Feb 2007

Physics Challenge for Teachers and Students Solutions to February 2007 Challenge ◗ Cash or Charge? Challenge: Three small identical coins of mass m each are connected by two light nonconducting strings of length d each. Each coin carries an unknown charge q. The coins are placed on a horizontal frictionless nonconducting surface as shown (the angle between the strings is very close to 180°). After the coins are released, they are observed to vibrate with period T. Find the charge q on each of the coins.

For small displacements, use the binomial approximations (1 + z)a  1 + az to get: U AC = ≈

Solution: Define the distances x and y as shown in the diagram below (drawn so that coin B moves vertically). x C

A y B d

q2 1 4πε0 2d 1 − ( y d )2 y2 ⎞ 1 q 2 ⎛⎜ ⎜⎜1 + 2 ⎟⎟⎟ . 4πε0 2d ⎜⎝ 2d ⎟⎠

Since the location of the center of mass will not move, coin B moves a distance yB = 2y/3 and the other two coins move y/3 in the other direction. The change in the potential energy as a function of yB is 1 ⎛ 1 9q 2 ⎞⎟⎟ 2 ΔU ≈ ⎜⎜⎜ ⎟ yB . 2 ⎜⎝ 4πε0 8d 3 ⎟⎠

If coin B is moving vertically with a speed vB, the vertical components of the velocities of coins A and C are vB/2 in the opposite direction to conserve momentum. For small displacements, the horizontal components of the velocities of coins A and C are negligible, so the total kinetic energy is

The total potential energy is 2

U = UAB + UBC + UAC, where UAB is the contribution due to the relative position of coins A and B, etc. Since the distances between coins A and B and coins B and C do not change, UAB = UBC = constant. The remaining term is U AC =

180

q2 1 q2 1 . = 4πε0 x 4πε0 2 d 2 − y 2

K ≈ 12 mvB2 + 2 ⋅ 12 m (vB 2) = 12 (3m 2) vB2 .

This can be treated as a one-dimensional system with an effective mass of meff  3m/2 and a spring constant: k≈

1 9q 2 . 4πε0 8d 3

The period is

THE PHYSICS TEACHER ◆ Vol. 45, 2007

m T = 2π eff ≈ 2π k

=

3m 2 1 9q 2 4πε0 8d 3

4π md 3 3 , q 1 4πε0

so the charge is q≈

4π md 3 3 . T 1 4πε0

(Contributed by Alan J. DeWeerd, University of Redlands, Redlands, CA) We would also like to recognize the following contributors: Richard A. Bachman (Potomac State College/ University of West Virginia, Keyser, WV) André Bellemans (Université Libre de Bruxelles, Belgium)

THE PHYSICS TEACHER ◆ Vol. 45, 2007

Marianne Breinig (The University of Tennessee, Knoxville, TN) David A. Cornell (Principia College, Elsah, IL) Don Easton (Lacombe, Alberta, Canada) Fredrick P. Gram (Cuyahoga Community College, Cleveland, OH) Jeff Melmed (Eastern Maine Community College, Bangor, ME) Carl E. Mungan (U. S. Naval Academy, Annapolis, MD) Michael Threapleton (Centralia College, Centralia, WA) Leo H. van den Raadt (Heemstede, The Netherlands)

Many thanks to all contributors and we hope to hear from you in the future! Please send correspondence to: Boris Korsunsky; [email protected]

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