Centripetal force

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An object that moves in a circular path undergoes a continuous acceleration towards the center of the circle. The net force that causes this acceleration is called a centripetal force (from Latin centrum "center" and petere "tend towards"). This term refers to the effect of the force (namely, to maintain the circular motion of the object); the origin of the centripetal force can be anything that causes a force to exist. An object can travel in a circle with a constant speed only if the net force acting on it is a centripetal force. (And if the object is traveling in a circle with a varying speed, the component of the net force along the radius is the centripetal force.)

In the case of 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; in the case of 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 axis of rotation. In the case of a spinning object, internal tensile stress gives the centripetal force that keeps the objects together in one piece.

Centripetal force must not be mixed up with centrifugal force. In an inertial reference frame (not rotating or accelerating), the centripetal force accelerates a particle in such a way that it moves along a circular path. In a corotating reference frame, a particle in circular motion appears to have zero velocity, if the rotation is not accounted for. The centripetal force is exactly cancelled by a centrifugal force that in this approach appears as a fictitious force. Centripetal forces are according to Newtonian mechanics true forces, while centrifugal forces only appear relative to rotating frames.

Centripetal force must not be confused with central force either.

Objects moving in a straight line with constant speed also have constant velocity. However, an object moving in an arc with constant speed has a changing direction of motion. As velocity is a vector of speed and direction, a changing direction implies a changing velocity. The rate of this change in velocity is the centripetal acceleration. Differentiating the velocity vector gives the direction of this acceleration towards the center of the circle.

Contents 1 Formula 2 Derivation 3 See also 4 References

Formula

The centripetal acceleration is given by

<math> \mathbf{a}_c = - \frac{v^2}{r} \hat{\mathbf{r}} = - \frac{v^2}{r} \frac{\mathbf{r}}{r} = - \omega^2 \mathbf{r}</math>

By Newton's second law of motion, as there is an acceleration there has to be a force in the direction of the acceleration. This is the centripetal force, and is equal to:

<math> \mathbf{F}_c = - \frac{m v^2}{r} \hat{\mathbf{r}} = - \frac{m v^2}{r} \frac{\mathbf{r}}{r} = - m \omega^2 \mathbf{r}</math>

(where m is mass, v is velocity, r is radius of the circle, and the minus sign denotes that the vector points to the center of the circle and ω = v / r is the angular velocity). In vector notation we can write:

<math> \boldsymbol F_c = m \boldsymbol\omega \times (\boldsymbol\omega \times \boldsymbol r )</math>,

where <math>\boldsymbol\omega</math> is the angular velocity vector of the rotation and <math>\boldsymbol r</math> is a vector from an arbitrary point on the rotation axis to the body (with mass <math>m</math>).

Derivation

Simply use a polar coordinate system, assume a constant radius, and differentiate twice.

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. In terms of Cartesian unit vectors:

<math>u_r = cos(\theta)u_x + sin(\theta)u_y \, </math>

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:

<math>v = R \frac {du_r}{dt} \, </math>

<math>v = R \frac{d\theta}{dt} u_\theta \, </math>

<math>v = R \omega u_\theta \, </math>

where ω is the angular velocity (just a short way of writing dθ/dt), u_θ is the unit vector that is perpendicular to u_r that points in the direction of increasing θ. In cartesian terms: u_θ = -sin(θ) u_x + cos(θ) u_y

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:

<math>a = R \left( \frac {d\omega}{dt} u_\theta - \omega^2 u_r \right) \, </math>

Thus, the radial component of the acceleration is:

<math>a_r = -\omega^2 R \, </math>

See also

• Centrifugal force
• Circular motion
• Coriolis force

References

• {{cite book

| author = Serway, Raymond A.; Jewett, John W.
| title = Physics for Scientists and Engineers
| edition = 6th ed.
| publisher = Brooks/Cole
| year = 2004
| id = ISBN 0534408427
}}

• {{cite book

| author = Tipler, Paul
| title = Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics
| edition = 5th ed.
| publisher = W. H. Freeman | year = 2004
| id = ISBN 0716708094
}}cs:Dostředivá síla

de:Zentripetalkraft eo:Centrifuga forto es:Fuerza centrípeta fi:Sentripetaalivoima it:Forza centripeta ja:回転運動 ko:구심력 no:Sentripetalkraft pl:Siła dośrodkowa sl:Centripetalna sila sv:Centripetalkraft