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Dynamics_Part66

Dynamics_Part66 -

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Unformatted text preview: Problem Set 9: Problem 3. Problem: A thin disk of mass m and radius r is mounted on a horizontal axle AB as shown. The plane of the disk is inclined at an angle to the vertical. The axle rotates with constant angular velocity . (a) Determine the angular momentum vector, HG . (b) Determine the angle, , between HG and the axle as a function of . Make a graph of your results for 0o 90o Solution: The first thing we must do is take care of kinematical considerations. For the problem at hand, we will have to transform from the given xyz coordinates to principal axis coordinates x y z . Also, we will need the inertia tensor for the thin disk. The figure below illustrates the relation between the xyz and x y z coordinate systems. . . . . . .. . ... . .... . ... . ... . ... . .. . . ..... . . .. . ......... . ........ . ......... . .......... . .. . .. .... .. . . ............ . ........... . . .. ............ . ............. ......... ...... ... . .... ........... . .. .......... . ..... ... ... . ............ . .... . .... . ............. ........... . .. ............. . ... . .......... .... .......... . .. .......... ....... ............. . .. . .... .. ...... ........... ....... ..... ......................................................................... .. ................................................................... . .... . ... . ..... . ... . ............................................................. ................................................... . . .... ............ . .. .... .. . ............ .............. ............ . ............. ............... ... ............ ............ .. .... .............. .. ............. .. .. .............. .. . ............ ........... ....... . . ...... ........ . . . .. y ........ . . . y A G x B x The unit vectors in the two coordinate systems are related as follows. i = cos i + sin j i = cos i - sin j and and j = - sin i + cos j j = sin i + cos j In the principal axis system, the inertia tensor for this thin disk is 1 2 0 0 2 mr 1 2 0 [I] = 0 4 mr 1 mr2 0 0 4 (a) The angular velocity vector is = i, wherefore = cos i - sin j ...
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