Francais | English | Espanõl

Centrifugal force

From Wikipedia, the free encyclopedia

Jump to: navigation, search

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.

  • A pseudo or "fictitious" centrifugal force appears when a rotating reference frame is used for analysis. The (true) frame acceleration is substituted by a (fictitious) centrifugal force that is exerted on all objects, and directed away from the axis of rotation.

ACCELERATION AND FORCE IN CIRCULAR MOTION by Peter Signell</ref>

Both of the above can be easily observed in action for a passenger riding in a car. If a car swerves around a corner, a passenger's body seems to move towards the outer edge of the car and then pushes against the door.

In the reference frame that is rotating together with the car (a model which those inside the car will often find natural), it looks like a force is pushing the passenger away from the center of the bend. This is not an actual force exerted by some other object, but results from ignoring the car's acceleration. A number of physicists treat it much as if it were a real force, as they find that it makes calculations simpler and gives correct results.

However, the force with which the passenger pushes against the door is very real. That force is called a reaction force because it results from passive interaction with the car which actively pushes against the body. As it is directed outward, it is a centrifugal force. Note that this real centrifugal force does not appear until the person touches the body of the car.

Contents

[edit] 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 path which the car follows; instead, the car's path curves to meet the passenger.

Once the car contacts the passenger, it then applies a sideways 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[1]. It is critical to realize that this centrifugal force acts upon the car, not the passenger.

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

<math>\mathbf{F}_\mathrm{centrifugal} \,</math> <math>= - m \mathbf{a}_\mathrm{centripetal} \,</math>
<math>= m \omega^2 \mathbf{r}_\perp \,</math>

where <math>m\,</math> is the mass of the rotating object.

The reactive centrifugal force[2] is as "real" as it can get as it can open the door of the car and even shatter DVD's[3]; however, it is rarely used in modern discussions[4][5].

[edit] Rotating reference frames

In the classical approach, the inertial frame remains the true reference for the laws of mechanics and analysis. 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.

<math>\mathbf{a}_\mathrm{rot}\,</math> <math>=\mathbf{a} - 2\mathbf{\omega \times v} - \mathbf{\omega \times (\omega \times r)} \,</math>
<math>=\mathbf{a + a_\mathrm{Coriolis} + a_\mathrm{centrifugal}} \,</math>

where <math>\mathbf{a}_\mathrm{rot}\,</math> is the acceleration relative to the rotating frame, <math>\mathbf{a}\,</math> is the acceleration relative to the inertial frame, <math>\mathbf{\omega}\,</math> is the angular velocity vector describing the rotation of the reference frame, <math>\mathbf{v}\,</math> is the velocity of the body relative to the rotating frame, and <math>\mathbf{r}\,</math> is a vector from an arbitrary point on the rotation axis to the body. A derivation can be found in the article fictitious force.

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

<math> \mathbf{a}_\textrm{centrifugal} = - \mathbf{\omega \times (\omega \times r)} = \omega^2 \mathbf{r}_\perp </math>

where <math>\mathbf{r_\perp}</math> is the component of <math>\mathbf{r}\,</math> perpendicular to the axis of rotation.

[edit] Derivation

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

<math> \left ( \frac{d}{dt} \right ) _i = \left ( \frac{d}{dt} \right ) _r + \vec \omega \times </math>

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 <math> \vec r </math> once gets you:

<math> \dot \vec r_i = \left ( \frac{d \vec r}{dt} \right ) _i = \left ( \frac{d \vec r}{dt} \right ) _r + \omega \times \vec r</math>

Putting <math> \dot \vec r_i </math> back into the transformation, you get:

<math> \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</math>
<math> \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 )</math>

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

<math> \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</math>
<math> \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</math>

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

<math> \vec a_i = \vec a_r + 2 \vec \omega \times \vec v_r + \vec \omega \times \left ( \vec \omega \times \vec r \right )</math>

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

<math> \vec a_r = \vec a_i + 2 \vec v_r \times \vec \omega + \vec \omega \times \left ( \vec r \times \vec \omega \right )</math>

This tells us that <math>\vec a_r</math>, the acceleration of some object at <math>\vec r</math> as observed by someone at rest in the rotating frame is equal to the acceleration, <math>\vec a_i</math>, as observed by an observer in the inertial, non-rotating frame, plus <math>2 \vec v_r \times \vec \omega</math>, which is the coriolis effect's contribution to the acceleration, and <math>\vec \omega \times \left ( \vec r \times \vec \omega \right )</math>, which is the centrifugal acceleration term.

[edit] Pseudo forces

An alternative way of dealing with a rotating frame of reference is to make Newton's laws of motion artificially valid by adding pseudo forces 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 centrifugal force, given by:

<math>\mathbf{F}_\mathrm{centrifugal} \,</math> <math>= m \mathbf{a}_\mathrm{centrifugal} \,</math>
<math>=m \omega^2 \mathbf{r}_\perp \,</math>

where <math>m\,</math> is the mass of the object.

This pseudo or fictitious[6] 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 second pseudo force, the "Coriolis force":

<math>\mathbf{F}_\mathrm{coriolis} = -2m \mathbf{\omega \times v} = -2m \omega^2 \mathbf{r}_\perp</math>

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 <math>-m \omega^2 \mathbf{r}_\perp</math> required to account for this apparent rotation is the sum of the centrifugal pseudo force (<math>m \omega^2 \mathbf{r}_\perp</math>) and the Coriolis force (<math>-2m \mathbf{\omega \times v} = -2m \omega^2 \mathbf{r}_\perp</math>). Since this centripetal force includes contributions from only pseudo forces, it has no reactive counterpart.

[edit] Potential energy

Image:Coriolis effect11.jpg

The fictitious centrifugal force can be described by a potential energy of the form

<math>E_p = -\frac{1}{2} m \omega^2 r_\perp^2</math>

This is useful, for example, in calculating the form of the water surface <math>h(r)\,</math> in a rotating bucket: requiring the potential energy per unit mass on the surface <math>gh(r) - \frac{1}{2}\omega^2 r^2\,</math> to be constant, we obtain the parabolic form <math>h(r) = \frac{\omega^2}{2g}r^2 + C</math> (where <math>C</math> is a constant).

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 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.

The coriolis force has no equivalent potential, as it acts perpendicular to the velocity vector and hence rotates the direction of motion, but does not change the energy of a body.

[edit] 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.

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.

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 only works 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.

[edit] 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 pseudo 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.

Because rotating frames are not vital for understanding mechanics, science teachers often de-emphasize the centrifugal forces that appear 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.

[edit] 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 make 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.

[edit] See also

[edit] References

[edit] Notes

<references/>da:Centrifugalkraft de:Zentrifugalkraft es:Fuerza centrífuga fi:Keskipakoisvoima fr:Force centrifuge he:כוח צנטריפוגלי it:Forza centrifuga ko:원심력 nl:Middelpuntvliedende kracht ja:遠心力 pl:Siła odśrodkowa ru:Центробежная сила sv:Centrifugalkraft tr:Merkezkaç vi:Lực ly tâm zh:離心力

Retrieved from "http://en.wikivisual.com/index.php/Centrifugal_force"
Personal tools