A well-known textbook problem treats the motion of a particle sliding frictionlessly on the surface of a sphere. An interesting variation is to consider what happens when kinetic friction is present.1 This problem can be solved exactly.
REFERENCES
1.
H.
Sarafian
, “How far down can you slide on a rough ball?
” AAPT Announcer
31
, 113
(Winter 2001
).Sarafian has analyzed this problem using Mathematica for a special issue of the Journal of Symbolic Computation to be published in late fall of 2003.2.
Many introductory textbooks now include an overview of Euler's method of numerical integration in a spreadsheet such as Excel. For example, see P.A. Tipler and G. Mosca, Physics for Scientists and Engineers, 5th ed. (Freeman, New York, 2003), Sec. 5–4.
3.
For many values of μ and there are two mathematical solutions of θ for which Only the smaller solution has physical significance for
4.
Integrating factors are discussed in standard differential equation texts, such as D.G. Zill and M.R. Cullen, Differential Equations with Boundary-Value Problems, 3rd ed. (PWS-Kent, Boston, 1993), Sec. 2.5.
5.
Students who have taken an introductory course in differential equations will recognize Eq. (A1)as a particular solution and Eq. (A2)as the complementary solution of the homogeneous equation corresponding to Eq. (5). If even this alternative approach (without the technical terminology) is too advanced, students could still be challenged to verify that Eq. (A3) satisfies both Eq. (5) and the initial conditions.
6.
Another reasonable choice of initial conditions has been adopted in
W.
Herreman
and H.
Pottel
, “Problem: The sliding of a mass down the surface of a solid sphere
,”Am. J. Phys.
56
, 351
(April 1988
).
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© 2003 American Association of Physics Teachers.
2003
American Association of Physics Teachers
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