Centripetal force

Overview

**Centripetal force**is a force that makes a body follow a curved path: it is always directed orthogonal

Orthogonality

Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

to the velocity of the body, toward the instantaneous center of curvature

Osculating circle

In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...

of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens. Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

'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."

The magnitude of the centripetal force on an object of mass

*m*moving at a speed

*v*along a path with radius of curvature

Osculating circle

In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...

*r*is:

where is the centripetal acceleration.

The direction of the force is toward the center of the circle in which the object is moving, or the osculating circle

Osculating circle

In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...

, the circle that best fits the local path of the object, if the path is not circular.

This force is also sometimes written in terms of the angular velocity

Angular velocity

In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

*ω*of the object about the center of the circle:

Expressed using the period for one revolution of the circle,

*T*, the equation becomes:

For a satellite

Satellite

In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....

in orbit

Orbit

In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

around a planet

Planet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

, the centripetal force is supplied by gravity.

Unanswered Questions

Discussions

Encyclopedia

**Centripetal force**is a force that makes a body follow a curved path: it is always directed orthogonal

Orthogonality

Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

to the velocity of the body, toward the instantaneous center of curvature

Osculating circle

of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens. Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

'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."

## Formula

The magnitude of the centripetal force on an object of mass*m*moving at a speed

*v*along a path with radius of curvature

Osculating circle

*r*is:

where is the centripetal acceleration.

The direction of the force is toward the center of the circle in which the object is moving, or the osculating circle

Osculating circle

, the circle that best fits the local path of the object, if the path is not circular.

This force is also sometimes written in terms of the angular velocity

Angular velocity

In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

*ω*of the object about the center of the circle:

Expressed using the period for one revolution of the circle,

*T*, the equation becomes:

## Sources of centripetal force

For a satelliteSatellite

In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....

in orbit

Orbit

In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

around a planet

Planet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

, the centripetal force is supplied by gravity. Some sources, including Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

, refer to the entire force as a centripetal force, even for eccentric orbits, for which gravity is not aligned with the direction to the center of curvature.

The gravitational force acts on each object toward the other, which is toward the center of mass

Center of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

of the two objects; for circular orbits, this center of gravity is the center of the circular orbits. For non-circular 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.

For an object swinging around on the end of a rope in a horizontal plane, the centripetal force on the object is supplied by the tension of the rope. For a spinning object, internal tensile stress provides the centripetal forces that make the parts of the object trace out circular motions.

The rope example is an example involving a 'pull' force. The centripetal force can also be supplied as a 'push' force such as in the case where the normal reaction of a wall supplies the centripetal force for a wall of death rider.

Another example of centripetal force arises in the helix which is traced out when a charged particle moves in a uniform magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

in the absence of other external forces. In this case, the magnetic force is the centripetal force which acts towards the helix axis.

## 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.#### Calculus derivation

In two dimensions the position vector which has magnitude (length) and directed at an angle above the x-axis can be expressed in Cartesian coordinates using the unit vectors and $\backslash hat\{y\}$:Assume uniform circular motion

Uniform circular motion

In physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. The distance of the body from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not constant: velocity, a vector quantity, depends...

, which requires three things.

- The object moves only on a circle.
- The radius of the circle does not change in time.
- The object moves with constant angular velocityAngular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

around the circle. Therefore where is time.

Now find the velocity

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

and acceleration

Acceleration

In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...

of the motion by taking derivatives of position with respect to time.

Notice that the term in parenthesis is the original expression of in Cartesian coordinates. Consequently,

The negative shows that the acceleration is pointed towards the center of the circle (opposite the radius), hence it is called "centripetal" (i.e. "center-seeking"). While objects naturally follow a straight path (due to inertia

Inertia

Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

), this centripetal acceleration describes the circular motion path caused by a centripetal force.

#### Derivation using vectors

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 right-hand rule

Right-hand rule

In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century....

) and has magnitude given by:

with

*θ*the angular position at time

*t*. In this subsection, d

*θ*/d

*t*is assumed constant, independent of time. The distance traveled

**ℓ**of the particle in time d

*t*along the circular path is

which, by properties of the vector cross product, has magnitude

*r*d

*θ*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, though the notation is slightly different.

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.

#### Example: The banked turn

The upper panel in 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. 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 lower panel 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

*m*

*g***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 surfaceGravitational acceleration

In physics, gravitational acceleration is the acceleration on an object caused by gravity. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass....

; the second is the upward normal force

Normal force

In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...

exerted by the road perpendicular to the road surface

**m****a**

_{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

Net force

In physics, net force is the total force acting on an object. It is calculated by vector addition of all forces that are actually acting on that object. Net force has the same effect on the translational motion of the object as all actual forces taken together...

on the ball resulting from vector addition of the normal force

Normal force

In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...

and the force of gravity. The resultant or net force

Net force

In physics, net force is the total force acting on an object. It is calculated by vector addition of all forces that are actually acting on that object. Net force has the same effect on the translational motion of the object as all actual forces taken together...

on the ball found by vector addition of the normal force

Normal force

In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...

exerted by the road and vertical force due to gravity must equal the centripetal force dictated by the need to travel a circular path. 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}| =

*m*|

**a**

_{n}|sin

*θ*. The vertical component of the force from the road must counteract the gravitational force: |

**F**

_{v}| =

*m*|

**a**

_{n}|cos

*θ*=

*m*|

**g**|, which implies |

**a**

_{n}|=|

**g**| / cos

*θ*. Substituting into the above formula for |

**F**

_{h}| yields a 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

Friction

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of 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.

These ideas apply to air flight as well. See the FAA pilot's manual.

### Nonuniform circular motion

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 systemPolar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....

.

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

Circular motion

In physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation , or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of...

, 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 x-axis and the unit vector, measured counterclockwise from the x-axis. 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

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

unit vectors in the

*x*and

*y*directions, denoted

**i**and

**j**respectively:

**u**_{r}= cos*θ***i**+ sin*θ***j**

and

**u**_{θ}= -sin*θ***i**+ cos*θ***j**.

We differentiate to find velocity:

where

*ω*is the angular velocity d

*θ*/d

*t*.

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:

and

where |

**v**| =

*R*ω 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.

### General planar motion

#### Polar coordinates

The above results can be derived perhaps more simply in polar coordinatesPolar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....

, 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. 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. 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 right-angled 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 d

**u**

_{ρ}has a component only perpendicular to

**u**

_{ρ}. When the trajectory

**r**(

*t*) rotates an amount d

*θ*,

**u**

_{ρ}, which points in the same direction as

**r**(

*t*), also rotates by d

*θ*. 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 d

*θ*,

**u**

_{θ}, which is orthogonal to

**r**(

*t*), also rotates by d

*θ*. See Figure 6. Therefore, the change d

**u**

_{θ}is orthogonal to

**u**

_{θ}and proportional to d

*θ*(see Figure 6):

Figure 6 shows the sign to be negative: to maintain orthogonality, if d

**u**

_{ρ}is positive with d

*θ*, then d

**u**

_{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:

As a particular example, if the particle moves in a circle of constant radius

*R*, then d

*ρ*/d

*t*= 0,

**v**=

**v**

_{θ}, and:

These results agree with those above for nonuniform circular motion. See also the article on non-uniform circular motion

Non-uniform circular motion

Non-uniform circular motion is any case in which an object moving in a circular path has a varying speed. Some examples of non-uniform circular motion include a roller coaster, a vertical pendulum, and a car riding over a hill. All of these situations include an object traveling at different...

. 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.

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**and

_{ρ}**u**and to the length |

_{θ}**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.

To deal directly with this issue, local coordinates are preferable, as discussed next.

#### Local coordinates

By local coordinates is meant a set of coordinates that travel with the particle,and have orientation determined by the path of the particle. 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*or

*nt-coordinates*, for

*normal-tangential*, 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.

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 d

*θ*/d

*s*:

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

Osculating circle

. See Figure 7.

Using these coordinates, the motion along the path is viewed as a succession of circular paths of ever-changing center, and at each position

*s*constitutes non-uniform circular motion

Non-uniform circular motion

Non-uniform circular motion is any case in which an object moving in a circular path has a varying speed. Some examples of non-uniform circular motion include a roller coaster, a vertical pendulum, and a car riding over a hill. All of these situations include an object traveling at different...

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 left-hand insert in Figure 7):

Consequently, the velocity and acceleration are:

and using the chain-rule of differentiation

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

:

with the tangential acceleration

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.

Extension of this approach to three dimensional space curves leads to the Frenet–Serret formulas.

##### Alternative approach

Looking at Figure 7, one might wonder whether adequate account has been taken of the difference in curvature between

*ρ*(

*s*) and

*ρ*(

*s*+ d

*s*) in computing the arc length as d

*s*=

*ρ*(

*s*)d

*θ*. 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:

Then an incremental displacement along the path d

*s*is described by:

where primes are introduced to denote derivatives with respect to

*s*. The magnitude of this displacement is d

*s*, showing that: (Eq. 1)

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

Orthogonality

Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

can be verified by showing that the vector dot product

Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

is zero. The unit magnitude of these vectors is a consequence of Eq. 1. Using the tangent vector, the angle

*θ*of the tangent to the curve is given by: and

The radius of curvature is introduced completely formally (without need for geometric interpretation) as:

The derivative of

*θ*can be found from that for sin

*θ*:

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*.

##### Example: circular motion

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*=

*ρ*π/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 time-dependence 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 non-uniform circular motion

Non-uniform circular motion

Non-uniform circular motion is any case in which an object moving in a circular path has a varying speed. Some examples of non-uniform circular motion include a roller coaster, a vertical pendulum, and a car riding over a hill. All of these situations include an object traveling at different...

.

## See also

- Fictitious forceFictitious forceA fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame....
- Centrifugal force
- Circular motionCircular motionIn physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation , or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of...
- Coriolis forceCoriolis effectIn physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...
- Reactive centrifugal forceReactive centrifugal forceIn classical mechanics, reactive centrifugal force is the reaction paired with centripetal force. A mass undergoing circular motion constantly accelerates toward the axis of rotation. This centripetal acceleration is caused by a force exerted on the mass by some other object. In accordance with...
- Bertrand theorem

- Example: circular motion
- Mechanics of planar particle motionMechanics of planar particle motionThis article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of planar motion are related to the motion of two spheres that are gravitationally attracted to one another, and the generalization of this problem to planetary motion....
- Frenet-Serret formulasFrenet-Serret formulasIn vector calculus, the Frenet–Serret formulas describe the kinematic properties of a particle which moves along a continuous, differentiable curve in three-dimensional Euclidean space R3...
- Orthogonal coordinatesOrthogonal coordinatesIn mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles . A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant...
- StaticsStaticsStatics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...
- KineticsKinetics (physics)In physics and engineering, kinetics is a term for the branch of classical mechanics that is concerned with the relationship between the motion of bodies and its causes, namely forces and torques...

- KinematicsKinematicsKinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....
- Applied mechanicsApplied mechanicsApplied mechanics is a branch of the physical sciences and the practical application of mechanics. Applied mechanics examines the response of bodies or systems of bodies to external forces...
- Analytical mechanicsAnalytical mechanicsAnalytical mechanics is a term used for a refined, mathematical form of classical mechanics, constructed from the 18th century onwards as a formulation of the subject as founded by Isaac Newton. Often the term vectorial mechanics is applied to the form based on Newton's work, to contrast it with...
- Dynamics (physics)
- Classical mechanicsClassical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

## Further reading

- Centripetal force vs. Centrifugal force, from an online Regents Exam physics tutorial by the Oswego City School District

## External links

- Notes from University of Winnipeg
- Notes from Physics and Astronomy HyperPhysics at Georgia State University; see also home page
- Notes from Britannica
- Notes from PhysicsNet
- NASA notes by David P. Stern
- Notes from U Texas.
- Analysis of smart yo-yo
- The Inuit yo-yo
- Kinematic Models for Design Digital Library (KMODDL)

Movies and photos of hundreds of working mechanical-systems models at Cornell University. Also includes an e-book library of classic texts on mechanical design and engineering.