FORCES SOME PHYSICISTS FORGET
While reading a fairly recent physics textbook I found a detailed discussion of fictitious or inertial forces. The textbook is PHYSICS For Scientists & Engineers, Third Edition, by Raymond Serway, James Madison University, Copyrights 1990, 1986, and 1982, Saunders Golden Sunburst Series, Saunders College Publishing, Philadelphia, Library of Congress Catalog Card Number 89-043325.
This is not meant to be a criticism of a book that is excellent in many ways and must have been a great challenge to write. It is still being used in teaching undergraduate physics. I use it only as an example of some prevalent ideas on the subject of inertial forces and inconsistencies rising from those ideas.
Section 6.3, MOTION IN ACCELERATED FRAMES, begins on page 135. According to Serway, an observer in an accelerating frame of reference must invoke fictitious forces (aka inertial forces) if he would use Newton's second law in his accelerated reference frame. Serway then claims that the invented forces appear real in the accelerated frame but are not forces resulting from interaction between the body and its environment, and therefore are not real forces. Serway’s claim in a positive form gives us a DEFINITION: A real force is one that results from the interaction of a body and its environment.
On page 136 Serway presents EXAMPLE 6.5 Linear Accelerometer. In that example a ball is suspended by a cord as a non-oscillating pendulum in a railroad boxcar that is accelerating, as illustrated by Figure 6.11(a). In order that we can see what is going on inside the boxcar, the facing side of the car has been removed. An arrow over the boxcar points rightward, indicating the direction of motion and acceleration. An observer stands upright on the ground near the tracks. The illustration depicts the cord displaced from the vertical by an angle theta (the ball to the left of the vertical from the point of suspension), the tension in the cord (T), and the downward gravitational force on the ball (mg). The observer sees the acceleration of the ball and boxcar to be the same and deduces that the horizontal component of tension (T) produces the acceleration, and that the vertical component of T balances the ball's weight. I do not believe that the omission of the horizontal component of T in that figure was an accidental oversight.
Immediately beneath Figure 6.11 (a) is Figure 6.11 (b), which depicts the same boxcar and suspended ball. An observer is now inside in the front of the boxcar, facing the rear, standing upright with arms and hands at sides, and looking at the suspended ball. The angle theta is the same. The horizontal component of T is labeled -ma and directed leftward from the ball.
For that second illustration Serway states that for the noninertial passenger in the car the sphere is at rest and there is no acceleration. That is why, writes Serway, the observer invokes fictitious force -ma to counter T's horizontal component, and thinks there is zero net force on the ball (italics his). Serway states that the interpretation of the situation is different in each of the two reference frames.
I suspect that Serway has not had daily experience commuting to work by rail. If he did, he certainly would know that anyone standing vertically with feet together and not holding on to anything will fall down in an accelerating railroad car. I do not believe the depicted position of that observer to be an accidental oversight or mistake. Were the observer to assume a spread-legs (in line with the direction of acceleration) braced position leaning forward (at about the angle theta), the observer would not fall over and would feel, resist, and know that there is a force of acceleration to be taken into account. Because it is well known that acceleration forces will move anything not secured in a boxcar, crates of heavy loads are often nailed to the wooden floors of boxcars to avoid damage to boxcar or contents.
Since in both illustrations the boxcar and track are horizontal we must assume that the engine pulling the train is producing the acceleration. That acceleration results from horizontal force generated through contact of the engine's drive wheels with the track. Couplings between the engine and subsequent cars transmit that force. The magnitude of that force is the product of the total mass of the train (and its contained objects) multiplied by the acceleration. Components of the train and the objects transported by the train (including observers) experience the same acceleration and, hence, a force in proportion to their individual masses.
Now although we might correctly think that acceleration of the train produces forces on its contents and passengers, we would be wrong to think that the acceleration of the train produces the motivating force in its engine. Clearly, it must be the engine that produces the force to accelerate the train and its contents. Equally clearly, if a reference frame containing one or more bodies is accelerating, forces are being applied to those bodies. Acceleration without force violates Newton’s second law. The notion of an accelerating reference frame devoid of mass bodies is meaningless. Only mass bodies can accelerate.
As with the train, the force causing acceleration of a reference frame is simultaneously applied to all objects in the reference frame. Because the source of that force is external to the bodies in the reference frame, that force is a part of the environment of those bodies. Therefore, observed forces on bodies in an accelerating reference frame do result from interaction between such bodies and their environment and, by the definition of a real force given above, those observed forces are real. The force that accelerates a reference frame is often a forgotten force.
Once the force that accelerates a reference frame is taken into account the apparent difference between an inertial and noninertial reference frame disappears. Observers in either reference frame find the same forces affecting the accelerometer. By not 'forgetting' the force created by the engine there is no need for postulating a 'fictitious force.'
If the accelerometer consisted, instead, of a vertically immobile, horizontal, helical compression spring, fastened at one end to the rear wall of the boxcar, and with a weight on the free end, both observers would see that the spring was compressed from its rest position. Both would recognize that a real force compressed the spring. They would recognize that the compressive force on the spring originated with the force on the train and its contents. By attaching a pointer to the free end of the spring and a calibrated scale behind the pointer and by then substituting various known weights either observer could verify Newton's second law, f=ma, without invoking fictitious forces. Note that the same device could be used to measure static forces where there is no acceleration.
On page 136 Serway presents EXAMPLE 6.9 Fictitious Force in a Rotating System. Two side by side illustrations depict a rotating turntable with a mass connected to the center of the table by a cord in figure 6.12. Illustration (a) has an observer standing next to the turntable, while illustration (b) has the observer in a reclining position along a radius on the turntable. We are subliminally asked to believe that the observer in (b) cannot feel, and need not resist, any force acting on him. According to Serway, that observer therefore invents a fictitious outward force mv²/2 (and calls it centrifugal force) to balance the force of tension in the cord.
Let us now imagine that the spring-and-weight force indicator is taken from the boxcar and fastened to the edge of the turntable, with the free end and weight resting on the turntable. Two opposing forces, at either end of the spring, would again compress the spring. The forces at the two ends of the spring would be designated centripetal force and centrifugal force by both observers. The observer as depicted on the turntable, without much source of centripetal force, would have to exert some effort to resist being thrown off the turntable.
There are amusement park turntable “rides” like that. Once the very smooth wooden turntable starts turning, almost everyone slides off. Few can position their exact center of gravity over the exact center of the turntable. If someone were to apply a pair of suction cups with handles to the turntable and hold on to the handles until the turntable reaches its steady angular velocity, the person would feel as if hanging from the rings in a gym and fighting against gravity. In that case, however, it would be centrifugal force being fought by the person’s centripetal force.
As with the linearly accelerating reference frame, the notion of an empty rotating frame is meaningless. Only the bodies in a rotating reference frame do the rotating. In the case of the rotating reference frame the force that has been forgotten is the force that put everything in that frame into rotation, creating their angular momentum, kinetic energy of rotation, and capability to exert forces on objects introduced into that reference frame.
Regarding the observer in the accelerating boxcar Serway wrote, in a disparaging sense, that the observer thought there was zero net force on the ball. According to Newton’s third law, for every force there is always an equal and opposite force. That law followed to its logical conclusion means that no matter how many forces act on a body, for each one there is always an equal and opposite force; and, therefore, the net force acting on any body is always zero. Surely Serway did not intend to slight Newton’s third law; however, that is what can inadvertently happen if one forgets the inertial force exerted by a body against an external force causing it to accelerate.
On page 816, Serway prepares to derive the circular orbital radius of a charged particle in a magnetic field. He writes, “Since the resultant force F in the radial direction has the magnitude of qvB, we can equate this to the required centrifugal force, which is the mass m multiplied by the centripetal acceleration v²/r.” At last centrifugal force becomes required and real, has a formula, and is equal and in opposition to centripetal force.
<font size=-1>[ This Message was edited by: Richard J. Hanak on 2002-08-09 21:31 ]</font>