Reactive centrifugal force

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A ball in circular motion held by a string tied to a fixed post.

In classical mechanics, reactive centrifugal force is the reaction paired with centripetal force. A mass undergoing circular motion constantly accelerates toward the center of rotation. This centripetal acceleration is caused by a force exerted on the mass by some other object. In accordance with Newton's third law of motion, the rotating mass exerts an equal and opposite force on that object, which is directed from that object toward the rotating mass. This reaction force is the "reactive centrifugal force" referred to in this article.[1][2][3][4]

The term "reactive centrifugal force", in relation to the reaction force to a centripetal force, is seldom referred to in modern physics and mechanics. The term "centrifugal force" usually refers to the fictitious force in a rotating reference frame that appears due to the rotation of the co-ordinate system.[5]

Paired forces[edit]

The figure at right (top panel) shows a ball in uniform circular motion held to its path by a massless string tied to an immovable post. In this system a centripetal force upon the ball provided by the string maintains the circular motion, and the reaction to it, usually called the reactive centrifugal force acts upon the string. In this model, the string is assumed massless and the rotational motion frictionless, so no propelling force is needed to keep the ball in circular motion.

Newton's first law requires that any body not moving in a straight line is subject to a force, and the free body diagram shows the force upon the ball (center panel) exerted by the string to maintain the ball in its circular motion.

Newton's third law of action and reaction states that if the string exerts an inward centripetal force on the ball, the ball will exert an equal but outward reaction upon the string, shown in the free body diagram of the string (lower panel) as the reactive centrifugal force.

The string transmits the reactive centrifugal force from the ball to the fixed post, pulling upon the post. Again according to Newton's third law, the post exerts a reaction upon the string, labeled the post reaction, pulling upon the string. The two forces upon the string are equal and opposite, exerting no net force upon the string (assuming that the string is massless), but placing the string under tension.

It should be noted, however, that the reason the post appears to be "immoveable" is because it is fixed to the earth. If the rotating ball was tethered to the mast of a boat, for example, the boat mass and ball would both experience rotation about a central point.

Applications[edit]

A centrifugal governor consisting of a pair of balls swinging around a rotating post. Forces due to gravity and from the rods that hold the balls add by vector addition to form the horizontal centripetal forces keeping the balls in circular motion around the post. The rods are in tension in reaction to the ball, and pull on the center post; the horizontal component of this pull is the reactive centrifugal force. The balls swing further out at higher rotational velocity, and the positions of the rods are used to control a throttle to regulate the engine speed.

Examples of rotating devices in which a reactive centrifugal force can be identified include the centrifugal governor and the centrifugal clutch.

  • A centrifugal governor regulates the speed of an engine by using spinning masses set in circular motion by the engine. Should the engine speed increase, the masses swing out to a larger radius and trigger a cut in the throttle. The net force on each spinning mass is the resultant of the downward force of gravity and the tension from the rod or cable supporting the mass. This net force provides the centripetal force necessary to maintain the circular orbit of the mass. The spinning mass exerts a reaction force on the support rod. This reaction force (directed along the rod) has a horizontal component that is the reactive centrifugal force exerted by the circling mass, and a vertical component.
A two-shoe centrifugal clutch. The motor spins the input shaft that makes the shoes go around, and the outer drum (removed) turns the output power shaft.
  • A centrifugal clutch is used in small engine-powered devices such as chain saws, go-karts and model helicopters. It allows the engine to start and idle without driving the device, but automatically and smoothly engages the drive as the engine speed rises. A spring is used to constrain the spinning clutch shoes. At low speeds, the spring provides the centripetal force to the shoes, which move to larger radius as the speed increases and the spring stretches under tension. At higher speeds, when the shoes can't move any further out to increase the spring tension, due to the outer drum, the drum provides some of the centripetal force that keeps the shoes moving in a circular path. The force of tension applied to the spring, and the outward force applied to the drum by the spinning shoes are the corresponding two reactive centrifugal force. The mutual force between the drum and the shoes provides the friction needed to engage the output drive shaft that is connected to the drum.

These devices are commonly analyzed in the frame of reference of the rotating mechanisms, using the fictitious force version of the concept of centrifugal force.

Relation to inertial centrifugal force[edit]

Reactive centrifugal force, being one-half of the reaction pair together with centripetal force, is a concept which applies in any reference frame. This distinguishes it from the inertial or fictitious centrifugal force, which appears only in rotating frames.

Reactive centrifugal force Inertial centrifugal force
Reference
frame
Any Only rotating frames
Exerted
by
Bodies moving in
curved paths
Acts as if emanating from the rotation axis,
it is a so-called fictitious force or d'Alembert force
Exerted
upon
The object(s) causing
the curved motion,
not upon the body in
curved motion
All bodies, moving or not;
if moving, coriolis force is present as well
Direction Opposite to the
centripetal force
causing curved path
Away from rotation axis,
regardless of path of body
Kinetic analysis Reaction to
the centripetal force
Included as a fictitious force in
Newton's second law
according to D'Alembert's principle

References[edit]

  1. ^ Delo E. Mook & Thomas Vargish (1987). Inside relativity. Princeton NJ: Princeton University Press. p. 47. ISBN 0-691-02520-7. 
  2. ^ J. S. Brar and R. K. Bansal (2004). A Text Book of Theory of Machines (3rd ed.). Firewall Media. p. 39. ISBN 9788170084181. 
  3. ^ De Volson Wood (1884). The elements of analytical mechanics: solids and fluids (4th ed.). J. Wiley & sons. p. 310. 
  4. ^ G. David Scott (1957). "Centrifugal Forces and Newton's Laws of Motion" 25. American Journal of Physics. p. 325. 
  5. ^ Vernon D. Barger and Martin G. Olsson (1973). Classical Mechanics: A Modern Perspective. McGraw-Hill. pp. 204–207.