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Old Jul 4th 2014, 09:01 PM   #1
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Gravitational Acceleration to a body offset from a tilted axis

I was trying to figure out the acceleration to a body on a tilted axis. Is it the same as if the body were on a hill? or would it be more like a free object have to rotate, so the inertia would just be mv? Assuming a short time interval, so the change in direction due to reaching the end of the arc doesn't affect the velocity...

So the vertical acceleration would be something like sin(angle from vertical) * g? and the acceleration around the axis would be cos(angle)?



Well that's just part of the problem...really I want to compute how much force the body has if the top of the axis of tilt is rotated in a circle.

So here's a sketch of the model...
A is the length of the axis from the pivot on the ground plane to the circle to rotate around.
B is the length of the arm that the mass (m) is attached to.
c represents the center of mass which is actually at length B from the axis.
r is the radius that the top of the axis moves through.

the arm the mass is on is 90 degree angle from the axel.



at some time T, the end of A moves around the circle in an angle(da)... from equilibrium this causes the mass (m) to move up. This is (sin(da) * B * r / A) (the angle must be less than 90 degrees... and really it's a continuous process so I'm sure some sort of limit integral should be used)... Hmm maybe there's two lengths tA (top A) and bA (bottom A) which is above and below the arm...

How much force is required to move A?
How much torque results on the axis A?
Does the speed matter? Like say A goes around the circle at 30RPM?
assume that the circle at the top is constrained by something, so disregard required force to keep the axis within the circle R
Edit: Maybe it's not torque I seek, more like... power; like a free spinning flywheel has a lot of torque but no horsepower... am more interested in the power of the falling mass (rotating mass)

I started to do a spreadsheet to get an idea of how far it moves (needs to move)
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Gravitational Acceleration to a body offset from a tilted axis-tilted-axis.jpg   Gravitational Acceleration to a body offset from a tilted axis-model.jpg  

Last edited by JDecker; Jul 5th 2014 at 07:03 AM.
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Old Jul 5th 2014, 06:55 AM   #2
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I guess my first illustration is from the wrong state... as the mass will be near the bottom of the circular path most of the time....
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Old Jul 5th 2014, 06:03 PM   #3
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If I understand you correctly, the best bet is to not change the coordinate system and resolve the forces in to components. In the tilted case you would have the weight pointing straight down and the centripetal force would be horizontal. These directions do not change, even if you are using a "tilted" coordinate system. For the tilted case, then, you would have to resolve both the weight and centripetal force.

-Dan
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Old Jul 5th 2014, 08:54 PM   #4
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Originally Posted by topsquark View Post
If I understand you correctly, the best bet is to not change the coordinate system and resolve the forces in to components. In the tilted case you would have the weight pointing straight down and the centripetal force would be horizontal. These directions do not change, even if you are using a "tilted" coordinate system. For the tilted case, then, you would have to resolve both the weight and centripetal force.

-Dan
Well... Like I said in the reply that the state of the tilted axis picture is sort of wrong... it's usually near the bottom of the curve...



And, isn't the centripetal just the stress on the rigid bar holding the mass? Much like stating, I'm not really interested in the force to keep the top end of A within the circle of rotation, just the required tangential force to move it...?

Sorry, I guess I'm asking in the wrong place, it's been 25 years since I studied calculus, and was hoping someone could help me to a full solution instead of a partial hint... hints are helpful, might lead me to a more complete solution, but I don't know where I'm overcomplicating it...
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Gravitational Acceleration to a body offset from a tilted axis-tilted-axis2.jpg  

Last edited by JDecker; Jul 5th 2014 at 09:08 PM.
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Old Jul 6th 2014, 05:26 AM   #5
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Is there a better question I can ask?

Is there something better than 'tilted axis' that I can call that to search for more information?
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Old Jul 6th 2014, 08:19 AM   #6
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Originally Posted by JDecker View Post
Is there something better than 'tilted axis' that I can call that to search for more information?
This is just a variation on the spinning top problem. You might try starting there. Another thought is to look up "precession."

-Dan
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Old Jul 6th 2014, 11:21 AM   #7
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Originally Posted by topsquark View Post
This is just a variation on the spinning top problem. You might try starting there. Another thought is to look up "precession."

-Dan
Ya sorta appears that way, but the speed is lower than 60RPM, it's not fast enough to be gyroscopic... especially with such a highly offset center of gravity
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