Recently, González-Cataldo *et al.*^{1} have presented the analysis of the motion of a block slipping on an arbitrary surface under the action of gravity, including the effects of friction. Using basic differential geometry, they achieved a general expression for determining the speed of the block, an expression depending on the tangential angle to the cross section of the surface. However, this analysis is a lengthy way to solve the problem. Here, we present a solution derived from the work-energy theorem. This method, previously used by Prior and Mele, gives an alternative didactic way to address the problem. In the present comment, the use of the so-called Volterra integral, which is covered in the literature, is the sole difficulty of the mathematical treatment.

Let *x* = *x*(*α*), *y* = *y*(*α*) be the parametric equations in rectangular coordinate that define the curve of the surface's cross section. The work-energy theorem yields the following relation:

where $f\u2192$ is the friction force, $ds\u2192$ is the tangent vector to the curve, whose value is the element of arc length $|ds\u2192|=(dx/d\alpha )2+(dy/d\alpha )2d\alpha $, *R* is the initial height, and *v*_{0} is the initial velocity. With the help of the free-body diagram in Fig. 1, we write the equation of motion according to Newton's second law

in which the explicit tangential and normal components are

where *φ* is the tangential angle, *v* is the speed, and $\kappa =|d\phi /ds|$ is the curvature. Equation (4) shows when the normal force becomes zero, so that the particle loses contact with the curve. The link between the parameter *α* and the tangential angle *φ* is set by

which is a Volterra integral equation of the second kind,^{4} with the Kernel

Taking into account that $ds/d\alpha =(dx/d\alpha )2+(dy/d\alpha )2$ and the definition of curvature *κ*, we find $K(\alpha )=2\mu |d\phi /d\alpha |$ and Eq. (6) becomes

where

If $d\phi /d\alpha $ is always positive, as in the case of concave downward curves, and if Eq. (5) can be inverted, such that it defines *α* in terms of *φ*, then we can write

The above equation leads to Mungan's result for the particular case of the sphere.^{3} The solution to Eq. (9) can be obtained by multiplying its derivative with respect to *φ* by the integrating factor $exp\u2009(\u22122\mu \phi )$, the approach that González-Cataldo *et al.* took in the commented paper. This gives

where $h(\phi )=Z0+2((y(\alpha )/R)\u22121)\u22122\mu (x(\phi )/R)$. It is well known that the release condition of the block is *N*(*φ*) = 0.^{1,2} From Eq. (4), it follows that:

where *φ _{s}* is the departure angle.

The analysis of Ref. 1 can be reproduced by applying the equations obtained here to the specific curves presented in Secs. III and IV of that paper. Alternatively, the solution can be obtained as shown in the Appendix.

The authors would like to acknowledge partial financial support from FONDECYT project 1170834. In addition, the authors thank one of the referees for suggesting the approach presented in the Appendix.

### APPENDIX: ALTENATIVE METHOD

Equation (4) shows that the normal force becomes zero, so that the particle loses contact with the curve, with *Z* from the Eq. (8), and $Z\u2261\u2009cos\u2009\phi /\kappa R$. Using the chain rule

Finally, from

it follows that Eq. (A3) can be integrated to get

in agreement with González-Cataldo Eq. (20).