Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path.^{[1]}^{[2]} The term centripetal force comes from the Latin words centrum ("center") and petere ("tend towards", "aim at"), signifying that the force is directed inward toward the center of curvature of the path. Isaac Newton's description was: "A centripetal force is that by which bodies are drawn or impelled, or in any way tend, towards a point as to a center."^{[3]}
Contents 
The magnitude of the centripetal force on an object of mass m moving at a speed v along a path with radius of curvature r is:^{[4]}
The direction of the force is toward the center of the square in which the object is moving, or the osculating circle, the circle that best fits the local path of the object, if the path is not circular.^{[5]} This force is also sometimes written in terms of the angular velocity ω of the object about the center of the circle:
For a satellite in orbit around a planet, the centripetal force is supplied by the gravitational attraction between the satellite and the planet. The gravitational force acts on each object toward the other, which is toward the center of mass of the two objects; for circular orbits, this center of gravity is the center of the circular orbits. For noncircular orbits or trajectories, only the component of gravitational force directed orthogonal to the path (toward the center of the osculating circle) is termed centripetal; the remaining component acts to speed up or slow down the satellite in its orbit.^{[6]} Alternatively, some sources, including Newton, refer to the entire gravitational force as centripetal, though it is not strictly centripetally directed when the orbit is not circular;^{[7]} the formulas above will not apply in such cases.
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 toward the axis of rotation. For a spinning object, internal tensile stress provides the centripetal forces that make the parts of the object move together in circular motions.
Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration.
Uniform circular motion refers to the case of constant rate of rotation. Here are two approaches to describing this case.
The circle in the left of Figure 2 shows an object moving on a circle at constant speed at two different times in its orbit. Its position is given by the vector R and its velocity by the vector v.
The velocity vector is always perpendicular to the position vector (since the velocity vector is always tangent to the circle of motion). 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
and, by analogy,
Setting these two equations equal and solving for a, we get
The angular rate of rotation in radians per second is:
Comparing the two circles in Figure 2 also shows that the acceleration points toward the center of the R circle. For example, in the left circle in Figure 2, 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.
Figure 3 shows the vector relationships for uniform circular motion. The rotation itself is represented by the vector Ω, which is normal to the plane of the orbit (using the righthand rule) and has magnitude given by:
with θ the angular position at time t. In this subsection, dθ/dt is assumed constant, independent of time. The distance traveled ℓ of the particle in time dt along the circular path is
which, by properties of the vector cross product, has magnitude rdθ and is in the direction tangent to the circular path.
Consequently,
In other words,
Differentiating with respect to time,
Lagrange's formula states:
Applying Lagrange's formula with the observation that Ω • r(t) = 0 at all times,
In words, the acceleration is pointing directly opposite to the radial displacement r at all times, and has a magnitude:
where vertical bars ... denote the vector magnitude, which in the case of r(t) is simply the radius R of the path. This result agrees with the previous section if the substitution is made for rate of rotation in terms of the period of rotation T:
When the rate of rotation is made constant in the analysis of nonuniform circular motion, that analysis agrees with this one.
A merit of the vector approach is that it is manifestly independent of any coordinate system.
Figure 4 shows a ball in circular motion on a banked curve. The curve is banked at an angle θ from the horizontal, and the surface of the road is considered to be slippery. The object is to find what angle the bank must have so the ball does not slide off the road.^{[8]} Intuition tells us that on a flat curve with no banking at all, the ball will simply slide off the road; while with a very steep banking, the ball will slide to the center unless it travels the curve rapidly.
Apart from any acceleration that might occur in the direction of the path, the right side of Figure 4 indicates the forces on the ball. There are two forces; one is the force of gravity vertically downward through the center of mass of the ball mg where m is the mass of the ball and g is the gravitational acceleration; the second is the upward normal force exerted by the road perpendicular to the road surface ma_{n}. The centripetal force demanded by the curved motion also is shown in Figure 4. This centripetal force is not a third force applied to the ball, but rather must be provided by the net force on the ball resulting from vector addition of the normal force and the force of gravity. The curved motion is maintained so long as this net force provides the centripetal force requisite to the motion.
The horizontal net force on the ball is the horizontal component of the force from the road, which has magnitude F_{h} = ma_{n}sinθ. The vertical component of the force from the road must counteract the gravitational force, that is F_{v} = ma_{n}cosθ = mg. Accordingly one finds the net horizontal force to be:
On the other hand, at velocity v on a circular path of radius R, kinematics says that the force needed to turn the ball continuously into the turn is the radially inward centripetal force F_{c} of magnitude:
Consequently the ball is in a stable path when the angle of the road is set to satisfy the condition:
or,
As the angle of bank θ approaches 90°, the tangent function approaches infinity, allowing larger values for v^{2}/R. In words, this equation states that for faster speeds (bigger v) the road must be banked more steeply (a larger value for θ), and for sharper turns (smaller R) the road also must be banked more steeply, which accords with intuition. When the angle θ does not satisfy the above condition, the horizontal component of force exerted by the road does not provide the correct centripetal force, and an additional frictional force tangential to the road surface is called upon to provide the difference. If friction cannot do this (that is, the coefficient of friction is exceeded), the ball slides to a different radius where the balance can be realized.^{[9]}^{[10]}
These ideas apply to air flight as well. See the FAA pilot's manual.^{[11]}
As a generalization of the uniform circular motion case, suppose the angular rate of rotation is not constant. The acceleration now has a tangential component, as shown in Figure 5. This case is used to demonstrate a derivation strategy based upon a polar coordinate system.
Let r(t) be a vector that describes the position of a point mass as a function of time. Since we are assuming 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. The direction of u_{r} is described by θ, the angle between the xaxis and the unit vector, measured counterclockwise from the xaxis. The other unit vector for polar coordinates, u_{θ} is perpendicular to u_{r} and points in the direction of increasing θ. These polar unit vectors can be expressed in terms of Cartesian unit vectors in the x and y directions, denoted i and j respectively:^{[12]}
and
We differentiate to find velocity:
where ω is the angular velocity dθ/dt.
This result for the velocity matches expectations that the velocity should be directed tangential to the circle, and that the magnitude of the velocity should be ωR. Differentiating again, and noting that
we find that the acceleration, a is:
Thus, the radial and tangential components of the acceleration are:
where v = Rω is the magnitude of the velocity (the speed).
These equations express mathematically that, in the case of an object that moves along a circular path with a changing speed, the acceleration of the body may be decomposed into a perpendicular component that changes the direction of motion (the centripetal acceleration), and a parallel, or tangential component, that changes the speed.
The above results can be derived perhaps more simply in polar coordinates, and at the same time extended to general motion within a plane, as shown next. Polar coordinates in the plane employ a radial unit vector u_{ρ} and an angular unit vector u_{θ}, as shown in Figure 6.^{[13]} A particle at position r is described by:
where the notation ρ is used to describe the distance of the path from the origin instead of R to emphasize that this distance is not fixed, but varies with time. The unit vector u_{ρ} travels with the particle and always points in the same direction as r(t). Unit vector u_{θ} also travels with the particle and stays orthogonal to u_{ρ}. Thus, u_{ρ} and u_{θ} form a local Cartesian coordinate system attached to the particle, and tied to the path traveled by the particle.^{[14]} By moving the unit vectors so their tails coincide, as seen in the circle at the left of Figure 6, it is seen that u_{ρ} and u_{θ} form a rightangled pair with tips on the unit circle that trace back and forth on the perimeter of this circle with the same angle θ(t) as r(t).
When the particle moves, its velocity is
To evaluate the velocity, the derivative of the unit vector u_{ρ} is needed. Because u_{ρ} is a unit vector, its magnitude is fixed, and it can change only in direction, that is, its change du_{ρ} has a component only perpendicular to u_{ρ}. When the trajectory r(t) rotates an amount dθ, u_{ρ}, which points in the same direction as r(t), also rotates by dθ. See Figure 6. Therefore the change in u_{ρ} is
or
In a similar fashion, the rate of change of u_{θ} is found. As with u_{ρ}, u_{θ} is a unit vector and can only rotate without changing size. To remain orthogonal to u_{ρ} while the trajectory r(t) rotates an amount dθ, u_{θ}, which is orthogonal to r(t), also rotates by dθ. See Figure 6. Therefore, the change du_{θ} is orthogonal to u_{θ} and proportional to dθ (see Figure 6):
Figure 6 shows the sign to be negative: to maintain orthogonality, if du_{ρ} is positive with dθ, then du_{θ} must decrease.
Substituting the derivative of u_{ρ} into the expression for velocity:
To obtain the acceleration, another time differentiation is done:
Substituting the derivatives of u_{ρ} and u_{θ}, the acceleration of the particle is:^{[15]}
As a particular example, if the particle moves in a circle of constant radius R, then dρ/dt = 0, v = v_{θ}, and:
These results agree with those above for nonuniform circular motion. See also the article on nonuniform 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.^{[16]}
For trajectories other than circular motion, for example, the more general trajectory envisioned in Figure 6, the instantaneous center of rotation and radius of curvature of the trajectory are related only indirectly to the coordinate system defined by u_{ρ} and u_{θ} and to the length r(t) = ρ. Consequently, in the general case, it is not straightforward to disentangle the centripetal and Euler terms from the above general acceleration equation.^{[17]} ^{[18]} To deal directly with this issue, local coordinates are preferable, as discussed next.
By local coordinates is meant a set of coordinates that travel with the particle, ^{[19]} and have orientation determined by the path of the particle.^{[20]} Unit vectors are formed as shown in Figure 7, both tangential and normal to the path. This coordinate system sometimes is referred to as intrinsic or path coordinates^{[21]}^{[22]} or ntcoordinates, for normaltangential, referring to these unit vectors. These coordinates are a very special example of a more general concept of local coordinates from the theory of differential forms.^{[23]}
Distance along the path of the particle is the arc length s, considered to be a known function of time.
A center of curvature is defined at each position s located a distance ρ (the radius of curvature) from the curve on a line along the normal u_{n} (s). The required distance ρ(s) at arc length s is defined in terms of the rate of rotation of the tangent to the curve, which in turn is determined by the path itself. If the orientation of the tangent relative to some starting position is θ(s), then ρ(s) is defined by the derivative dθ/ds:
The radius of curvature usually is taken as positive (that is, as an absolute value), while the curvature κ is a signed quantity.
A geometric approach to finding the center of curvature and the radius of curvature uses a limiting process leading to the osculating circle.^{[24]}^{[25]} See Figure 7.
Using these coordinates, the motion along the path is viewed as a succession of circular paths of everchanging center, and at each position s constitutes nonuniform circular motion at that position with radius ρ. The local value of the angular rate of rotation then is given by:
with the local speed v given by:
As for the other examples above, because unit vectors cannot change magnitude, their rate of change is always perpendicular to their direction (see the lefthand insert in Figure 7):^{[26]}
Consequently, the velocity and acceleration are:^{[25]}^{[27]}^{[28]}
and using the chainrule of differentiation:
In this local coordinate system the acceleration resembles the expression for nonuniform circular motion with the local radius ρ(s), and the centripetal acceleration is identified as the second term.^{[29]}
Extension of this approach to three dimensional space curves leads to the FrenetSerret formulas.^{[30]}^{[31]}
Looking at Figure 7, one might wonder whether adequate account has been taken of the difference in curvature between ρ(s) and ρ(s + ds) in computing the arc length as ds = ρ(s)dθ. Reassurance on this point can be found using a more formal approach outlined below. This approach also makes connection with the article on curvature.
To introduce the unit vectors of the local coordinate system, one approach is to begin in Cartesian coordinates and describe the local coordinates in terms of these Cartesian coordinates. In terms of arc length s let the path be described as:^{[32]}
Then an incremental displacement along the path ds is described by:
where primes are introduced to denote derivatives with respect to s. The magnitude of this displacement is ds, showing that:^{[33]}
This displacement is necessarily tangent to the curve at s, showing that the unit vector tangent to the curve is:
while the outward unit vector normal to the curve is
Orthogonality can be verified by showing the vector dot product is zero. The unit magnitude of these vectors is a consequence of Eq. 1. Using the tangent vector, the angle of the tangent to the curve, say θ, is given by:
The radius of curvature is introduced completely formally (without need for geometric interpretation) as:
The derivative of θ can be found from that for sinθ:
Now:
in which the denominator is unity. With this formula for the derivative of the sine, the radius of curvature becomes:
where the equivalence of the forms stems from differentiation of Eq. 1:
With these results, the acceleration can be found:
as can be verified by taking the dot product with the unit vectors u_{t}(s) and u_{n}(s). This result for acceleration is the same as that for circular motion based on the radius ρ. Using this coordinate system in the inertial frame, it is easy to identify the force normal to the trajectory as the centripetal force and that parallel to the trajectory as the tangential force. From a qualitative standpoint, the path can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius.
This result for acceleration agrees with that found earlier. However, in this approach the question of the change in radius of curvature with s is handled completely formally, consistent with a geometric interpretation, but not relying upon it, thereby avoiding any questions Figure 7 might suggest about neglecting the variation in ρ.
To illustrate the above formulas, let x, y be given as:
Then:
which can be recognized as a circular path around the origin with radius α. The position s = 0 corresponds to [α, 0], or 3 o'clock. To use the above formalism the derivatives are needed:
With these results one can verify that:
The unit vectors also can be found:
which serve to show that s = 0 is located at position [ρ, 0] and s = ρπ/2 at [0, ρ], which agrees with the original expressions for x and y. In other words, s is measured counterclockwise around the circle from 3 o'clock. Also, the derivatives of these vectors can be found:
To obtain velocity and acceleration, a timedependence for s is necessary. For counterclockwise motion at variable speed v(t):
where v(t) is the speed and t is time, and s(t = 0) = 0. Then:
where it already is established that α = ρ. This acceleration is the standard result for nonuniform circular motion.

Centripetal force (or acceleration) is a type of force that acts on any body that revolves around a centre. (circular motion). This force contributes to keeping the body in circular motion. This force is always directed towards the centre.
The opposite force (by Isaac Newton's third law of motion) is called centrifugal force. This is the force that acts on the body in a direction away from the centre, which contributes to making the body try to fly away. When you hold a rope with a heavy object attached to it, and rotate it around, the rope becomes tight and keeps the body from flying away. This is caused by centripetal force. An example is a roller coaster which uses centripetal force to keep the carts going in a circular motion.
