Explicit derivation of the relativistic mass-energy relation for internal kinetic and potential energies of a composite system

A straightforward derivation of relativistic expressions for the mechanical momentum, kinetic and total energies, and mass-energy equivalence (including potential energy) which does not require any knowledge of the energy-momentum relation for electromagnetic waves or consideration of elastic collisions, but is directly based on Newton's second law and Lorentz's transformations, is presented in this paper. The existence of an invariant force is shown to be important for the validity of the relativistic mechanics.

Nonplanar distortions and strain energies of polycyclic aromatic hydrocarbons

On the basis of HF/6-31G(d) optimized structures, the nonplanar distortions of 135 polycyclic aromatic hydrocarbons (PAHs) have been classified as splitting (S-) and arching (A-) distortions. Three bay structures are proposed as the structural origin of S-distortion. Due to the limitation of sample molecules, a set of universal motifs for molecules containing A-distortions is not available; however, a set of motifs and parameters are developed for the semiquantitative estimation of the nonplanar strain energies of PAHs containing the corannulene structure, and the differences between the E, values from quantum calculations and those from these estimations vary from -5.60 to 5.51 kcal/mol. The above results are fundamentally important for the understanding of nonplanar distortion of PAHs and fullerenes, and this method can also be employed to semiquantitatively estimate strain energies of such molecules containing hundreds of carbon atoms.

Geodesics without differential equations: general relativistic calculations for introductory modern physics classes

Introductory courses covering modem physics sometimes introduce some elementary ideas from general relativity, though the idea of a geodesic is generally limited to shortest Euclidean length on a curved surface of two spatial dimensions rather than extremal aging in spacetime. It is shown that Epstein charts provide a simple geometric picture of geodesics in one space and one time dimension and that for a hypothetical uniform gravitational field, geodesics are straight lines on a planar diagram. This means that the properties of geodesics in a uniform field can be calculated with only a knowledge of elementary geometry and trigonometry, thus making the calculation of some basic results of general relativity accessible to students even in an algebra-based survey course on physics.

Strain energies due to nonplanar distortion of fullerenes and their dependence on structural motifs

Accurate strain energies due to nonplanar distortion of 114 isolated pentagon rule (IPR) fullerenes with 60-102 carbon atoms have been calculated based on B3LYP/6-31G(d) optimized structures. The calculated values of strain energy due to nonplanar distortion (E-np) are reproduced by three simple schemes based upon counts of 8, 16, and 30 distinct structural motifs composed of hexagons and pentagons. Using C-180 (I-h) and CN (I-h) (N is very large) as test molecules, the intrinsic limitations of the motif model based on six-membered rings (6-MRs) as the central unit have been discussed. On the basis of the relationship between the contributions of motifs to E-np and the number of five-membered rings (5-MRs) in motifs, we found that IPR fullerenes with dispersed 5-MRs present smaller nonplanar distortions.