Reactive centrifugal force

From Wikipedia, the free encyclopedia

It has been suggested that this article or section be merged into Centripetal force. (Discuss)

For the pseudoforce that appears with a noninertial, rotating frame of reference, see Centrifugal force.

A reactive centrifugal force refers to a force which is generated by a rotation and acts away from the axis of rotation. It is the reaction force to a centripetal force.

A mass undergoing circular motion constantly accelerates toward the center of the circle. This centripetal acceleration is caused by a centripetal force, which is exerted on the mass by some other object. In accordance with Newton's Third Law of Motion, the mass exerts an equal and opposite force on the object. This is the real or "reactive" centrifugal force: it is directed away from the center of rotation, and is exerted by the rotating mass on the object which imposes the centripetal acceleration.^[1] ^[2] Although this sense was used by Isaac Newton, it is only occasionally used in modern discussions.

1 Reactive centrifugal force
- 1.1 Example: The turning car
2 Confusion and misconceptions
3 Fictitious forces
4 See also
5 References
6 External links

[edit] Reactive centrifugal force

Figure 1:– 1: Ball in uniform circular motion held by string tied to post; 2: Centripetal force on ball; 3: Reactive centrifugal force on string and post; 4: Centripetal force on string and ball; 5: Combination of forces on string; the string is in tension 6: Combination of forces on post–the force indicated by the blue arrow is applied by the post-hole that holds the post.

Figure 1 (upper left) shows a ball in uniform circular motion held to its path by a massless string tied to a post stuck in the ground. Newton's second law requires that any body not moving in a straight line is subject to a force, and indeed, panel 2 shows the ball is subjected to a centripetal force by the string. Newton's third law states that if the string exerts a centripetal force on the ball, the ball will exert an equal reaction upon the string and post, the reactive centrifugal force shown in panel 3. Panel 4 shows the string and ball combined, both in uniform circular motion. The centripetal force of panel 2 is applied by the post to the end of the string. The string is looked at separately in panel 5, and is subject to the reactive centrifugal force at one end that balances the centripetal force at the other end. The string is subject to zero net force and so does not move radially. As an idealization, it is supposed that the string has negligible mass, so we can ignore the forces on the string to maintain its circular motion. (Panel 5 also shows the string is under tension.) Panel 6 shows the post separately. Due to Newton's third law, the post is subject to the reactive centrifugal force, a reaction to the centripetal force of panel 4, and also is subject to the centripetal force exerted by the post-hole on the post. The net force on the post is zero, so it does not move.

Figure 2: Car with passenger making a turn. The road exerts a centripetal force on the car to force a curved path. Apparently the passenger is facing backwards; hopefully they are not driving.

Figure 3: Exploded view showing force components. Each object is subject to a net inward force that is the difference between the outward reactive centrifugal force and an inward centripetal force. This net inward force is the centripetal force upon that object necessary for it to make the turn. (Torque is ignored here, for simplicity.)

[edit] Example: The turning car

See also: Centripetal force#Example: The banked turn

Here is a more picturesque example: A car with a passenger inside driving around a curve.^[3] See Figure 2. The road exerts a centripetal force upon the car, forcing the car to make the turn. This force is called a centripetal ("center seeking") force because its vector changes direction to continue to point toward the center (precisely, the center of curvature) of the car's arc as the car traverses it.

Viewed from an inertial frame of reference, the passenger's inertia resists acceleration, keeping the passenger moving with constant speed and direction as the car begins to turn; that is, failing interference by the car, the passenger is going to travel a straight line, not go around the turn. From this point of view, the passenger does not gravitate toward the outside of the path the car follows; instead, the car's path curves to meet the passenger.

The car contacts the passenger through the car seat. The car seat applies a sideways force to accelerate the passenger around the turn with the car. If the car seat is slippery, the passenger slides against the wall of the car, and the car wall exerts inward force upon the passenger.

If the car is acting upon the passenger, then the passenger must be acting upon the car with an equal and opposite force. See Figure 3. 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 seat, not upon the passenger.^[4]

Figure 3 also shows that the car seat is subjected to a centripetal force applied by the car itself. This centripetal force exceeds the reactive centrifugal force exerted on the seat by the passenger, causing the seat to move around the turn under a net inward force.

Figure 3 shows the car body is subject to the centripetal force from the road of Figure 1, and also is subject to the reactive centrifugal force exerted by the car seat. The inward force on the car body exceeds the outward centrifugal force due to the cart seat by enough to make the car body follow the turn.

The sketch in Figure 3 shows the forces are not colinear, so they exert a torque on the various parts. Although torque is a likely real-world consideration (the car tends to tip over as it rounds the turn), it simply is ignored here to focus upon the reactive centrifugal force.

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

$\mathbf{F}_\mathrm{centrifugal} \,$	$= - m \mathbf{a}_\mathrm{centripetal} \,$
	$= m \omega^2 \mathbf{r}_\perp \,$

where $m$ is the mass of the passenger, $ω$ the rotational speed (in radians per unit time), and r the radius vector outward from the axis of rotation to the passenger.

The reactive centrifugal force^[4] is a real force, but the term is rarely used in modern discussions.

[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. 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).

[edit] Fictitious forces

Main article: Fictitious force

See also: Centrifugal force and Coriolis force

	Real force	Pseudo force
Reference frame	Any	Only rotating reference frames
Exerted by	Bodies moving along curved paths	Acts as if exerted by the rotation axis of the frame of reference
Exerted upon	The object imposing curved motion	All bodies
Direction	Away from the center of curvature	Away from the rotation axis of the frame of reference

An inertial (also known as fictitious or pseudo) centrifugal force appears when a rotating reference frame is used for analysis. To make Newton's laws of motion valid in such a frame, any true force on a mass must be supplemented by a (fictitious) centrifugal force that is directed away from the axis of rotation and a coriolis force that bends a moving objects path.

Returning to the example of the car turning, if we consider the reference frame that is rotating together with the car (a model which those inside the car will often find natural), it feels as if a 'magical' force is pushing the passenger away from the center of the bend. In a rotating frame this is ascribed to a fictitious force (as opposed to an actual force exerted by another object), an inertial force called centrifugal force. The centrifugal force is necessary because the car's acceleration is hidden from observers in the reference frame moving with the car. Nevertheless, in the rotating frame, this type of force appears as real and can simplify some calculations.^{[citation needed]} Fictitious forces do not appear, however, in inertial frames of reference. See the articles on centrifugal force and fictitious force for examples.