Ten monomers (rods) with a length of 1 each were connected head to tail using the ideal chain model. For the radius of gyration calculation, each rod was considered a point centered at the middle of the rod, as shown in the above animations.
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The distance between where rods are joined is always 1, but this is not true for
the centers of rods; a sharp angle between two connected rods could puts the centers
of the rods close together, and in the above animation, the points represent centers
of rods, not ends of rods.
The points representing the centers of the rods are as follows:
- -0.55, 0.68, 0.94
- -1.08, 1.27, 1.27
- -1.86, 1.67, 1.69
- -2.21, 1.95, 2.35
- -2.21, 2.25, 2.47
- -2.06, 2.49, 2.56
- -1.80, 2.52, 3.50
- -1.89, 3.09, 3.90
- -1.87, 3.20, 3.54
- -2.12, 3.00, 3.29
The center of gravity, calculated by taking the average of the x, y, and z values, and is shown by the red sphere (center of gravity at -1.765, 2.212, 2.551). The animation shows three different rotations (each rotation is along an axis that runs parallel to either the x, y, or z axis.
The above formula is for the general case where each "point" could have a different mass, but for this example, each monomer rod has the same mass, so all mi values are equal, and can be substituted with the constant m. The m moves outside of the summation for the summation in the numerator. In the denominator, the summation changes to m multiplied by the number of monomers; this happens because when the m moves out of the summation, it leaves behind the number 1, and when the summation is done, there is a string of 1's. This will be added later and it is mentioned in Summation.
The radius of gyration for the example above is 1.34. This was calculated using the Rg Excel Spreadsheet.
Comparison of two 10mers- two 10mers are creating using a lattice with z=6 (possible movements include up, down, left, right, forward, backward).
