NationMaster - Encyclopedia: Centrifugal acceleration

Centrifugal force (from Latin centrum "center" and fugere "to flee") is a term which may refer to two different forces which are related to rotation. Both of them are oriented away from the axis of rotation, but the object on which they are exerted differs. Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ... In physics, the Newtonian definition of force is the rate of change of momentum. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ...

The reactive centrifugal force is the reaction to the centripetal force. This is equal in magnitude to the centripetal force, directed away from the center of rotation, and is exerted by the rotating object upon the object which exerts the centripetal force. As it is an actual force, it is always present, independent of the choice of reference frame.

The fictitious centrifugal force appears when a rotating reference frame is used for analyzing the system. The centrifugal force is exerted on all objects, and directed away from the axis of rotation.

Both of the above can be observed in action on a passenger riding in a car. If the car swerves around a corner, the passenger's body pushes against the outer edge of the car. This is the reactive centrifugal force, which is called a reaction force because it results from passive interaction with the car which actively pushes against the body. In classical physics, a reaction is defined by Newtons third law that can be formulated like Whenever one body exerts a force upon a second body, the second body exerts on the first body, a force that is equal in magnitude, and opposite in direction, to the force exerted... Centripetal force is the force pulling of an object toward the center of a circular path as the object goes around the circle. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... A fictitious force is a force used to explain acceleration in a non-inertial frame of reference, such as a rotating frame. ...

Using a reference frame which is fixed relative to the car (a model which those inside the car will often find natural) and while ignoring its rotation, it looks like an external force is pulling the passenger out of the car. This is the fictitious centrifugal force, so called because it is not an actual force exerted by some other object.

1 Reactive centrifugal force
2 Rotating reference frames
- 2.1 Derivation
3 Fictitious centrifugal force
- 3.1 Potential energy of the fictitious centrifugal force
4 Centrifugal forces in statics
5 Confusion and misconceptions
6 Applications
7 See also
8 References and external links

Reactive centrifugal force

When viewed from an inertial frame of reference, the application of Newton's laws of motion is simple. The passenger's inertia resists acceleration, keeping the passenger moving with constant speed and direction as the car begins to turn. From this point of view, the passenger does not gravitate toward the outside of the car; instead, the car curves to meet the passenger. An inertial frame is a coordinate system defined by the non-accelerated motion of objects with a common direction and speed (as opposed to a non-inertial reference frame). ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Acceleration is the time rate of change of velocity, and at any point on a v-t graph, it is given by the slope of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. ...

Once the car contacts the passenger, it then applies a sidewise force to accelerate him or her around the turn with the car. This force is called a centripetal ("center seeking") force because its vector changes direction to continue to point toward the center of the car's arc as the car traverses it.

If the car is acting upon the passenger, then the passenger must be acting upon the car with an equal and opposite force. Being opposite, this reaction force is directed away from the center, therefore centrifugal. It is critical to realize that this centrifugal force acts upon the car, not the passenger. Basing the physics on an inertial reference frame, there is no mystery force pulling the passenger outward.

The centrifugal reaction force with which the passenger pushes back against the door of the car is trivial and is simply given by:

$mathbf{F}_mathrm{centrifugal} ,$	$= - m mathbf{a}_mathrm{centripetal} ,$
	$= m omega^2 mathbf{r}_perp ,$

where $m,$ is the mass of the rotating object.

Rotating reference frames

In the classical approach, the inertial frame remains the true reference for the laws of mechanics. When using a rotating reference frame, the laws of physics are mapped from the most convenient inertial frame to that rotating frame. Assuming a constant rotation speed, this is achieved by adding to every object two coordinate accelerations which correct for the rotation of the coordinate axes. A rotating frame of reference is a coordinate system that describes how physics appears when measured against a hypothetical network of rigid rulers extending from a rotating body. ...

$mathbf{a}_mathrm{rot},$	$=mathbf{a} - 2mathbf{omega times v} - mathbf{omega times (omega times r)} ,$
	$=mathbf{a + a_mathrm{Coriolis} + a_mathrm{centrifugal}} ,$

where $mathbf{a}_mathrm{rot},$ is the acceleration relative to the rotating frame, $mathbf{a},$ is the acceleration relative to the inertial frame, $mathbf{omega},$ is the angular velocity vector describing the rotation of the reference frame, $mathbf{v},$ is the velocity of the body relative to the rotating frame, and $mathbf{r},$ is a vector from an arbitrary point on the rotation axis to the body. A derivation can be found in the article fictitious force. Angular velocity describes the speed of rotation. ... A fictitious force is a force used to explain acceleration in a non-inertial frame of reference, such as a rotating frame. ...

The last term is the centrifugal acceleration, so we have:

$mathbf{a}_textrm{centrifugal} = - mathbf{omega times (omega times r)} = omega^2 mathbf{r}_perp$

where $mathbf{r_perp}$ is the component of $mathbf{r},$ perpendicular to the axis of rotation.

Derivation

If we have two frames, one inertial and one rotating with a constant angular velocity $vec omega$ , a time derivative of a vector in the rotating frame, $left ( frac{d}{dt} right ) _r$ , is transformed to the time derivative in the inertial frame, $left ( frac{d}{dt} right ) _i$ , by the following relation:

$left ( frac{d}{dt} right ) _i = left ( frac{d}{dt} right ) _r + vec omega times$

This relationship is one between two operators. Now, acceleration is the second derivative of position with respect to time. So, applying the above transformation to the position vector $vec r$ once gets you: In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...

$dot vec r_i = left ( frac{d vec r}{dt} right ) _i = left ( frac{d vec r}{dt} right ) _r + omega times vec r$

Putting $dot vec r_i$ back into the transformation, you get:

$ddot vec r_i = left ( frac{d dot vec r}{dt} right ) _i = left ( frac{d dot vec r}{dt} right ) _r + omega times dot vec r$

$ddot vec r_i = left ( frac{d^2 vec r}{dt^2} right ) _i = left ( frac{d}{dt} right ) _r left ( left ( frac{d vec r}{dt} right ) _r + omega times vec r right ) + vec omega times left ( left ( frac{d vec r}{dt} right ) _r + omega times vec r right )$

Because $vec omega$ is a contant vector - that is the rotating reference frame is rotating constantly in the same direction - its time derivative is zero. So, simplifying:

$ddot vec r_i = left ( frac{d^2 vec r}{dt^2} right ) _i = left ( frac{d^2 vec r}{dt^2} right ) _r + omega times left ( frac{d vec r}{dt} right ) _r + vec omega times left ( frac{d vec r}{dt} right ) _r + omega times omega times vec r$

$ddot vec r_i = left ( frac{d^2 vec r}{dt^2} right ) _i = left ( frac{d^2 vec r}{dt^2} right ) _r + 2 vec omega times left ( frac{d vec r}{dt} right ) _r + omega times omega times vec r$

Finally, putting in $vec a$ for $left ( frac{d^2 vec r}{dt^2} right )$ and $vec v_r$ for $left ( frac{d vec r}{dt} right ) _r$ , we get the following:

$vec a_i = vec a_r + 2 vec omega times vec v_r + vec omega times left ( vec omega times vec r right )$

Moving things to the other side, but reversing one cross-product in each term, you find:

$vec a_r = vec a_i + 2 vec v_r times vec omega + vec omega times left ( vec r times vec omega right )$

This tells us that $vec a_r$ , the acceleration of some object at $vec r$ as observed by someone at rest in the rotating frame is equal to the acceleration, $vec a_i$ , as observed by an observer in the inertial, non-rotating frame, plus $2 vec v_r times vec omega$ , which is the coriolis effect's contribution to the acceleration, and $vec omega times left ( vec r times vec omega right )$ , which is the centrifugal acceleration term. This low pressure system over Iceland spins counter-clockwise due to the Coriolis effect. ...

Fictitious centrifugal force

An alternative way of dealing with a rotating frame of reference is to make Newton's laws of motion artificially valid by adding fictitious forces that are pretended to be the cause of the above acceleration terms. In particular, the centrifugal acceleration is added to the motion of every object, and attributed to a fictitious centrifugal force, given by:

$mathbf{F}_mathrm{centrifugal} ,$	$= m mathbf{a}_mathrm{centrifugal} ,$
	$=m omega^2 mathbf{r}_perp ,$

where $m,$ is the mass of the object.

This centrifugal force is a sufficient correction to Newton's second law only if the body is stationary in the rotating frame. For bodies that move with respect to the rotating frame it must be supplemented with a fictitious Coriolis force. For example, a body that is stationary relative to the non-rotating frame, will be rotating when viewed from the rotating frame. The centripetal force of $-m omega^2 mathbf{r}_perp$ required to account for this apparent rotation is the sum of the centrifugal force ( $m omega^2 mathbf{r}_perp$ ) and the Coriolis force ( $-2m mathbf{omega times v} = -2m omega^2 mathbf{r}_perp$ ). Since this centripetal force includes contributions from only fictitious forces, it has no reactive counterpart. In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ...

Potential energy of the fictitious centrifugal force

The fictitious centrifugal force can be described by a potential energy of the form Potential energy is stored energy. ...

$E_p = -frac{3}{2} m omega^2 r_perp^2$

This is useful, for example, in calculating the form of the water surface $h(r),$ in a rotating bucket: requiring the potential energy per unit mass on the surface $gh(r) - frac{1}{2}omega^2 r^2,$ to be constant, we obtain the parabolic form $h(r) = frac{omega^2}{2g}r^2 + C$ (where $C$ is a constant). A parabola A parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a cone, and a plane tangent to the cone or parallel to some plane tangent to the cone. ...

Similarly, the potential energy of the centrifugal force is often used in the calculation of the height of the tides on the Earth (where the centrifugal force is included to account for the rotation of the Earth around the Earth-Moon center of mass). The tide is the regular rising and falling of the oceans surface caused by changes in gravitational forces external to the Earth. ...

The principle of operation of the centrifuge also can be simply understood in terms of this expression for the potential energy, which shows that it is favorable energetically when the volume far from the axis of rotation is occupied by the heavier substance. A laboratory centrifuge tabletop centrifuge A centrifuge is a piece of equipment that puts a substance in rotation around a fixed axis in order for the centrifugal force to separate a fluid from a fluid or from a solid substance. ...

Centrifugal forces in statics

Consider a ball that swings around a stationary pivot to which it is tethered by a light, strong rope. There is tension in the rope, pulling inwards on the ball (the centripetal force) and simultaneously pulling outwards on the pivot (the reactive centrifugal force). The tension is real, so these two forces still exist if we move to a corotating frame. However, in the rotating frame there is also a fictitious centrifugal force that pulls outwards on the ball. It is distinct from the reactive centrifugal force that pulls outward on the pivot. Two characters in Napoleon Dynamite get ready to play tetherball. ...

When solving statics problems in a rotating frame (e.g. when calculating the internal stresses in a flywheel) it is convenient to think of the fictitious centrifugal force as being transmitted through the rope and becoming the pull on the pivot. In statics one often considers a force "the same" before and after it has been conveyed by a structural element, so according to this view the reaction force on the pivot is the fictitious force. Statics is the branch of physics concerned with physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at rest under the action of external forces of equilibrium. ... Spoked flywheel A flywheel is a heavy rotating disk used as a repository for angular momentum. ...

This identification often leads to confusion about the "fictitious" nature of the centrifugal force, because the pull on the pivot is a perfectly real force. The confusion can be resolved by noting that the distinction between fictitious and real forces depends on the frame of reference that one chooses for the laws of physics. On the other hand, considering the reaction force to be the fictitious force is only valid in statics, that is, once we have decided to always use that particular reference frame in which the entire system is stationary. The convenience of viewing a transmitted force as the same as the original force comes at the cost of a meaningful distinction between whether a force is real or fictitious. Statics is the branch of physics concerned with physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at rest under the action of external forces of equilibrium. ...

Confusion and misconceptions

Centrifugal force can be a confusing term because it is used (or misused) in more than one instance, and because sloppy labeling can obscure which forces are acting upon which objects in a system (which is true for physics in general). When diagramming forces in a system, one must describe each object separately, attaching only those forces acting upon it (not forces that it exerts upon other objects).

One can avoid dealing with fictitious forces entirely by analyzing systems using inertial frames of reference for the physics; and when convenient, one simply maps to a rotating frame without forgetting about the frame rotation, as shown above. Such is standard practice in mechanics textbooks. An inertial frame is a coordinate system defined by the non-accelerated motion of objects with a common direction and speed (as opposed to a non-inertial reference frame). ...

Because rotating frames are not vital for understanding mechanics, science teachers often de-emphasize the fictitious centrifugal force that appears to exist in a rotating reference frame. However, in their zeal to stamp out the misunderstanding of the term in this one case, they may try to expunge it from the language entirely.

Applications

A Centrifugal governor regulates the speed of an engine by using spinning masses that respond to centrifugal force generated by the engine. If the engine increases in speed, the masses move and trigger a cut in the throttle.
Centrifugal forces can be used to generate artificial gravity. Proposals have been made to have gravity generated in space stations designed to rotate. The Mars Gravity Biosatellite will use study the effects of Mars level gravity on mice with simulated gravity from centrifugal force.
Centrifuges are used in science and industry to separate substances by their relative masses.
Some Amusement park rides makes use of centrifugal forces. For instance, a Gravitron’s spin forces riders against a wall and allows riders to be elevated above the machine’s floor in defiance of Earth’s gravity.

A centrifugal governor controls the speed of an engine by regulating the amount of fuel admitted, so as to maintain a near constant speed whatever the load or fuel supply conditions. ... In an engine, the throttle is the mechanism by which the engines power is increased or decreased. ... Artificial gravity is a simulation of gravity in outer space or free-fall. ... The Mars Gravity Biosatellite project is a joint venture of the Georgia Institute of Technology, the University of Queensland, and MIT. The projects mission is to investigate the effects of [Mars-level gravity on mammals. ... For the Roman god, see Mars (mythology). ... A laboratory centrifuge tabletop centrifuge A centrifuge is a piece of equipment that puts a substance in rotation around a fixed axis in order for the centrifugal force to separate a fluid from a fluid or from a solid substance. ... Amusement park is the more generic term for a collection of rides and other entertainment attractions assembled for the purpose of entertaining a fairly large group of people. ... Types of amusement rides include: Bumper cars Carousels Dark rides Ferris wheels Freefall towers Hall of mirrors Log flumes Loop-O-Plane Motion platforms Observation towers Octopus (ride) Roller coasters Scenic railways switchbacks Horror train Shoot-the-Chutes The Zipper Tilt-A-Whirl See also Amusement park Closed rides and... The Gravitron is an amusement ride which involves fast rotations around an axis creating centrifugal force. ...

References and external links

Newton's description in Principia
Centrifugal force - Columbia electronic encyclopedia
Centrifugal Force - from ScienceWorld
Java applet demonstrating centrifugal and Coriolis forces
M. Alonso and E.J. Finn, Fundamental university physics, Addison-Wesley

Categories: Force | Mechanics ScienceWorld, also known as Eric Weissteins World of Science, is a web site that opened to the general public in January 2002. ...

Results from FactBites:

Centrifugal force - Wikipedia, the free encyclopedia (1629 words)
	Centrifugal force (from Latin centrum "center" and fugere "to flee") is a term which may refer to two different forces which are related to rotation.
	Similarly, the potential energy of the centrifugal force is often used in the calculation of the height of the tides on the Earth (where the centrifugal force is included to account for the rotation of the Earth around the Earth-Moon center of mass).
	Centrifuges are used in science and industry to separate substances by their relative masses.

Centrifuge - Wikipedia, the free encyclopedia (694 words)
	The acceleration is often quoted in multiples of g, the acceleration due to gravity at the Earth's surface.
	Centrifuges are used in the attraction Mission: SPACE, located at Epcot in Walt Disney World, which propels riders using a combination of a centrifuge and a motion simulator to simulate the feeling of going into space.
	The heavier isotopes of uranium (uranium-238) in the uranium hexafluoride gas tend to concentrate at the walls of the centrifuge as it spins, while the desired uranium-235 isotopes are extracted and concentrated with a scoop selectively placed inside the centrifuge.

More results at FactBites »

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