An inward-directed radial force required to keep an object on a circular path; gravity, for example, provides a centripetal force, causing the Moon to orbit the Earth. Corresponding to a centripetal force is a centripetal acceleration due to that force, and acting in the same direction as it. For an object moving in a circle at a constant speed, the rate of change of velocity (the direction changes continuously, but magnitude is constant) is the centripetal acceleration and is directed towards the centre of the circle. The term centrifugal force, denoting an outward-acting force, is common in everyday use, but is best avoided. It is a fictitious force; such a force balancing the centripetal force is applicable only to rotating observers, for whom the object appears at rest.
The centripetal force is the external force required to make the body move in a circular path with uniform speed and directed towards the center. Any physical force (gravity, electrostatics, tension, friction, etc.) can be used to supply the centripetal force.
Basic idea
Objects moving in a straight line with constant speed have constant velocity and require no force to do so, since they experience no acceleration (see Inertia). More precisely, the centripetal acceleration is given by
where ω = v / r is the angular velocity. (We assume that the origin of is the center of the circle.)
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 exactly
where the formula has been written in several equivalent ways;
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 speed of the velocity.
Examples
For an orbiting satellite, the centripetal force is supplied by the gravitational attraction between the satellite and its primary, and acts toward the center of mass which lies in the satellite's primary.
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.
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 two objects. Examples of central forces include the gravitational force between two masses and the electrostatic force between two charges. Central forces are physical forces, whereas the centripetal force is not. However, central forces are often used to meet the centripetal force requirement.
Geometric derivation (without calculus)
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.
The velocity vector is always perpendicular to the position vector (since the velocity vector is always tangent to the circle);
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 circle. For example, in the left circle in Figure 1, the position vector pointing at 12 o'clock has a velocity vector pointing at 9 o'clock, which (switching to the circle on the right) has an acceleration vector pointing at 6 o'clock. In terms of Cartesian unit vectors where θ = ωt:
Note: unlike in cartesian coordinates where the unit vectors are constants, in polar coordinates the direction of the unit vectors depend on the angle between the x-axis and the point being described; the angle θ.
So we differentiate to find velocity:
where ω is the angular velocity (just a short way of writing dθ/dt), uθ is the unit vector that is perpendicular to ur that points in the direction of increasing θ. In cartesian terms: uθ = −sin(θ) ux + cos(θ) uy
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:
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