Torque, mean square

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

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Figure 3-3 Examples of (a) cup-and-bob viscosity (torque vs. speed) and (b) viscosity versus shear rate. (Lines are least-mean-square fit.)...

The intermolecular potential consists of the sum of Eqs. (176), (177), (178), and (179). This simulation was done for 216 and 512 molecules but again only the autocorrelation functions for 512 molecules are discussed here. This potential is the strongest angular dependent potential we considered. The results from this potential indicate that it is slightly stronger than that in real liquid carbon monoxide. For example the mean square torque/TV2), for this simulation is 36 x 10-28 (dyne-cm)2 51 and the experimental value is 21 x 10-28 (dyne-cm)2. If this potential is taken seriously, then it should be pointed out that this small discrepancy in torques could be easily removed by using a smaller quadrupole moment. This would be a well justified step since experimental quadrupole moments for carbon monoxide range from 0.5 x 10-26 to 2.43 x 10-26 esu.49... 

The mean square torque is another test of the pair potentials used. The calculated mean square torques are very potential dependent they range from 7 x 10"31 to 36 x 10"28 (dyne-cm)2. The experimental values51 of the mean square torque in solid CO at 68°K and in liquid CO at 77.5°K are 19 x 10 28 and 21 x 10 28 (dyne-cm)2, respectively. Therefore, the Stockmayer potential clearly does not represent the noncentral forces in liquid CO, i.e., this potential is much too weak. On the other hand, the noncentral part of the modified Stockmayer potential is too strong. However, as pointed out previously, this problem can easily be solved by using a smaller quadrupole moment. The mean square torques from the other two potentials agree quite favorably with the experimental values. We conclude from the above that the quadrupole-quadrupole interaction can easily account for observed mean square torques in liquid CO. 

The mean square torque is taken from computer experiments. Nevertheless, it could have been found from the infrared bandshapes. Likewise the integral in this expression can be found from the experimental spin rotation relaxation time, or it can be found directly from the computer experiment as it is here. The memory function equation can be solved for this memory. The corresponding angular momentum correlation function has the same form as v /(0 in Eq. (302) with... 

How efficient is the described representation of the ArCC>2 potential To answer this question the above PES along with a few empirical potentials have been used to derive a number of properties, such as the ground vibrational state and dissociation energy of the complex, ground state rotational constants, the mean square torque, the interaction second virial coefficients, diffusion coefficients, mixture viscosities, thermal conductivities, the NMR relaxation cross sections, and many others [47]. Overall, the ab initio surface provided very good simulations of the empirical estimates of all studied properties. The only parameters that were not accurately reproduced were the interaction second virial coefficients. It is important that its performance proved comparable to the best empirical surface 3A of Bohac, Marshall and Miller [48], This fact must be greeted with satisfaction since no empirical adjustments were performed for the ab initio surface. 

A. De Santis, R. Frattini, and M, Sampoli. Rayleigh spectra of fluid Nj Mean square torques and induced contributions to the fourth moment. Chem. Phys. Lett., 774 71-73... 

R.L.Armstrong, S.M.Blumenfeld, and C.G.Gray. Infrared spectra, rotational correlation functions and intermolecular mean squared torques in compressed gaseous methane. Can.J.Phys., 46 1331,1968. 

The method was illustrated by application to the Isotopic COs (38) (Table JL), Friedman and Kimel (39) also evaluated mean square torques for CO In the liquid phase but their calculated values are much smaller than those derived by Gordon from measurements of the third and fourth moments of IR and Raman spectra. The most likely reason for the discrepancy is that Friedman s assumption of a spherically symmetric external force field for CO Is not justified. 

Table 1. Predicted VPIE and Changes In Mean-Squared Torques of the Isotopes of Liquid CO at 77 K. (26) ...

Thus the moments of the frequency spectrum may be identified as coefficients in the time power series of C,(t). Results for the first coefficients have been given for different molecular symmetries (Ic). If is the mean squared torque associated with rotation of the symmetry axis, one has ... 

Experimental determinations of mean squared torques values are few. Typical values have been reported in references (19b) and (20). 

Here, is the mean squared torque and for convenience we shall call /4(kT)2 the mean squared torque factor. The best fitted Gaussian approximation to the experimental memory function is also included in figure 10 (x X x). 

In figures 11 and 12 the density and temperature of the mean squared torque factor is reproduced which resembles the behaviour of 6 shown in figures 8 and 9. The decrease of the mean squared torque factor with temperature at constant density results mainly from the 2... 

Fig. 4. Mean square torque < 17 > on a CS2 molecule as function of time. 

Descriptive Statistics (e.g., mean, median, variance, standard deviation) Hypothesis Testing (e.g., paired and un-paired t-tests chi-squared test) Principles of Statics (e.g., forces moments couples torques free-body diagrams)... 

Both agonist and antagonist muscles contribute (unequally) to the net torque developed about a joint In fact, for any given joint in the body, there are many more muscles crossing the joint than there are dof prescribing joint movement llie knee, for example, has at most 6 dof, yet there are at least 14 muscles that actuate this joint One consequence of this arrangement is that the force developed by each muscle carmot be determined uniquely. Specifically, there are more unknown musculotendinous actuator forces than net actuator torques exerted about the knee that is, m > n in Eq. (6.9), which means that the matrix of muscle moment arms is not square and therefore not invertible. This is the so-called indeterminate problem in biomechanics, and virtually all attempts to solve it are based on the aj Ucation of optimization theory (see also the Qiap. 5 by Manal and Buchanan). 

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