There is considerable interest in the mechanics of carbon nanostructures, such as carbon nanotubes and fullerenes , and the manner of their interactions at the intennolecular level. Medical applications i.n clud e the use of carbon .nanotub e. for tar etcd drug a nd gen e delivery, for which issues relating to ll1c acceptan ce and containmen t of dru g or genes are n ot prop erly understood . A sph eroid is an ellipsoid with rwo equal axes and the general spheroidal shape inclu des a wide vadcty of possible m olecu lar onfigurations su ch a spheres. capped cylindrica l rubes and ellipsoids of revo lution , and therefore the detcm1imlliou of U1e interaction forces for this general shape may have many applications. Phenomena such as the suction of fullerenes into carbon nanotubes due to the van der Waals interatomic interactions and ultra-low friction of a molecule moving inside a carbon nanon.tbe give rise to th e possibi lity of constructing nanoscaled oscillators with frequenci es in the gigahcn z range . This paper models the mechanics of such a system by empl oy.ing a six-twe)ve Len nard-Janes potential taken over two surfaces assumed to be composed of mean distributi on of atom s over the two idealized surfaces of an 'open-ended semi-infinite circular cylinder and a spheroid. Following the methodology of previous work with spherical surfaces, the acceptance energy and suction energy for spheroidal molecules are given and the special case of spherical molecules is also reproduced to validate the method. The results for elliptical molecules are novel and cannot be validated experimentally at this stage, but the results for the special case of spherical molecules are given and shown to be in good agreement with published molecular dynamical simulations. Finally, a general numerical-analytical procedure is proposed to calculate the Lennard-Janes potential for any axially symmetric surface, and the prior results obtained for the spheroid are used to validate the procedure .