
Centripetal force (from
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
''centrum'', "center" and ''petere'', "to seek") is the
force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
that makes a body follow a curved
path. The direction of the centripetal force is always
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
to the motion of the body and towards the fixed point of the instantaneous
center of curvature of the path.
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
coined the term, describing it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In
Newtonian mechanics
Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
# A body r ...
, gravity provides the centripetal force causing astronomical
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
s.
One common example involving centripetal force is the case in which a body moves with uniform speed along a circular path. The centripetal force is directed at right angles to the motion and also along the radius towards the centre of the circular path.
The mathematical description was derived in 1659 by the Dutch physicist
Christiaan Huygens
Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
.
Formula
From the
kinematics
In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics.
Kinematics is concerned with s ...
of curved motion it is known that an object moving at
tangential speed ''v'' along a path with
radius of curvature ''r'' accelerates toward the center of curvature at a rate
Here,
is the
centripetal acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are vector quantities (in that they have magn ...
and
is the
difference between the velocity vectors at
and
.
By
Newton's second law, the cause of acceleration is a net force acting on the object, which is proportional to its mass ''m'' and its acceleration. The force, usually referred to as a ''centripetal force'', has a magnitude
and is, like centripetal acceleration, directed toward the center of curvature of the object's trajectory.
Derivation

The centripetal acceleration can be inferred from the diagram of the velocity vectors at two instances. In the case of uniform circular motion the velocities have constant magnitude. Because each one is perpendicular to its respective position vector, simple vector subtraction implies two similar isosceles triangles with congruent angles – one comprising a
base of
and a
leg
A leg is a weight-bearing and locomotive anatomical structure, usually having a columnar shape. During locomotion, legs function as "extensible struts". The combination of movements at all joints can be modeled as a single, linear element cap ...
length of
, and the other a
base of
(position vector
difference) and a
leg
A leg is a weight-bearing and locomotive anatomical structure, usually having a columnar shape. During locomotion, legs function as "extensible struts". The combination of movements at all joints can be modeled as a single, linear element cap ...
length of
:
Therefore,
can be substituted with
:
The direction of the force is toward the center of the circle 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).
The speed in the formula is squared, so twice the speed needs four times the force, at a given radius.
This force is also sometimes written in terms of the
angular velocity
In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...
''ω'' of the object about the center of the circle, related to the tangential velocity by the formula
so that
Expressed using the
orbital period ''T'' for one revolution of the circle,
the equation becomes
In particle accelerators, velocity can be very high (close to the speed of light in vacuum) so the same rest mass now exerts greater inertia (relativistic mass) thereby requiring greater force for the same centripetal acceleration, so the equation becomes:
where
is the
Lorentz factor
The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
.
Thus the centripetal force is given by:
which is the rate of change of
relativistic momentum .
Sources

In the case of an object that is 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. 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
The wall of death, motordrome, velodrome or well of death is a Traveling carnival, carnival sideshow featuring a silo- or barrel-shaped wooden cylinder, typically ranging from in diameter and made of wooden planks, inside which motorcyclists, o ...
or a
Rotor rider.
Newton's idea of a centripetal force corresponds to what is nowadays referred to as a
central force. When a
satellite
A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scient ...
is in
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
around a
planet
A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
, gravity is considered to be a centripetal force even though in the case of eccentric orbits, the gravitational force is directed towards the focus, and not towards the instantaneous center of curvature.
Another example of centripetal force arises in the helix that is traced out when a charged particle moves in a uniform
magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
in the absence of other external forces. In this case, the magnetic force is the centripetal force that 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
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
using the
unit vectors and
:
The assumption of
uniform circular motion 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 velocity
In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...
around the circle. Therefore,
where
is time.
The
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
and
acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
of the motion are the first and second derivatives of position with respect to time:
The term in parentheses is the original expression of
in
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
. Consequently,
The negative sign 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 is the natural tendency of objects in motion to stay in motion and objects at rest to stay at rest, unless a force causes the velocity to change. It is one of the fundamental principles in classical physics, and described by Isaac Newto ...
), this centripetal acceleration describes the circular motion path caused by a centripetal force.
Derivation using vectors

The image at right shows the vector relationships for uniform circular motion. The rotation itself is represented by the angular velocity vector Ω, which is normal to the plane of the orbit (using the
right-hand rule
In mathematics and physics, the right-hand rule is a Convention (norm), convention and a mnemonic, utilized to define the orientation (vector space), orientation of Cartesian coordinate system, axes in three-dimensional space and to determine the ...
) 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 dℓ of the particle in time d''t'' along the circular path is
:
which, by properties of the
vector cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
, 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 the image at right 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 objective 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 the image above 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
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag (physics), drag). This is the steady gain in speed caused exclusively by gravitational attraction. All bodi ...
; the second is the upward
normal force
In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts. In this instance '' normal'' is used in the geometric sense and means perpendicular, as opposed to the meanin ...
exerted by the road at a right angle to the road surface ''m''a
n. The centripetal force demanded by the curved motion is also shown above. 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
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
of the
normal force
In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts. In this instance '' normal'' is used in the geometric sense and means perpendicular, as opposed to the meanin ...
and the
force of gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
. The resultant or
net force on the ball found by
vector addition
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
of the
normal force
In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts. In this instance '' normal'' is used in the geometric sense and means perpendicular, as opposed to the meanin ...
exerted by the road and vertical force due to
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
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 . The vertical component of the force from the road must counteract the gravitational force: , which implies . Substituting into the above formula for 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
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
approaches infinity, allowing larger values for , v,
2/''r''. In words, this equation states that for greater 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 is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
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 the image at right. This case is used to demonstrate a derivation strategy based on a
polar coordinate system
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are
*the point's distance from a reference point called the ''pole'', and
*the point's direction from ...
.
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 , where ''R'' is a constant (the radius of the circle) and u
r is the
unit vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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 unit vectors in the ''x'' and ''y'' directions, denoted
and
respectively:
and
One can differentiate to find velocity:
where is the angular velocity .
This result for the velocity matches expectations that the velocity should be directed tangentially to the circle, and that the magnitude of the velocity should be . Differentiating again, and noting that
we find that the acceleration, a is:
Thus, the radial and tangential components of the acceleration are:
and
where 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
In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the ...
, that changes the speed.
General planar motion
Polar coordinates
The above results can be derived perhaps more simply in
polar coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are
*the point's distance from a reference ...
, 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 above.
[Although the polar coordinate system moves with the particle, the observer does not. The description of the particle motion remains a description from the stationary observer's point of view.] 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 travelled by the particle.
[Notice that this local coordinate system is not autonomous; for example, its rotation in time is dictated by the trajectory traced by the particle. The radial vector r(''t'') does not represent the radius of curvature of the path.] By moving the unit vectors so their tails coincide, as seen in the circle at the left of the image above, 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 du
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 image above. 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 image above. Therefore, the change du
θ is orthogonal to u
θ and proportional to d''θ'' (see image above):
:
The equation above shows the sign to be negative: to maintain orthogonality, if du
ρ is positive with d''θ'', 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:
:
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:
where
These results agree with those above for
nonuniform circular motion. See also the article on
non-uniform 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.
For trajectories other than circular motion, for example, the more general trajectory envisioned in the image above, 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.
[See, for example, ] To deal directly with this issue, local coordinates are preferable, as discussed next.
Local coordinates

Local coordinates mean a set of coordinates that travel with the particle,
[The ''observer'' of the motion along the curve is using these local coordinates to describe the motion from the observer's ''frame of reference'', that is, from a stationary point of view. In other words, although the local coordinate system moves with the particle, the observer does not. A change in coordinate system used by the observer is only a change in their ''description'' of observations, and does not mean that the observer has changed their state of motion, and ''vice versa''.] and have orientation determined by the path of the particle.
Unit vectors are formed as shown in the image at right, 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
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
'' ''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.
[The osculating circle at a given point ''P'' on a curve is the limiting circle of a sequence of circles that pass through ''P'' and two other points on the curve, ''Q'' and ''R'', on either side of ''P'', as ''Q'' and ''R'' approach ''P''. See the online text by Lamb: ] See image above.
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 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 the image above):
:
Consequently, the velocity and acceleration are:
:
and using the chain-rule of differentiation:
: 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.
Extending this approach to three dimensional space curves leads to the Frenet–Serret formulas.
= Alternative approach
=
Looking at the image above, 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
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
.
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:[The article on ]curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
treats a more general case where the curve is parametrized by an arbitrary variable (denoted ''t''), rather than by the arc length ''s''.
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 a 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
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically ...
can be verified by showing that the vector dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
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 ut(''s'') and un(''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 the image above 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 can also be found:
:
which serve to show that ''s'' = 0 is located at position 'ρ'', 0and ''s'' = ''ρ''π/2 at , ''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.
See also
* Analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...
* Applied mechanics
* Bertrand theorem
* Central force
* Centrifugal force
* Circular motion
* Classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
* Coriolis force
* Dynamics (physics)
* Eskimo yo-yo
* Example: circular motion
* Fictitious force
* Frenet-Serret formulas
* History of centrifugal and centripetal forces
* Kinematics
In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics.
Kinematics is concerned with s ...
* Kinetics
* Orthogonal coordinates
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
* Reactive centrifugal force
* Statics
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in mechanical equilibrium, equilibrium with its environment ...
Notes and references
Further reading
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*
Centripetal force
vs
from an online Regents Exam physics tutorial by the Oswego City School District
External links
{{DEFAULTSORT:Centripetal Force
Force
Mechanics
Kinematics
Rotation
Acceleration
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