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You are not logged in. #1 20071030 06:23:51
angular acceleration as pivot moves from centre of mass, weirdness.imagine a long bar horizontally placed in the air, pivoting it at its centre its not going to accelerate (angular) at all since weight acts through the pivot radians doesn't have a unit, so this can represent angular acceleration, so units atleast are fine. differentiating with respect to x you get: units of this: thats also fine. equating this to 0 for the stationary points gives: (allowing negative x to mean the other side of the centre of mass) the units here: so thats also fine, x should be in metres, units match up fine here. so, the maximum angular acceleration that can be achieved for a horizontal bar by pivoting it and allowing its weight to act is when x = ±√(I/M) giving a value of: units of this: so units are fine the minimum acceleration occurs at x = 0, ±∞, for which a = 0 (loose terminology of infinity)  this is just for horizontal position of bar, however the acceleration is proportional to cosθ where θ is angle made with horizontal so if one pivoted position gives larger acceleration at horizontal then another position, it will always have larger acceleration at any angle than the the position, so the acceleration at horizontal, and the time it would talk to swing past vertical are directly related. at some angle theta, the acceleration at the point in time at which the bar makes that angle would be (generalising the above coincidently) I do not think it is possible to get an exact formula for the time it would take to pass the vertical state, since acceleration and angle are dependant on eachover, perhaps someone can correct me if this is not true.  Now, is it just me or does this seem just a tad bit weird? I dunno it makes sense thinking about it, but at the same time seems unnatural. I need to quiz my physics teacher on this tomorrow and see if i can do an experiment to justify it  in both cases for distance for maximum acceleration, and maximum acceleraiton, mass does not matter since moment of inertia is Kgm^2 and mass Kg, I/M and M/I are functions of area with no mass, so both of these things are only effected by the shape of the object and its density field Last edited by lucadeltodesco (20071030 07:13:01) The Beginning Of All Things To End. The End Of All Things To Come. #2 20071030 07:53:32
Re: angular acceleration as pivot moves from centre of mass, weirdness.using a very small timestep in a RK4 integrator of 0.0005s ive approximated the time it would take for the 'bar' or whatever it may be to rotate through vertical from horizontal as distance of pivot from centre of mass changes. Last edited by lucadeltodesco (20071030 07:57:29) The Beginning Of All Things To End. The End Of All Things To Come. #3 20071030 08:06:43
Re: angular acceleration as pivot moves from centre of mass, weirdness.heres another interesting thing: The Beginning Of All Things To End. The End Of All Things To Come. #4 20071030 22:21:15
Re: angular acceleration as pivot moves from centre of mass, weirdness.hmm, perhaps i can get some help with this, i've had a go at approximating a function for the time it will take to pass through vertical for the bar being horizontal, and i've found that this equation does it almost perfectly, the only find tuning needed is in the first constant Last edited by lucadeltodesco (20071030 22:21:39) The Beginning Of All Things To End. The End Of All Things To Come. #5 20071113 00:34:54
Re: angular acceleration as pivot moves from centre of mass, weirdness.i employed the help of one of my maths teachers who came up with this: i've tried integrating this numerically ontop of my previous graphs of integrating numerically the system of equations i started with and its correct. it can't be solved analytically, and plugged into wolfram's integrator gives: where F is the incomplete elliptic integral of the first kind o.O Last edited by lucadeltodesco (20071113 00:37:03) The Beginning Of All Things To End. The End Of All Things To Come. 