Nonconservation of Angular Momentum of a Weight on a String

Several physicists have asked me for a pdf copy of my 1963 publication

Nonconversation of Angular Momentum of a Weight on a String

In addition to the pdf copy, below is a a more readable version because the print on in the pdf is hard to read.

In a separate post, I tell you the heretofore untold story about the origin, discussion, and testing of this publication.

Nonconservation of Angular Momentum of a Weight on a String

American Journal of Physics, Vol 31, No. 4, 304-305, April, 1963.

by Edwin X Berry, University of Nevada

An oversight in simple mechanics has occasionally occurred in the analysis of the problem of having a weight on the end of a string which is winding up on a steady finger. Several authors of meteorological publications have used this example to illustrate the conservation of angular momentum, concluding that the speed of the weight increases as the string winds up.

This error appeared in the April 1960 issue of Scientific American [1] and was later corrected [2]. It is included in a recent book by Battan [3] and was not mentioned in the reviews [4, 5]. It remains in the third edition of a text by Byers [6].

A second look at the problem shows the angular momentum of the mass-string system is not conserved at all. Angular momentum is defined about a fixed point of reference and the conservation of angular momentum has meaning only with respect to that point.

Using the center of the finger as a reference, one finds a torque due to string tension opposing the rotation. Clearly angular momentum cannot be conserved.

The common observation of an increase in the “speed” of rotation is the detection of the increase in angular velocity, which increases because the radius shortens and not because of any change in the speed of the weight.

The correct interpretation is easily had by applying the conservation of energy: No work is done on the weight because the finger is stationary. Therefore, the kinetic energy of the weight is constant and so is its speed. If the speed of the weight were to increase, we would be getting energy while performing no work.

Fig1
Fig. 1. Coordinates that give easy solution to the angular momentum and trajectory of the mass on a string as it winds up on a finger.

 

Using the coordinates shown in Fig. 1, where r is the instantaneous length of the string, R is the radius of the finger, and q is the angle, we have by the conservation of energy

(1)   \begin{equation*}  r \frac{d\theta}{dt} = constant \end{equation*}

Then the angular momentum is just proportional to r and goes to zero as the string winds up. The equations of motion are easily derived by using the relation

(2)   \begin{equation*}  r = {r_0} - R\theta \end{equation*}

A proper illustration of the conservation of angular momentum is that of having the string exit through a hole in a smooth table. Here the radius may be shortened only by pulling on the end of the string. Angular momentum is conserved and the speed of the weight increases, but the resultant increase in kinetic energy is just equal to the work done by pulling on the string. This correct illustration was used by Rossby [7] in a classic meteorological paper two decades ago.

References

  1. C. L. Stong, Sci. Am. 202, 183 (April 1960).
  2. C. L. Stong, Sci. Am. 202, 186 (June 1960).
  3. L. J. Battan, The Nature of Violent Storms (Doubleday and Company, New York, 1961), pp 43, 44.
  4. F. W. Swan, Am. J. Phys, 30, 77 (1962)
  5. E. M. Brooks, Bull. Am. Meteorol. Soc. 43, 287, (1962).
  6. H. R. Byers, General Meteorology (McGraw-Hill Book Company, Inc., New York, 1959), p. 198.
  7. C. G. Rossby, “The Scientific Basis of Modern Meteorology.” U.S. Yearbook of Agriculture, “Climate and Man” (U.S. Government Printing Office, Washington, D.C., 1941), p. 507.

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