centripetal force: Definition from Answers.com

Concept

Most people have heard of centripetal and centrifugal force. Though it may be somewhat difficult to keep track of which is which, chances are anyone who has heard of the two concepts remembers that one is the tendency of objects in rotation to move inward, and the other is the tendency of rotating objects to move outward. It may come as a surprise, then, to learn that there is no such thing, strictly speaking, as centrifugal (outward) force. There is only centripetal (inward) force and the inertia that makes objects in rotation under certain situations move outward, for example, a car making a turn, the movement of a roller coaster—even the spinning of a centrifuge.

How It Works

Like many other principles in physics, centripetal force ultimately goes back to a few simple precepts relating to the basics of motion. Consider an object in uniform circular motion: an object moves around the center of a circle so that its speed is constant or unchanging.

The formula for speed—or rather, average speed—is distance divided by time; hence, people say, for instance, "miles (or kilometers) per hour." In the case of an object making a circle, distance is equal to the circumference, or distance around, the circle. From geometry, we know that the formula for calculating the circumference of a circle is 2πr, where r is the radius, or the distance from the circumference to the center. The figure π may be rendered as 3.141592 …, though in fact, it is an irrational number: the decimal figures continue forever without repetition or pattern.

From the above, it can be discerned that the formula for the average speed of an object moving around a circle is 2πr divided by time. Furthermore, we can see that there is a proportional relationship between radius and average speed. If the radius of a circle is doubled, but an object at the circle's periphery makes one complete revolution in the same amount of time as before, this means that the average speed has doubled as well. This can be shown by setting up two circles, one with a radius of 2, the other with a radius of 4, and using some arbitrary period of time—say, 2 seconds.

The above conclusion carries with it an interesting implication with regard to speeds at different points along the radius of a circle. Rather than comparing two points moving around the circumferences of two different circles—one twice as big as the other—in the same period of time, these two points could be on the same circle: one at the periphery, and one exactly halfway along the radius. Assuming they both traveled a complete circle in the same period of time, the proportional relationship described earlier would apply. This means, then, that the further out on the circle one goes, the greater the average speed.

Velocity = Speed + Direction

Speed is a scalar, meaning that it has magnitude but no specific direction; by contrast, velocity is a vector—a quantity with both a magnitude (that is, speed) and a direction. For an object in circular motion, the direction of velocity is the same as that in which the object is moving at any given point. Consider the example of the city of Atlanta, Georgia, and Interstate-285, one of several instances in which a city is surrounded by a "loop" highway. Local traffic reporters avoid giving mere directional coordinates for spots on that highway (for instance, "southbound on 285"), because the area where traffic moves south depends on whether one is moving clockwise or counterclockwise. Hence, reporters usually say "southbound on the outer loop."

As with cars on I-285, the direction of the velocity vector for an object moving around a circle is a function entirely of its position and the direction of movement—clockwise or counter-clockwise—for the circle itself. The direction of the individual velocity vector at any given point may be described as tangential; that is, describing a tangent, or a line that touches the circle at just one point. (By definition, a tangent line cannot intersect the circle.)

It follows, then, that the direction of an object in movement around a circle is changing; hence, its velocity is also changing—and this in turn means that it is experiencing acceleration. As with the subject of centripetal force and "centrifugal force," most people have a mistaken view of acceleration, believing that it refers only to an increase in speed. In fact, acceleration is a change in velocity, and can thus refer either to a change in speed or direction. Nor must that change be a positive one; in other words, an object undergoing a reduction in speed is also experiencing acceleration.

The acceleration of an object in rotational motion is always toward the center of the circle. This may appear to go against common sense, which should indicate that acceleration moves in the same direction as velocity, but it can, in fact, be proven in a number of ways. One method would be by the addition of vectors, but a "hands-on" demonstration may be more enlightening than an abstract geometrical proof.

It is possible to make a simple accelerometer, a device for measuring acceleration, with a lit candle inside a glass. The candle should be standing at a 90°-angle to the bottom of the glass, attached to it by hot wax as you would affix a burning candle to a plate. When you hold the candle level, the flame points upward; but if you spin the glass in a circle, the flame will point toward the center of that circle—in the direction of acceleration.

Mass × Acceleration = Force

Since we have shown that acceleration exists for an object spinning around a circle, it is then possible for us to prove that the object experiences some type of force. The proof for this assertion lies in the second law of motion, which defines force as the product of mass and acceleration: hence, where there is acceleration and mass, there must be force. Force is always in the direction of acceleration, and therefore the force is directed toward the center of the circle.

In the above paragraph, we assumed the existence of mass, since all along the discussion has concerned an object spinning around a circle. By definition, an object—that is, an item of matter, rather than an imaginary point—possesses mass. Mass is a measure of inertia, which can be explained by the first law of motion: an object in motion tends to remain in motion, at the same speed and in the same direction (that is, at the same velocity) unless or until some outside force acts on it. This tendency to maintain velocity is inertia. Put another way, it is inertia that causes an object standing still to remain motionless, and likewise, it is inertia which dictates that a moving object will "try" to keep moving.

Centripetal Force

Now that we have established the existence of a force in rotational motion, it is possible to give it a name: centripetal force, or the force that causes an object in uniform circular motion to move toward the center of the circular path. This is not a "new" kind of force; it is merely force as applied in circular or rotational motion, and it is absolutely essential. Hence, physicists speak of a "centripetal force requirement": in the absence of centripetal force, an object simply cannot turn. Instead, it will move in a straight line.

The Latin roots of centripetal together mean "seeking the center." What, then, of centrifugal, a word that means "fleeing the center"? It would be correct to say that there is such a thing as centrifugal motion; but centrifugal force is quite a different matter. The difference between centripetal force and a mere centrifugal tendency—a result of inertia rather than of force—can be explained by referring to a familiar example.

Real-Life Applications

Riding in a Car

When you are riding in a car and the car accelerates, your body tends to move backward against the seat. Likewise, if the car stops suddenly, your body tends to move forward, in the direction of the dashboard. Note the language here: "tends to move" rather than "is pushed." To say that something is pushed would imply that a force has been applied, yet what is at work here is not a force, but inertia—the tendency of an object in motion to remain in motion, and an object at rest to remain at rest.

A car that is not moving is, by definition, at rest, and so is the rider. Once the car begins moving, thus experiencing a change in velocity, the rider's body still tends to remain in the fixed position. Hence, it is not a force that has pushed the rider backward against the seat; rather, force has pushed the car forward, and the seat moves up to meet the rider's back. When stopping, once again, there is a sudden change in velocity from a certain value down to zero. The rider, meanwhile, is continuing to move forward due to inertia, and thus, his or her body has a tendency to keep moving in the direction of the now-stationary dashboard.

This may seem a bit too simple to anyone who has studied inertia, but because the human mind has such a strong inclination to perceive inertia as a force in itself, it needs to be clarified in the most basic terms. This habit is similar to the experience you have when sitting in a vehicle that is standing still, while another vehicle alongside moves backward. In the first split-second of awareness, your mind tends to interpret the backward motion of the other car as forward motion on the part of the car in which you are sitting—even though your own car is standing still.

Now we will consider the effects of centripetal force, as well as the illusion of centrifugal force. When a car turns to the left, it is undergoing a form of rotation, describing a 90°-angle or one-quarter of a circle. Once again, your body experiences inertia, since it was in motion along with the car at the beginning of the turn, and thus you tend to move forward. The car, at the same time, has largely overcome its own inertia and moved into the leftward turn. Thus the car door itself is moving to the left. As the door meets the right side of your body, you have the sensation of being pushed outward against the door, but in fact what has happened is that the door has moved inward.

The illusion of centrifugal force is so deeply ingrained in the popular imagination that it warrants further discussion below. But while on the subject of riding in an automobile, we need to examine another illustration of centripetal force. It should be noted in this context that for a car to make a turn at all, there must be friction between the tires and the road. Friction is the force that resists motion when the surface of one object comes into contact with the surface of another; yet ironically, while opposing motion, friction also makes relative motion possible.

Suppose, then, that a driver applies the brakes while making a turn. This now adds a force tangential, or at a right angle, to the centripetal force. If this force is greater than the centripetal force—that is, if the car is moving too fast—the vehicle will slide forward rather than making the turn. The results, as anyone who has ever been in this situation will attest, can be disastrous.

The above highlights the significance of the centripetal force requirement: without a sufficient degree of centripetal force, an object simply cannot turn. Curves are usually banked to maximize centripetal force, meaning that the roadway tilts inward in the direction of the curve. This banking causes a change in velocity, and hence, in acceleration, resulting in an additional quantity known as reaction force, which provides the vehicle with the centripetal force necessary for making the turn.

The formula for calculating the angle at which a curve should be banked takes into account the car's speed and the angle of the curve, but does not include the mass of the vehicle itself. As a result, highway departments post signs stating the speed at which vehicles should make the turn, but these signs do not need to include specific statements regarding the weight of given models.

The Centrifuge

To return to the subject of "centrifugal force"—which, as noted earlier, is really just centrifugal motion—you might ask, "If there is no such thing as centrifugal force, how does a centrifuge work?" Used widely in medicine and a variety of sciences, a centrifuge is a device that separates particles within a liquid. One application, for instance, is to separate red blood cells from plasma.

Typically a centrifuge consists of a base; a rotating tube perpendicular to the base; and two vials attached by movable centrifuge arms to the rotating tube. The movable arms are hinged at the top of the rotating tube, and thus can move upward at an angle approaching 90° to the tube. When the tube begins to spin, centripetal force pulls the material in the vials toward the center.

Materials that are denser have greater inertia, and thus are less responsive to centripetal force. Hence, they seem to be pushed outward, but in fact what has happened is that the less dense material has been pulled inward. This leads to the separation of components, for instance, with plasma on the top and red blood cells on the bottom. Again, the plasma is not as dense, and thus is more easily pulled toward the center of rotation, whereas the red blood cells respond less, and consequently remain on the bottom.

The centrifuge was invented in 1883 by Carl de Laval (1845-1913), a Swedish engineer, who used it to separate cream from milk. During the 1920s, the chemist Theodor Svedberg (1884-1971), who was also Swedish, improved on Laval's work to create the ultracentrifuge, used for separating very small particles of similar weight.

In a typical ultracentrifuge, the vials are no larger than 0.2 in (0.6 cm) in diameter, and these may rotate at speeds of up to 230,000 revolutions per minute. Most centrifuges in use by industry rotate in a range between 1,000 and 15,000 revolutions per minute, but others with scientific applications rotate at a much higher rate, and can produce a force more than 25,000 times as great as that of gravity.

In 1994, researchers at the University of Colorado created a sort of super-centrifuge for simulating stresses applied to dams and other large structures. The instrument has just one centrifuge arm, measuring 19.69 ft (6 m), attached to which is a swinging basket containing a scale model of the structure to be tested. If the model is 1/50 the size of the actual structure, then the centrifuge is set to create a centripetal force 50 times that of gravity.

The Colorado centrifuge has also been used to test the effects of explosions on buildings. Because the combination of forces—centripetal, gravity, and that of the explosion itself—is so great, it takes a very small quantity of explosive to measure the effects of a blast on a model of the building.

Industrial uses of the centrifuge include that for which Laval invented it—separation of cream from milk—as well as the separation of impurities from other substances. Water can be removed from oil or jet fuel with a centrifuge, and likewise, waste-management agencies use it to separate solid materials from waste water prior to purifying the water itself.

Closer to home, a washing machine on spin cycle is a type of centrifuge. As the wet clothes spin, the water in them tends to move outward, separating from the clothes themselves. An even simpler, more down-to-earth centrifuge can be created by tying a fairly heavy weight to a rope and swinging it above one's head: once again, the weight behaves as though it were pushed outward, though in fact, it is only responding to inertia.

Roller Coasters and Centripetal Force

People ride roller coasters, of course, for the thrill they experience, but that thrill has more to do with centripetal force than with speed. By the late twentieth century, roller coasters capable of speeds above 90 MPH (144 km/h) began to appear in amusement parks around America; but prior to that time, the actual speeds of a roller coaster were not particularly impressive. Seldom, if ever, did they exceed that of a car moving down the highway. On the other hand, the acceleration and centripetal force generated on a roller coaster are high, conveying a sense of weightlessness (and sometimes the opposite of weightlessness) that is memorable indeed.

Few parts of a roller coaster ride are straight and flat—usually just those segments that mark the end of one ride and the beginning of another. The rest of the track is generally composed of dips and hills, banked turns, and in some cases, clothoid loops. The latter refers to a geometric shape known as a clothoid, rather like a teardrop upside-down.

Because of its shape, the clothoid has a much smaller radius at the top than at the bottom—a key factor in the operation of the roller coaster ride through these loops. In days past, roller-coaster designers used perfectly circular loops, which allowed cars to enter them at speeds that were too high, built too much force and resulted in injuries for riders. Eventually, engineers recognized the clothoid as a means of providing a safe, fun ride.

As you move into the clothoid loop, then up, then over, and down, you are constantly changing position. Speed, too, is changing. On the way up the loop, the roller coaster slows due to a decrease in kinetic energy, or the energy that an object possesses by virtue of its movement. At the top of the loop, the roller coaster has gained a great deal of potential energy, or the energy an object possesses by virtue of its position, and its kinetic energy is at zero. But once it starts going down the other side, kinetic energy—and with it speed—increases rapidly once again.

Throughout the ride, you experience two forces, gravity, or weight, and the force (due to motion) of the roller coaster itself, known as normal force. Like kinetic and potential energy—which rise and fall correspondingly with dips and hills—normal force and gravitational force are locked in a sort of "competition" throughout the roller-coaster rider. For the coaster to have its proper effect, normal force must exceed that of gravity in most places.

The increase in normal force on a roller-coaster ride can be attributed to acceleration and centripetal motion, which cause you to experience something other than gravity. Hence, at the top of a loop, you feel lighter than normal, and at the bottom, heavier. In fact, there has been no real change in your weight: it is, like the idea of "centrifugal force" discussed earlier, a matter of perception.

Where to Learn More

Aylesworth, Thomas G. Science at the Ball Game. New York: Walker, 1977.

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

Buller, Laura and Ron Taylor. Forces of Nature. Illustrations by John Hutchinson and Stan North. New York: Marshall Cavendish, 1990.

"Centrifugal Force—Rotational Motion." National Aeronautics and Space Administration (Web site). <http://observe.ivv.nasa.gov/nasa/space/centrifugal/centrifugal3.html> (March 5, 2001).

"Circular and Satellite Motion" (Web site). <http://www.glenbrook.k12.il.us/gbssci/phys/Class/circles/circtoc.html> (March 5, 2001).

Cobb, Vicki. Why Doesn't the Earth Fall Up? And Other Not Such Dumb Questions About Motion. Illustrated by Ted Enik. New York: Lodestar Books, 1988.

Lefkowitz, R. J. Push! Pull! Stop! Go! A Book About Forces and Motion. Illustrated by June Goldsborough. New York: Parents' Magazine Press, 1975.

"Rotational Motion." Physics Department, University of Guelph (Web site). <http://www.physics.uoguelph.ca/tutorials/torque/> (March 4, 2001).

Schaefer, Lola M. Circular Movement. Mankato, MN: Pebble Books, 2000.

Snedden, Robert. Forces. Des Plaines, IL: Heinemann Library, 1999.

Whyman, Kathryn. Forces in Action. New York: Gloucester Press, 1986.

Not to be confused with Centrifugal force.

Roller coaster cars are forced through a loop by the track applying a centripetal force on them. The reactive centrifugal force of the cars, associated with their inertia, holds them on the track.

Figure 1: A simple example corresponding to uniform circular motion. A ball is tethered to a rotational axis and is rotating counterclockwise around the specified path at a constant angular rate ω. The velocity of the ball is a vector tangential to the orbit, and is continuously changing direction, a change requiring a radially inward directed centripetal force. The centripetal force is provided by the tether, which is in a state of tension.

Centripetal force is a force that makes a body follow a curved, as opposed to straight, 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] and is found in the Principia.^[4] Any force (gravitational, electromagnetic, etc.) or combination of forces can act to provide a centripetal force. An example for the case of uniform circular motion is shown in Figure 1.

1 Formula
2 Sources of centripetal force
3 Analysis of several cases
4 See also
5 Notes and references
6 Further reading
7 External links

Formula

The magnitude of the centrifugal force on an object of mass m moving at a speed v along a path with radius of curvature r is:^[5]

$F = \frac{m v^2}{r}$

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.^[6] This force is also sometimes written in terms of the angular velocity ω of the object about the center of the circle:

F = m r ω 2

Sources of centripetal force

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 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.^[7] 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;^[8] 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.

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.

Geometric derivation

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Figure 2: Left circle: The particle's orbit – particle moves in a circle and velocity is tangent to orbit; Right circle: a "velocity circle"; velocity vectors are brought together so tails coincide: because velocity is a constant in uniform motion, the tip of the velocity vector describes a circle, and acceleration is tangent to the velocity circle. That means the acceleration is radially inward in the left-hand circle showing the orbit.

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

$T = \frac{2\pi|\mathbf{R}|}{|\mathbf{v}|}$

and, by analogy,

$T = \frac{2\pi |\mathbf{v}|}{|\mathbf{a}|}$

Setting these two equations equal and solving for |a|, we get

$|\mathbf{a}| = \frac{|\mathbf{v}|^{2}}{|\mathbf{R}|}$

The angular rate of rotation in radians per second is:

$\omega = \frac {2 \pi} {T} \$

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.

Derivation using vectors

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Figure 3: Vector relationships for uniform circular motion; vector Ω representing the rotation is normal to the plane of the orbit with polarity determined by the right-hand rule and magnitude dθ /dt.

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) and has magnitude given by:

$|\mathbf{\Omega}| = \frac {\mathrm{d} \theta } {\mathrm{d}t} = \omega \ ,$

with θ the angular position at time t. In this subsection, dθ/dt is assumed constant, independent of time. The displacement ℓ of the particle in time dt along the circular path is

$\mathrm{d}\boldsymbol{\ell} = \mathbf {\Omega} \times \mathbf{r}(t) \mathrm{d}t \ ,$

which, by properties of the vector cross product, has magnitude rdθ and is in the direction tangent to the circular path.

Consequently,

$\frac {\mathrm{d} \mathbf{r}}{\mathrm{d}t} = \frac {\mathbf{r}(t + \mathrm{d}t)-\mathbf{r}(t)}{\mathrm{d}t} = \frac{\mathrm{d} \boldsymbol{\ell}}{\mathrm{d}t} \ .$

In other words,

$\mathbf{v}\ \stackrel{\mathrm{def}}{ = }\ \frac {\mathrm{d} \mathbf{r}}{\mathrm{d}t} = \frac {\mathrm{d}\mathbf{\boldsymbol{\ell}}}{\mathrm{d}t} = \mathbf {\Omega} \times \mathbf{r}(t)\ .$

Differentiating with respect to time,

$\mathbf{a}\ \stackrel{\mathrm{def}}{ = }\ \frac {\mathrm{d} \mathbf{v}} {d\mathrm{t}} = \mathbf {\Omega} \times \frac{\mathrm{d} \mathbf{r}(t)}{\mathrm{d}t} = \mathbf{\Omega} \times \left[ \mathbf {\Omega} \times \mathbf{r}(t)\right] \ .$

Lagrange's formula states:

$\mathbf{a} \times \left ( \mathbf{b} \times \mathbf{c} \right ) = \mathbf{b} \times \left ( \mathbf{a} \cdot \mathbf{c} \right ) - \mathbf{c} \times \left ( \mathbf{a} \cdot \mathbf{b} \right ) \ .$

Applying Lagrange's formula with the observation that Ω • r(t) = 0 at all times,

$\mathbf{a} = - {|\mathbf{\Omega|}}^2 \mathbf{r}(t) \ .$

In words, the acceleration is pointing directly opposite to the radial displacement r at all times, and has a magnitude:

$|\mathbf{a}| = |\mathbf{r}(t)| \left ( \frac {\mathrm{d} \theta}{\mathrm{d}t} \right) ^2 = R {\omega}^2\$

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:

$\frac {\mathrm{d} \theta }{\mathrm{d}t} = \omega = \frac {2 \pi } {T} = \frac {|\mathbf{v}|}{R} \ .$

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

Main article: Banked turn

Figure 4: Left panel: Ball on a banked circular track moving with constant speed v; Right panel: Forces on the ball. The resultant or net force on the ball found by vector addition of the normal force exerted by the road and vertical force due to gravity must equal the centripetal force dictated by the need to travel a circular path.

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.^[9] 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| = m|a_n|sinθ. The vertical component of the force from the road must counteract the gravitational force, that is |F_v| = m|a_n|cosθ = m|g|. Accordingly one finds the net horizontal force to be:

$|\mathbf{F}_\mathrm{h}| = m |\mathbf{g}| \frac { \mathrm{sin}\ \theta}{ \mathrm {cos}\ \theta} = m|\mathbf{g}| \mathrm{tan}\ \theta \ .$

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:

$|\mathbf{F}_\mathrm{c}| = m |\mathbf{a}_\mathrm{c}| = \frac{m|\mathbf{v}|^2}{R} \ .$

Consequently the ball is in a stable path when the angle of the road is set to satisfy the condition:

$m |\mathbf{g}| \mathrm{tan}\ \theta = \frac{m|\mathbf{v}|^2}{R} \ ,$

or,

$\mathrm{tan}\ \theta = \frac {|\mathbf{v}|^2} {|\mathbf{g}|R} \ .$

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.^[10]^[11]

These ideas apply to air flight as well. See the FAA pilot's manual.^[12]

Nonuniform circular motion

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Figure 5: Velocity and acceleration for nonuniform circular motion: the velocity vector is tangential to the orbit, but the acceleration vector is not radially inward because of its tangential component a_θ that increases the rate of rotation: dω / dt = | a_θ| / R.

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 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 i and j respectively:^[13]

u_r = cosθ i + sinθ j

and

u_θ = sinθ i + cosθ j.

We differentiate to find velocity:

$\mathbf{v} = R \frac {\mathrm{d} \mathbf{u}_\mathrm{r}}{\mathrm{d}t} = R \frac {\mathrm{d}}{\mathrm{d}t} \left( \mathrm{cos}\ \theta \ \mathbf{i} + \mathrm{sin}\ \theta \ \mathbf{j}\right)$

$= R \frac {d \theta} {dt} \left( -\mathrm{sin}\ \theta \ \mathbf{i} + \mathrm{cos}\ \theta \ \mathbf{j}\right)\,$

$= R \frac{\mathrm{d}\theta}{\mathrm{d}t} \mathbf{u}_\mathrm{\theta} \,$

$= \omega R \mathbf{u}_\mathrm{\theta} \,$

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

${\frac {\mathrm{d}\mathbf{u}_\mathrm{\theta}}{\mathrm{d}t} = -\frac{\mathrm{d}\theta}{\mathrm{d}t} \mathbf{u}_\mathrm{r} = - \omega \mathbf{u}_\mathrm{r}} \ ,$

we find that the acceleration, a is:

$\mathbf{a} = R \left( \frac {\mathrm{d}\omega}{\mathrm{d}t} \mathbf{u}_\mathrm{\theta} - \omega^2 \mathbf{u}_\mathrm{r} \right) \ .$

Thus, the radial and tangential components of the acceleration are:

$\mathbf{a}_{\mathrm{r}} = - \omega^{2} R \ \mathbf{u}_\mathrm{r} = - \frac{|\mathbf{v}|^{2}}{ R} \ \mathbf{u}_\mathrm{r} \$ and $\ \mathbf{a}_{\mathrm{\theta}} = R \ \frac {\mathrm{d}\omega}{\mathrm{d}t} \ \mathbf{u}_\mathrm{\theta} = \frac {\mathrm{d} | \mathbf{v} | }{\mathrm{d}t} \ \mathbf{u}_\mathrm{\theta} \ ,$

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.

General planar motion

Figure 6: Polar unit vectors at two times t and t + dt for a particle with trajectory r ( t ); on the left the unit vectors u_ρ and u_θ at the two times are moved so their tails all meet, and are shown to trace an arc of a unit radius circle. Their rotation in time dt is dθ, just the same angle as the rotation of the trajectory r ( t ).

Polar coordinates

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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.^[14] A particle at position r is described by:

$\mathbf{r} = \rho \mathbf{u}_{\rho} \ ,$

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.^[15] 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

$\mathbf{v} = \frac {\mathrm{d} \rho }{\mathrm{d}t} \mathbf{u}_{\rho} + \rho \frac {\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} \ .$

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

$\mathrm{d} \mathbf{u}_{\rho} = \mathbf{u}_{\theta} \mathrm{d}\theta \ ,$

$\frac {\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} = \mathbf{u}_{\theta} \frac {\mathrm{d}\theta}{\mathrm{d}t} \ .$

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):

$\frac{\mathrm{d} \mathbf{u}_{\theta}}{\mathrm{d}t} = -\frac {\mathrm{d} \theta} {\mathrm{d}t} \mathbf{u}_{\rho} \ .$

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:

$\mathbf{v} = \frac {\mathrm{d} \rho }{\mathrm{d}t} \mathbf{u}_{\rho} + \rho \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t} = \mathbf{v}_{\rho} + \mathbf{v}_{\theta} \ .$

To obtain the acceleration, another time differentiation is done:

$\mathbf{a} = \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2} \mathbf{u}_{\rho} + \frac {\mathrm{d} \rho }{\mathrm{d}t} \frac{\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} + \frac {\mathrm{d} \rho}{\mathrm{d}t} \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \frac{\mathrm{d} \mathbf{u}_{\theta}}{\mathrm{d}t} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \mathbf{u}_{\theta} \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2} \ .$

Substituting the derivatives of u_ρ and u_θ, the acceleration of the particle is:^[16]

$\mathbf{a} = \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2} \mathbf{u}_{\rho} + 2\frac {\mathrm{d} \rho}{\mathrm{d}t} \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t}-\rho \mathbf{u}_{\rho}\left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 + \rho \mathbf{u}_{\theta} \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2} \ ,$

$= \mathbf{u}_{\rho} \left[ \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2}-\rho\left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 \right] + \mathbf{u}_{\theta}\left[ 2\frac {\mathrm{d} \rho}{\mathrm{d}t} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2}\right] \$

$= \mathbf{u}_{\rho} \left[ \frac {\mathrm{d}|\mathbf{v}_{\rho}|}{\mathrm{d}t}-\frac{|\mathbf{v}_{\theta}|^2}{\rho}\right] + \mathbf{u}_{\theta}\left[ \frac{2}{\rho}|\mathbf{v}_{\rho}||\mathbf{v}_{\theta}| + \rho\frac{\mathrm{d}}{\mathrm{d}t}\frac{|\mathbf{v}_{\theta}|}{\rho}\right] \ .$

As a particular example, if the particle moves in a circle of constant radius R, then dρ/dt = 0, v = v_θ, and:

$\mathbf{a} = \mathbf{u}_{\rho} \left[ -\rho\left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 \right] + \mathbf{u}_{\theta}\left[ \rho \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2}\right] \$

$= \mathbf{u}_{\rho} \left[ -\frac{|\mathbf{v}|^2}{R}\right] + \mathbf{u}_{\theta}\left[ \frac {\mathrm{d} |\mathbf{v}|} {\mathrm{d}t}\right] \ .$

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.^[17]

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.^[18] ^[19] To deal directly with this issue, local coordinates are preferable, as discussed next.

Local coordinates

Figure 7: Local coordinate system for planar motion on a curve. Two different positions are shown for distances s and s + ds along the curve. At each position s, unit vector u_n points along the outward normal to the curve and unit vector u_t is tangential to the path. The radius of curvature of the path is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the osculating circle at position s. The unit circle on the left shows the rotation of the unit vectors with s.

By local coordinates is meant a set of coordinates that travel with the particle, ^[20] and have orientation determined by the path of the particle.^[21] 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^[22]^[23] 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.^[24]

Distance along the path of the particle is the arc length s, considered to be a known function of time.

$s = s(t) \ .$

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:

$\frac{1} {\rho (s)} = \kappa (s) = \frac {\mathrm{d}\theta}{\mathrm{d}s}\ .$

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.^[25]^[26] 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 at that position with radius ρ. The local value of the angular rate of rotation then is given by:

$\omega(s) = \frac{\mathrm{d}\theta}{\mathrm{d}t} = \frac{\mathrm{d}\theta}{\mathrm{d}s} \frac {\mathrm{d}s}{\mathrm{d}t} = \frac{1}{\rho(s)}\ \frac {\mathrm{d}s}{\mathrm{d}t} = \frac{v(s)}{\rho(s)}\ ,$

with the local speed v given by:

$v(s) = \frac {\mathrm{d}s}{\mathrm{d}t}\ .$

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):^[27]

$\frac{d\mathbf{u}_\mathrm{n}(s)}{ds} = \mathbf{u}_\mathrm{t}(s)\frac{d\theta}{ds} = \mathbf{u}_\mathrm{t}(s)\frac{1}{\rho} \ ;$ $\frac{d\mathbf{u}_\mathrm{t}(s)}{\mathrm{d}s} = -\mathbf{u}_\mathrm{n}(s)\frac{\mathrm{d}\theta}{\mathrm{d}s} = - \mathbf{u}_\mathrm{n}(s)\frac{1}{\rho} \ .$

Consequently, the velocity and acceleration are:^[26]^[28]^[29]

$\mathbf{v}(t) = v \mathbf{u}_\mathrm{t}(s)\ ;$

and using the chain-rule of differentiation:

$\mathbf{a}(t) = \frac{\mathrm{d}v}{\mathrm{d}t} \mathbf{u}_\mathrm{t}(s) - \frac{v^2}{\rho}\mathbf{u}_\mathrm{n}(s) \ ;$ with the tangential acceleration $\frac{\mathrm{\mathrm{d}}v}{\mathrm{\mathrm{d}}t} = \frac{\mathrm{d}v}{\mathrm{d}s}\ \frac{\mathrm{d}s}{\mathrm{d}t} = \frac{\mathrm{d}v}{\mathrm{d}s}\ v \ .$

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.^[30]

Extension of this approach to three dimensional space curves leads to the Frenet-Serret formulas.^[31]^[32]

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 + 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:^[33]

$\mathbf{r}(s) = \left[ x(s),\ y(s) \right] \ .$

Then an incremental displacement along the path ds is described by:

$\mathrm{d}\mathbf{r}(s) = \left[ \mathrm{d}x(s),\ \mathrm{d}y(s) \right] = \left[ x'(s),\ y'(s) \right] \mathrm{d}s \ ,$

where primes are introduced to denote derivatives with respect to s. The magnitude of this displacement is ds, showing that:^[34]

$\left[ x'(s)^2 + y'(s)^2 \right] = 1 \ .$ (Eq. 1)

This displacement is necessarily tangent to the curve at s, showing that the unit vector tangent to the curve is:

$\mathbf{u}_\mathrm{t}(s) = \left[ x'(s), \ y'(s) \right] \ ,$

while the outward unit vector normal to the curve is

$\mathbf{u}_\mathrm{n}(s) = \left[ y'(s),\ -x'(s) \right] \ ,$

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:

$\sin \theta = \frac{y'(s)}{\sqrt{x'(s)^2 + y'(s)^2}} = y'(s) \ ;$ and $\cos \theta = \frac{x'(s)}{\sqrt{x'(s)^2 + y'(s)^2}} = x'(s) \ .$

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

$\frac{1}{\rho} = \frac{\mathrm{d}\theta}{\mathrm{d}s}\ .$

The derivative of θ can be found from that for sinθ:

$\frac{\mathrm{d} \sin\theta}{\mathrm{d}s} = \cos \theta \frac {\mathrm{d}\theta}{\mathrm{d}s} = \frac{1}{\rho} \cos \theta \ = \frac{1}{\rho} x'(s)\ .$

Now:

$\frac{\mathrm{d} \sin \theta }{\mathrm{d}s} = \frac{\mathrm{d}}{\mathrm{d}s} \frac{y'(s)}{\sqrt{x'(s)^2 + y'(s)^2}}$ $= \frac{y''(s)x'(s)^2-y'(s)x'(s)x''(s)} {\left(x'(s)^2 + y'(s)^2\right)^{3/2}}\ ,$

in which the denominator is unity. With this formula for the derivative of the sine, the radius of curvature becomes:

$\frac {\mathrm{d}\theta}{\mathrm{d}s} = \frac{1}{\rho} = y''(s)x'(s) - y'(s)x''(s)\$ $= \frac{y''(s)}{x'(s)} = -\frac{x''(s)}{y'(s)} \ ,$

where the equivalence of the forms stems from differentiation of Eq. 1:

$x'(s)x''(s) + y'(s)y''(s) = 0 \ .$

With these results, the acceleration can be found:

$\mathbf{a}(s) = \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{v}(s)$ $= \frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\mathrm{d}s}{\mathrm{d}t} \left( x'(s), \ y'(s) \right) \right]\$

$= \left(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\right)\mathbf{u}_\mathrm{t}(s) + \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right) ^2 \left(x''(s),\ y''(s) \right)$

$= \left(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\right)\mathbf{u}_\mathrm{t}(s) - \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right) ^2 \frac{1}{\rho} \mathbf{u}_\mathrm{n}(s) \ ,$

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

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To illustrate the above formulas, let x, y be given as:

$x = \alpha \cos \frac{s}{\alpha} \ ; \ y = \alpha \sin\frac{s}{\alpha} \ .$

Then:

$x^2 + y^2 = \alpha^2 \ ,$

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:

$y^{\prime}(s) = \cos \frac{s}{\alpha} \ ; \ x^{\prime}(s) = -\sin \frac{s}{\alpha} \ ,$

$y^{\prime\prime}(s) = -\frac{1}{\alpha}\sin\frac{s}{\alpha} \ ; \ x^{\prime\prime}(s) = -\frac{1}{\alpha}\cos \frac{s}{\alpha} \ .$

With these results one can verify that:

$x^{\prime}(s)^2 + y^{\prime}(s)^2 = 1 \ ; \ \frac{1}{\rho} = y^{\prime\prime}(s)x^{\prime}(s)-y^{\prime}(s)x^{\prime\prime}(s) = \frac{1}{\alpha} \ .$

The unit vectors also can be found:

$\mathbf{u}_\mathrm{t}(s) = \left[-\sin\frac{s}{\alpha} \ , \ \cos\frac{s}{\alpha} \right] \ ; \ \mathbf{u}_\mathrm{n}(s) = \left[\cos\frac{s}{\alpha} \ , \ \sin\frac{s}{\alpha} \right] \ ,$

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:

$\frac{\mathrm{d}}{\mathrm{d}s}\mathbf{u}_\mathrm{t}(s) = -\frac{1}{\alpha} \left[\cos\frac{s}{\alpha} \ , \ \sin\frac{s}{\alpha} \right] = -\frac{1}{\alpha}\mathbf{u}_\mathrm{n}(s) \ ;$

$\ \frac{\mathrm{d}}{\mathrm{d}s}\mathbf{u}_\mathrm{n}(s) = \frac{1}{\alpha} \left[-\sin\frac{s}{\alpha} \ , \ \cos\frac{s}{\alpha} \right] = \frac{1}{\alpha}\mathbf{u}_\mathrm{t}(s) \ .$

To obtain velocity and acceleration, a time-dependence for s is necessary. For counterclockwise motion at variable speed v(t):

$s(t) = \int_0^t \ dt^{\prime} \ v(t^{\prime}) \ ,$

where v(t) is the speed and t is time, and s(t = 0) = 0. Then:

$\mathbf{v} = v(t)\mathbf{u}_\mathrm{t}(s) \ ,$

$\mathbf{a} = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s) + v\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s) = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s)-v\frac{1}{\alpha}\mathbf{u}_\mathrm{n}(s)\frac{\mathrm{d}s}{\mathrm{d}t}$

$\mathbf{a} = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s)-\frac{v^2}{\alpha}\mathbf{u}_\mathrm{n}(s) \ ,$

where it already is established that α = ρ. This acceleration is the standard result for non-uniform circular motion.

Notes and references

^ Russelkl C Hibbeler (2009). "Equations of Motion: Normal and tangential coordinates". Engineering Mechanics: Dynamics (12 ed.). Prentice Hall. p. 131. ISBN 0136077919. http://books.google.com/books?id=tOFRjXB-XvMC&pg=PA131.
^ Paul Allen Tipler, Gene Mosca (2003). Physics for scientists and engineers (5th ed.). Macmillan. p. 129. ISBN 0716783398. http://books.google.com/books?id=2HRFckqcBNoC&pg=PA129.
^ Felix Klein, Arnold Sommerfeld (2008). The Theory of the Top (Reprint with translators' notes of 1897 ed.). p. 232. ISBN 0817647201. http://books.google.com/books?id=xdxGF918uI8C&pg=PA232.
^ Andrew Motte translation (1729) of Newton's Principia (1687): Of the invention of Centripetal Forces, p. 114. A discussion may be found in Johan Christiaan Boudri (2002). What was mechanical about mechanics: The concept of force between metaphysics and mechanics from Newton to Lagrange. Springer. p. 63. ISBN 1402002335. http://books.google.com/books?id=jLMsr3pIaOgC&pg=PA63. . Newton was concerned primarily with planetary motion.
^ Chris Carter (2001). Facts and Practice for A-Level: Physics. Oxford University Press US. p. 30. ISBN 9780199147687. http://books.google.com/books?id=xF1gTP-OoxQC&pg=PA30&dq=centripetal-force+mass+angular-velocity+formula+circle+radius&lr=&as_brr=3&ei=PmJASvC8JpTCkAS13eCFDw.
^ Eugene Lommel and George William Myers (1900). Experimental physics. K. Paul, Trench, Trübner & Co. p. 63. http://books.google.com/books?id=4BMPAAAAYAAJ&pg=PA63&dq=centripetal-force+osculating-circle&lr=&as_brr=3&ei=gmNASs2tCYXWlQTXieiADw.
^ Johnnie T. Dennis (2003). The Complete Idiot's Guide to Physics. Alpha Books. p. 91. ISBN 9781592570812. http://books.google.com/books?id=P1hL1EwElX4C&pg=PA91&dq=centripetal+component+gravity&lr=&as_brr=3&ei=VWpASuruKY-gkQTc7bzxDg.
^ George Bernard Benedek and Felix Villars (2000). Physics, with Illustrative Examples from Medicine and Biology: Mechanics. Springer. p. 52. ISBN 9780387987699. http://books.google.com/books?id=GeALYXiy9sMC&pg=PA52&dq=gravity+%22centripetal+force%22+intitle:mechanics&lr=&as_brr=3&ei=W7EtSqu7Bo62zATZnrSOBw.
^ Lawrence S. Lerner (1997). Physics for Scientists and Engineers. Boston: Jones & Bartlett Publishers. p. 128. ISBN 0867204796. http://books.google.com/books?id=kJOnAvimS44C&pg=PA129&dq=centripetal+%22banked+curve%22&lr=&as_brr=0&sig=0ueAq7G5l2R3ausiXue0CPW_1dM#PPA128,M1.
^ Arthur Beiser (2004). Schaum's Outline of Applied Physics. New York: McGraw-Hill Professional. p. 103. ISBN 0071426116. http://books.google.com/books?id=soKguvJDgmsC&pg=PA103&dq=friction+%22banked+turn%22&lr=&as_brr=0&sig=hMYfCzJHm6Ni4Noq5v5NRvvPSyQ.
^ Alan Darbyshire (2003). Mechanical Engineering: BTEC National Option Units. Oxford: Newnes. p. 56. ISBN 0750657618. http://books.google.com/books?id=fzfXLGpElZ0C&pg=PA57&dq=centripetal+%22banked+curve%22&lr=&as_brr=0&sig=hbaHu8Xt_uTGvd5b1DG01vCYFF8#PPA56,M1.
^ Federal Aviation Administration (2007). Pilot's Encyclopedia of Aeronautical Knowledge. Oklahoma City OK: Skyhorse Publishing Inc.. Figure 3-21. ISBN 1602390347. http://books.google.com/books?id=m5V04SXE4zQC&pg=PT33&lpg=PT33&dq=+%22angle+of+bank%22&source=web&ots=iYTi_mZAra&sig=ytjcmr9RStdIdgZzaiBJJ-wxjts&hl=en#PPT32,M1.
^ Note: unlike the Cartesian unit vectors i and j, which are constant, in polar coordinates the direction of the unit vectors u_r and u_θ depend on θ, and so in general have non-zero time derivatives.
^ 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.
^ Notice that this local coordinate system is not autonomous; for example, its rotation in time is dictated by the trajectory traced by the particle. Note also that the radial vector r(t) does not represent the radius of curvature of the path.
^ John Robert Taylor (2005). Classical Mechanics. Sausalito CA: University Science Books. p. pp.28-29. ISBN 189138922X. http://books.google.com/books?id=P1kCtNr-pJsC&printsec=index&dq=isbn=189138922X&lr=&as_brr=0&source=gbs_toc_s&cad=1#PPA29,M1.
^ Cornelius Lanczos (1986). The Variational Principles of Mechanics. New York: Courier Dover Publications. p. 103. ISBN 0486650677. http://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA103&dq=%22Euler+force%22&lr=&as_brr=0&sig=UV46Q9NIrYWwn5EmYpPv-LPuZd0#PPA103,M1.
^ See, for example, Howard D. Curtis (2005). Orbital Mechanics for Engineering Students. Butterworth-Heinemann. p. 5. ISBN 0750661690. http://books.google.com/books?id=6aO9aGNBAgIC&pg=PA193&dq=orbit+%22coordinate+system%22&lr=&as_brr=0&sig=p5hZldx_U1CnV0Ggc29YBLgLj9k#PPA5,M1.
^ S. Y. Lee (2004). Accelerator physics (2nd Edition ed.). Hackensack NJ: World Scientific. p. 37. ISBN 981256182X. http://books.google.com/books?id=VTc8Sdld5S8C&pg=PA37&dq=orbit+%22coordinate+system%22&lr=&as_brr=0&sig=h5GU58FVOzEGclxsJkYQuVvtWkU.
^ 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.
^ Zhilin Li & Kazufumi Ito (2006). The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains. Philadelphia: Society for Industrial and Applied Mathematics. p. 16. ISBN 0898716098. http://books.google.com/books?id=_E084AX-iO8C&pg=PA16&dq=%22local+coordinates%22&lr=&as_brr=0&sig=ACfU3U2p_S2c7vRzd1vabU9WhIBJXk8ESw.
^ K L Kumar (2003). Engineering Mechanics. New Delhi: Tata McGraw-Hill. p. 339. ISBN 0070494738. http://books.google.com/books?id=QabMJsCf2zgC&pg=PA339&dq=%22path+coordinates%22&lr=&as_brr=0&sig=ACfU3U1ZlP_syppme85cv4pimhLxyUOLug.
^ Lakshmana C. Rao, J. Lakshminarasimhan, Raju Sethuraman & SM Sivakuma (2004). Engineering Dynamics: Statics and Dynamics. Prentice Hall of India. p. 133. ISBN 8120321898. http://books.google.com/books?id=F7gaa1ShPKIC&pg=PA134&dq=%22path+coordinates%22&lr=&as_brr=0&sig=ACfU3U0PT2mGvAHroVJFVXGB46y6zLWaGA#PPA132,M1.
^ Shigeyuki Morita (2001). Geometry of Differential Forms. American Mathematical Society. p. 1. ISBN 0821810456. http://books.google.com/books?id=5N33Of2RzjsC&pg=PA1&dq=%22local+coordinates%22&lr=&as_brr=0&sig=ACfU3U3dnL01bMDu8d0GCmCC9eI717lsPA.
^ 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: Horace Lamb (1897). An Elementary Course of Infinitesimal Calculus. University Press. p. 406. http://books.google.com/books?id=eDM6AAAAMAAJ&pg=PA406&dq=%22osculating+circle%22&lr=&as_brr=0.
^ ^a ^b Guang Chen & Fook Fah Yap (2003). An Introduction to Planar Dynamics (3rd Edition ed.). Central Learning Asia/Thomson Learning Asia. p. 34. ISBN 9812435689. http://books.google.com/books?id=xt09XiZBzPEC&pg=PA34&dq=motion+%22center+of+curvature%22&lr=&as_brr=0&sig=ACfU3U2lKY09hG88_XSHtU9H_xuaXXdGlA.
^ R. Douglas Gregory (2006). Classical Mechanics: An Undergraduate Text. Cambridge University Press. p. 20. ISBN 0521826780. http://books.google.com/books?id=uAfUQmQbzOkC&pg=RA1-PA18&dq=particle+curve+normal+tangent&lr=&as_brr=0&sig=ACfU3U3_WR6_esuEz-mUMmOXuZabQY6Now#PRA1-PA20,M1.
^ Edmund Taylor Whittaker & William McCrea (1988). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: with an introduction to the problem of three bodies (4rth Edition ed.). Cambridge University Press. p. 20. ISBN 0521358833. http://books.google.com/books?id=epH1hCB7N2MC&pg=PA20&vq=radius+of+curvature&dq=particle+movement+%22radius+of+curvature%22+acceleration+-soap&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U3cKHnTYJNja9o0t_Iw2VeRSGEWCg.
^ Jerry H. Ginsberg (2007). Engineering Dynamics. Cambridge University Press. p. 33. ISBN 0521883032. http://books.google.com/books?id=je0W8N5oXd4C&pg=PA723&dq=osculating+%22planar+motion%22&lr=&as_brr=0&sig=ACfU3U0Yuca2DhshUowHlkVtw_bRSR-qww#PPA33,M1.
^ Joseph F. Shelley (1990). 800 solved problems in vector mechanics for engineers: Dynamics. McGraw-Hill Professional. p. 47. ISBN 0070566879. http://books.google.com/books?id=ByNrVgf041MC&pg=PA46&dq=particle+movement+%22radius+of+curvature%22+acceleration+-soap&lr=&as_brr=0&sig=ACfU3U3xRX2OpTCS7_OF87Yi07fQmymg7A#PPA47,M1.
^ Larry C. Andrews & Ronald L. Phillips (2003). Mathematical Techniques for Engineers and Scientists. SPIE Press. p. 164. ISBN 0819445061. http://books.google.com/books?id=MwrDfvrQyWYC&pg=PA164&dq=particle+%22planar+motion%22&lr=&as_brr=0&sig=ACfU3U2LpH6ofhuuC2UiED0pf38wbspY8A#PPA164,M1.
^ Ch V Ramana Murthy & NC Srinivas (2001). Applied Mathematics. New Delhi: S. Chand & Co.. p. 337. ISBN 81-219-2082-5. http://books.google.com/books?id=Q0Pvv4vWOlQC&pg=PA337&vq=frenet&dq=isbn=8121920825&source=gbs_search_s&sig=ACfU3U3S5vGMS-NnraAEmpBf6B9bB2wK6A.
^ The article on curvature treats a more general case where the curve is parametrized by an arbitrary variable (denoted t), rather than by the arc length s.
^ Ahmed A. Shabana, Khaled E. Zaazaa, Hiroyuki Sugiyama (2007). Railroad Vehicle Dynamics: A Computational Approach. CRC Press. p. 91. ISBN 1420045814. http://books.google.com/books?id=YgIDSQT0FaUC&pg=RA1-PA207&dq=%22generalized+coordinate%22&lr=&as_brr=0&sig=ACfU3U2tosoLUEAUNkGu2x8TTtuxLfeLGQ#PRA1-PA91,M1.

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.

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centripetal force

Contents

Formula

Sources of centripetal force

Analysis of several cases

Uniform circular motion

Geometric derivation

Derivation using vectors

Example: The banked turn

Nonuniform circular motion

General planar motion

Polar coordinates

Local coordinates

Alternative approach

Example: circular motion

See also

Notes and references

Further reading

External links