers, 6th ed., Brooks/Cole. ISBN 0-534-40842-7. 
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics, 5th ed., W. H. Freeman. ISBN 0-7167-0809-4. 
• Centripetal force vs. Centrifugal force, from an online Regents Exam physics tutorial by the Oswego City School District