The motion of a block sliding on a curve is a well studied problem for flat and circular surfaces, but the necessary conditions for the block to leave the surface deserve a deeper treatment. In this article, we generalize this problem to an arbitrary surface, including the effects of friction, and provide a general expression to determine under what conditions a particle will leave the surface. An explicit integral form for the speed is given, which is analytically integrable for some cases. We demonstrate general criteria to determine the critical speed at which the particle immediately leaves the surface. Three curves, a circle, a cycloid, and a catenary, are analyzed in detail, revealing several interesting features.

1.
M.
Alonso
and
E. J.
Finn
,
Fundamental University Physics: Mechanics
(
Addison-Wesley
,
Reading, MA
,
1967
), vol. 1.
2.
C. H.
Holbrow
, “
Archaeology of a bookstack: Some major introductory physics texts of the last 150 years
,”
Phys. Today
52
(
3
),
50
56
(
1999
).
3.
C. E.
Mungan
, “
Sliding on the surface of a rough sphere
,”
Phys. Teach.
41
,
326
328
(
2003
).
4.
T.
Prior
and
E. J.
Mele
, “
A block slipping on a sphere with friction: Exact and perturbative solutions
,”
Am. J. Phys.
75
,
423
426
(
2007
).
5.
O. L.
de Lange
,
J.
Pierrus
,
T.
Prior
, and
E. J.
Mele
, “
Comment on ‘A block slipping on a sphere with friction: Exact and perturbative solutions,’ by Tom Prior and E. J. Mele [Am. J. Phys. 75, 423–426 (2007)]
,”
Am. J. Phys.
76
,
92
95
(
2008
).
6.
G. E.
Hite
, “
The sled race
,”
Am. J. Phys.
72
,
1055
1058
(
2004
).
7.
W.
Kłobus
, “
Motion on a vertical loop with friction
,”
Am. J. Phys.
79
,
913
918
(
2011
).
8.
M. P.
do Carmo
,
Differential Geometry of Curves and Surfaces
(
Pearson
,
London
,
1976
).
9.
W.
Whewell
,
Of the Intrinsic Equation of a Curve, and its Application
(
Cambridge Philosophical Transactions
,
Cambridge, England
,
1849
), vol. VIII.
10.

When the surface is concave upwards, b̂·ẑ=1 and the remain-on-the-surface condition, N(s)0, implies that v2(s)/gRH(s). Therefore, the initial speed must satisfy v02/gRH(0). Because g·n̂(0)<0 in this case, we have H(0)0 and therefore v02/gRH(0) for all concave-upwards surfaces. The end result is that the initial speed has no restrictions—the particle always stays on the surface if the surface is concave upwards.

11.

When φ is a decreasing function (the curve is concave upwards) we have κ=dφ/ds and (b̂·ẑ)=1, so that v2(φ)=v02e2μ(φφ0)2ge2μφφ0φ(sinφμcosφ)e2μφκ(φ)dφ.

12.
W.
Greiner
,
Classical Mechanics: Point Particles and Relativity
(
Springer
,
Berlin, Heidelberg
,
2004
), chap. 20, problem 20.10.
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