We solve analytically the differential equations for a skier on a hemispherical hill and for a particle on a loop-the-loop track when the hill or track is endowed with a coefficient of kinetic friction μ. For each problem, we determine the exact “phase diagram” in the two-dimensional parameter plane.
I. INTRODUCTION
Two classic homework exercises in an elementary mechanics course are the skier on a hemispherical hill (Fig. 1) and the particle on a loop-the-loop track (Fig. 2).1 Both problems illustrate nicely the use of conservation of energy (to find the speed as a function of height) followed by (to find the normal force).
Skier on a hill of quarter-circular cross section. The horizontal portion of the hill is frictionless; the circular portion has a coefficient of kinetic friction μ.
Skier on a hill of quarter-circular cross section. The horizontal portion of the hill is frictionless; the circular portion has a coefficient of kinetic friction μ.
Particle on a loop-the-loop. The horizontal portion of the track is frictionless; the circular portion has a coefficient of kinetic friction μ.
Particle on a loop-the-loop. The horizontal portion of the track is frictionless; the circular portion has a coefficient of kinetic friction μ.
It is interesting to consider what happens when the hill or track is endowed with a coefficient of kinetic friction μ. Somewhat surprisingly, the exact differential equations turn out to be analytically solvable.2–11 Our purpose here is to provide a unified treatment of the two problems, using only elementary methods that are easily accessible to undergraduates (e.g., linear first-order differential equations). Though most of our results have been obtained previously—as we shall document in detail—they are somewhat scattered in the literature. It may thus be of some modest value to have a complete elementary derivation collected in one place.
The skier and loop-the-loop problems give rise to very similar differential equations, which differ only by some sign changes. However, these sign changes lead to significant differences in the qualitative interpretation of the solutions. Since the skier problem turns out to be somewhat simpler, we treat it first and give a complete solution; in particular, we determine the exact phase diagram in the two-dimensional parameter plane. For the loop-the-loop, we solve the differential equations only up to the first time (if any) that the particle halts or completes one cycle of the loop, so we obtain only a partial phase diagram. The full phase diagram will (as we explain later) contain an infinite sequence of bifurcations, and we leave its computation to a reader who wishes to take up where we have left off.
II. SKIER ON A HEMISPHERICAL HILL
When , by contrast, the normal force is no longer a decreasing function of θ, nor is it guaranteed to reach zero within the interval . Indeed, , so the normal force is initially increasing.
The curves for μ = 1 and , 0.4, and 0.45, and , 0.6, and 0.8. The skier halts when , or flies off the hill when (shown as a dotted curve), whichever happens first. The critical curve is . The dot indicates the point (here ).
The curves for μ = 1 and , 0.4, and 0.45, and , 0.6, and 0.8. The skier halts when , or flies off the hill when (shown as a dotted curve), whichever happens first. The critical curve is . The dot indicates the point (here ).
The curve that forms the boundary between the “halt” phase and the “fly-off” phase.
The curve that forms the boundary between the “halt” phase and the “fly-off” phase.
In this way, we have obtained a phase diagram that divides the plane into three possible qualitative behaviors:
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For , the skier halts after a finite time at some angle : This angle is an increasing function of λ that runs from 0 to as λ runs from 0 to .
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For , the skier comes to rest asymptotically as at the angle .20
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For , the skier flies off the hill at some angle : This angle is a decreasing function of λ that tends to 0 as .
The curve thus forms the boundary between the “halt” phase and the “fly-off” phase (see again Fig. 4).21 In particular, the skier always either halts or flies off; she never reaches angle .
Some typical curves of for all three scenarios are shown in Fig. 3. Note, in particular, that when and ; and note the fundamental qualitative difference between the curves for , which reach the Λ = 0 axis, and those for , which do not.22
Some typical curves of as a function of λ are shown in Fig. 5, and some typical curves of as a function of λ are shown in Fig. 6. Please note the discontinuous change in behavior as the phase boundary is crossed: (the dotted curve in Fig. 6) is much larger than (the dashed curve in Fig. 5).23 This is a very simple example of sensitive dependence to initial conditions, giving rise to a discontinuous phase transition—a phenomenon pointed out already by James Clerk Maxwell in 1876.24
as a function of λ in the halt phase , for , 1, and 1.5. The endpoints lie on the dashed curve, defined parametrically by and .
as a function of λ in the halt phase , for , 1, and 1.5. The endpoints lie on the dashed curve, defined parametrically by and .
as a function of λ in the fly-off phase , for μ = 0, 0.5, 1, and 1.5. The endpoints lie on the dotted curve, corresponding to from above. For μ = 0, we have the closed-form solution (9).
as a function of λ in the fly-off phase , for μ = 0, 0.5, 1, and 1.5. The endpoints lie on the dotted curve, corresponding to from above. For μ = 0, we have the closed-form solution (9).
Since the proofs of all the previous claims involve some slightly intricate calculus, we relegate them to Appendix A in the supplementary material.25
Let us remark, finally, that by the same methods one can study the more general problem in which the coefficient of kinetic friction is an arbitrary function of the position along the hill: The Eq. (5) is still a first-order inhomogeneous linear differential equation for the unknown function —albeit now one with nonconstant coefficients—so can still be solved by the method of integrating factors (though the result may not be analytically expressible in terms of elementary functions). We leave it to interested readers to pursue this generalization.
Some recent related articles are Refs. 10, 11, and 26, which study a particle sliding down an arbitrary curve in the presence of kinetic friction; Ref. 27, which uses the Lagrangian formalism with Lagrange multipliers to analyze a particle sliding without friction down an arbitrary concave curve; and Ref. 28, which studies a ball rolling (initially without slipping, later with sliding and kinetic friction) on an arbitrary curve in the presence of gravity, including an experimental realization.
III. PARTICLE ON LOOP-THE-LOOP TRACK
The loop-the-loop problem is more complicated than the skier, for three reasons: The particle can cycle around the track; it can reverse direction; and it can halt due to static friction. Each time the particle reverses direction, we need to apply Eq. (22) with a new value for ; this repeated switching between different equations seems quite complicated, and probably needs to be handled by numerical solution.29 To simplify matters, we will here follow the block only until it first reaches or falls off the track; we, therefore, have .
The solution (25) must, therefore, be supplemented by the two inequalities and . (Please note that, unlike in the skier problem, both of these inequalities point in the same direction; this radically changes the nature of the qualitative analysis.) The block comes instantaneously to rest when or falls off the track when , whichever happens first; if neither happens for , then the block completes one full cycle of the loop-the-loop. Now, the inequality is the more stringent one in the lower half of the loop-the-loop (that is, modulo ), while the inequality is the more stringent one in the upper half of the loop-the-loop (that is, modulo ). Therefore, the block can come instantaneously to rest only in the lower half of the loop-the-loop, and it can fall off the track only in the upper half of the loop-the-loop.
In the presence of friction ( ), the analysis proceeds as follows:
- The first step is to determine the conditions under which the particle halts in the first quadrant ( ). The particle halts at angle θ when , i.e., in case the initial velocity satisfiesSinceis an increasing function of θ in the interval (as is intuitively clear: To reach a larger angle, more initial velocity is needed). In particular, the particle reaches with if and only if
- If the particle reaches angle without halting, the next step is to determine the conditions under which the particle flies off in the second or third quadrant ( ). The particle flies off at angle θ when , i.e., in case the initial velocity satisfiesNote that . Sincewe see that is an increasing function of θ in the interval from to π and then a decreasing function in the interval from π to . The first of these facts is again intuitively clear: To survive to a larger angle without flying off, more initial velocity is needed. The second fact implies that if the particle reaches angle π without flying off—that is, if
then it also reaches angle without flying off. This is intuitively clear when there is no friction, but not so obvious in the presence of friction. This implies—analogously to what happens in the skier problem—a discontinuous change of behavior as λ passes through . See Fig. 7 for plots of and versus θ for some selected values of μ.
- If the particle reaches angle π (and hence also angle ) without halting or flying off, the next step is to determine what happens in the fourth quadrant ( ). The particle halts at angle θ in case λ equals the quantity defined in Eq. (31). From Eq. (32), we see that is negative at and positive at , with a unique zero at . So is decreasing in the interval and increasing in the interval . Its maximum value in the interval , therefore, lies either at or at . Since we are in the situation , the only relevant question is whether λ is larger than or not. If it is, then the particle reaches angle without halting. If it is not, then the particle halts at some angle in the interval , namely, the unique angle where . The first of these cases always occurs when , i.e., when . (See Appendix B in the supplementary material25 for the proof that there is a unique such value .) When , then there is a “halt in fourth quadrant” phase at and a “survive to angle ” phase at . We record the formula
- If the particle survives to angle , then it has there a forward velocity corresponding to a value,
The functions (black) and (green or gray) vs θ for some selected values of μ. The dominant (respectively, subdominant) condition is shown as a solid (respectively, dotted) curve. A horizontal dashed line is shown at . The curve in the bottom-left panel corresponds to the value where . From Eq. (32), we see that is increasing for , decreasing for , and increasing for . From Eq. (35), we see that is increasing for and decreasing for . The two curves cross at and .
The functions (black) and (green or gray) vs θ for some selected values of μ. The dominant (respectively, subdominant) condition is shown as a solid (respectively, dotted) curve. A horizontal dashed line is shown at . The curve in the bottom-left panel corresponds to the value where . From Eq. (32), we see that is increasing for , decreasing for , and increasing for . From Eq. (35), we see that is increasing for and decreasing for . The two curves cross at and .
Since in the survive to angle phase, we have : Thus, the kinetic energy is reduced by at least a factor at each revolution. The subsequent motion can then be found by repeating the foregoing analysis with λ replaced by .
The resulting phase diagram is shown in Fig. 8. Since grows extremely rapidly with μ, we have used instead of λ on the vertical axis, to compress the plot. This phase diagram agrees with the one found by Kłobus (Ref. 8, Fig. 2); the value of also agrees with his. All three phase boundaries are increasing functions of μ: See Appendices B1–B3 in the supplementary material.25
Phase diagram for the loop-the-loop problem, up to the first time that the particle reaches or . The vertical axis shows . The three boundary curves are, from bottom to top, , , and , shown respectively in solid black, dashed blue, and dotted red. The particle either halts in the first quadrant (Q1), flies off the second quadrant (Q2), halts in the fourth quadrant (Q4), or survives to angle .
Phase diagram for the loop-the-loop problem, up to the first time that the particle reaches or . The vertical axis shows . The three boundary curves are, from bottom to top, , , and , shown respectively in solid black, dashed blue, and dotted red. The particle either halts in the first quadrant (Q1), flies off the second quadrant (Q2), halts in the fourth quadrant (Q4), or survives to angle .
Of course, this phase diagram only follows the particle up to the first time that it reaches or . A more complete analysis would show that the phase “survives to angle ” is itself divided into sub-phases “halts in the first quadrant” ( ), “flies off the second quadrant” ( ), “halts in the fourth quadrant” ( ), and “survives to angle ”; and this latter phase is further divided into sub-phases; and so on infinitely. We leave it to interested readers to work out the details of this infinite sequence of bifurcations.
ACKNOWLEDGMENTS
The authors are extremely grateful to three referees for their detailed and helpful comments on several versions of this paper.