Dynamics_Part76

Dynamics_Part76 - Thus, in standard vector form, we have 2...

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Unformatted text preview: Thus, in standard vector form, we have 2 2 HG = - mc2 sin i + mc2 ( + cos ) k 5 5 For steady precession, we know that , and are constant. Thus, inspection of the equation above shows that HG does not vary with time in the rotating xyz frame. Using the Coriolis Theorem, we find d HG dHG = + HG dt dt = 0+ i sin - - 2 mc2 sin 5 j 0 0 k cos 2 mc2 ( + cos ) 5 = 2 2 2 mc sin ( + cos ) - mc2 2 sin cos j 5 5 Simplifying this result by canceling the terms proportional to 2 , we conclude that 2 HG = mc2 sin j 5 (b) The center of mass of the sphere rotates in a horizontal circular path with angular velocity . As illustrated in the figure below, the radius, R, of the circular path is R= A sin + c sin .. .. .. .... ..... ... ..... . . ... .... . .. .. . .. ... . . ... .... . .... ... . . ... ... . ..... .... . . .. ... . . . . ... ... . . . ..... ..... .... . .. ... .... .... . ....... ... ... ... .. . . ... ... . . ... . ... . . ... ... . . ... ... . ... ... . . ... ... . ... ... . . ... ... . ... ... . . ... ... ... ... . . . ... ... ... ... . . . ... ... ... ... . . ... ... . ... ... . . ... .... .. ... .... . . . . ... .... .. ... ... .... . . . ................................... ... ............................... .............. .. . . ..... ...... .. .. .. .. ..... .... . ......... . .. . . . .. ....... . ................................. ............. . ...................................................... . . ................................................. .... . ............................................................... . . .. . ...................................................................... .. .......... ............................................................................................. ...................................................................................... . .. . .. .. .. . ....... . ........... . . . . . . .. . .... ... . .................................................................................. ............................................................................................ ...... . ...................................... .. ..... . .. . . .. ... . . . . ............................................ . . . ... .................. ............ .. . . .............. ................................................................................. .. . ............................................................................................. .................. ................... ............................................................................................................................. ..................................................... ...................................................................................... ... . . . . . . .. .. .. .. .. . . ................................................ .. . .. ....... . . . . .. . . .. . ............ .. ............. .. . ............ . . ............ . . .. . . . . . . .. . ...... ...... B sin c G c sin The acceleration of the center of mass of the sphere is purely centrifugal so that a = ( sin + c sin )2 er Thus, the acceleration vector is a = ( sin + c sin )2 cos i + ( sin + c sin )2 sin k Note that the direction of the x axis is to the left, so that the sphere's acceleration is toward the precession axis, as it must be. Turning to the forces acting on the sphere, there are two. Referring to the figure below, we see that the first is the tension in cord AB, T, and the second is the sphere's weight, mg. T - .... .. . .... .. .. ... . .. . . .. ... .... .. .. ..... . .. ... .......... . ... ........ . . . ...... ... B . ...... .. .. .. ........................ ... ...... ........................ ..................... ................................ .......... .. ............. ......... ........ ......... . ....... .... . . .. ............................................... .. . ................. ... ....... .. .. .......................... . ....... .............................................................. ....................................................... .... . . .. .............................................................. ...................... .................... . . .... ........ .... .. . . ........................................... ..... ............................... .. . ............................................................... ................................................................. .................................................................... ............ . .. .. .................................................................... ........................................................................ . . . .. ... ... . .. .......................................................................... . . ................ ............................................... ...................................................G....................... .. ......................................................................... . . .................. ........................... . . .... ................................................................... .................................................................. ................................................................ . . . .. .. . ................................................. ....................... ................................ .. ... .. ..... .. . .. . .............................................................. .... .. ................................................................. . .. ........................................................................... . .... ...................................................... .... . .... ..................................................... ....... ................. . ... ............................................. .. . . .. ..................................... ......................... .... .... .. ................ . . .. ........... ... .. .. . . . . .. ... . ... .. .. . . . . where er = i cos + k sin z x mg ...
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This note was uploaded on 05/12/2010 for the course AME 301 taught by Professor Shiflett during the Spring '06 term at USC.

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