Centrifugal

Centrifugal force (from Latin centrum "center" and fugere "to avoid") is one of the fictitious forces that appears to act on an object when its motion is viewed from a rotating frame of reference.

Centripetal force

The force that maintains circular motion is called centripetal force. If no force is exerted on an object, it moves in a straight line at a constant speed. To make the object deviate from that straight path into in a circular one, a centripetal ("center seeking") force must be exerted at right angles to the object's velocity, directed toward the center of the circle. Since this causes a change in the direction of the object's velocity, the centripetal force causes a corresponding centripetal acceleration, also toward the center.

Main article: Centripetal force

Examples

A classic example of these forces in action is a passenger riding in a car. The car is initially heading along a straight line but then swerves around a corner. If we keep track of the passenger's motion relative to the car, the passenger's body is apparently pushed towards the outer edge of the car. This is the result attributed to the centrifugal force. It is called a fictitious force because it does not reult from interaction with another object, but is merely an artifact of our considering the movement relative to the car, rather than relative to the road.

Inertial frame of reference

When viewed from an inertial frame of reference, what is really happening is that the passenger's inertia resists any change of motion and keeps the passenger moving along the initial straight line of motion. From this point of view, the only reason that the passenger is pushed to the outside of the car is that the person is still travelling in a straight line, and the car has accelerated. Once the passenger hits the door of the car, the car is then able to apply the centripetal force on the passenger to accelerate him or her around the turn with the car. Friction between the seat of the car and the seat of the passenger's pants is also a component of the centripetal force, and at lower speeds, where passengers do not slide, friction accounts for all of it. In turn, the passenger also exerts a reaction force upon the door: some people also call this force the centrifugal force.

Rotating frame of reference

Viewed from a frame of reference that co-rotates with the car, the passenger remains stationary. Because the centripetal force exerted by the door of the car still exists independently of the reference frame, it appears that Newton's second law has been broken: A net force acts on the passenger, yet he does not accelerate. Newton's second law indeed does not hold in rotating frames of reference, but it can still be used for calculations if we add a correction term, which takes the form of a force directed away from the center of rotation. This is the centrifugal force; it is minus the centripetal force and given by:

,

where ω is the vector angular velocity of the rotation and r is a vector from an arbitrary point on the rotation axis to the body (with mass m) experiencing the centripetal force. See Fictitious force for a derivation.

The centrifugal force is a sufficient correction term in Newton's second law only if the body being considered is stationary in the rotating frame. For bodies that move with respect to the rotating frame it must be supplemented with the Coriolis force.

The reaction to the centripetal force

Whenever a body is stationary in a rotating frame, there must be some force that cancels out the centrifugal force, or it would be seen to accelerate away from the center. This is the centripetal force, and it is always a real force. By Newton's Third Law there must be an equal and opposite reaction to the centripetal force, applying to the object that exerts the centripetal force. This reaction force is also a real force, but because it has the same magnitude and direction as the centrifugal force (either of them is minus the centripetal force) it is easy to confuse the two.

For example, 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 reaction 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 centrifugal force that pulls outwards on the ball. It is distinct from the reaction force that pulls outward on the pivot.

When solving statics problems in the rotating frame (e.g. when calculating the internal stesses in a flywheel) it is convenient to think of the 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 centrifugal 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 is only relevant if we plan to switch between different frames of reference. On the other hand, considering the reaction force to be the centrifugal 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.

Confusion and misconceptions

Centrifugal force can be a confusing subject. Because rotating frames are not vital for understanding mechanics, teachers of science often de-emphasize the centrifugal force when teaching about circular motion, and instead emphasize the central role (quite literally) of the centripetal force, since it is the force responsible for maintaining circular motion and centripetal acceleration. It is usually highly recommended that students of elementary mechanics think about rotational physics using only inertial frames of reference, thus avoiding the need to think about centrifugal force.

Some commonly encountered misconceptions about centrifugal force include:

• An object that moves in a circle creates a centrifugal force.
  No matter how objects move, there are never any centrifugal force in an intertial frame of reference. Centrifugal forces always occur when objects are viewed in a rotating frame of reference, irrespective of whether the object follows the frame's rotation. (But if the object is not stationary the centrifugal force is not the whole story; see Coriolis force).

• It is always wrong to speak about centrifugal force.
  No, centrifugal force is a valid concept in rotating frames. This misconception is very common and is often caused by misunderstanding a teacher's explanation that in an intertial frame there is no centrifugal force (especially if the teacher does not mention rotating frames at all). Another possible cause is the term "fictitious" which one could easily think means "unscientific superstition".

• Centrifugal force is not fictitious but real.
  The centrifugal force describes an effect that actually happens when a system is viewed from a rotating frame. However, in this context "real" is a technical term which denotes properties that the centrifugal force does not have. That the force is not real does not mean that it is not valid.

• Centrifugal force is just another word for inertia.
  Centrifugal force can reasonably be said to be caused by intertia, but the two words do not mean the same. Inertia is the principle that undisturbed bodies tend to move in straight lines; it is not a quantity that can be calculated with. The centrifugal force, on the other hand, is vector quantity with a certain direction and magnitude, with the dimension of a force (i.e., mass times acceleration).

• The centrifugal force is the reaction force to the centripetal force.
  This is a sometimes useful mental shorthand within the rotating frame only, as described above. It is not true at all when a rotating movement is described in an intertial frame.

External links

• Centrifugal Force - from ScienceWorld
• Java applet demonstrating centrifugal and Coriolis forces