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Old Aug 12th 2015, 10:26 AM   #1
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Question Circular motion in two planes problem

Hi everybody,
I have a problem that has got me puzzled as to how to tackle it. The problem reads as follows -

A small bead can slide without friction on a circular hoop that is in a vertical plane and has a radius of 0.100m. The hoop rotates at a constant rate of 4.00rev/s about a vertical diameter.

a) Find the angle a at which the bead is in vertical equilibrium.
(Of course it has a radial acceleration towards the axis).

I am not sure how to tackle this one as it seems to involve movement in two planes ??
I would appreciate any help you can offer on how to go about solving this one.
Look forward to hearing from you.
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Old Aug 12th 2015, 10:56 AM   #2
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This is easiest using polar coordinates. The acceleration of the bead is indeed toward the axis of rotation. For the bead to be in equilibrium the component of force acting along on the bead in the direction of the wire due to that acceleration must be equal in magnitude and opposite in direction to the component of the bead's weight acting along the wire.
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Old Aug 12th 2015, 11:09 AM   #3
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Hi ChipB,
Thanks for taking the time to reply. However I am afraid you have lost me somewhat. You say ..
For the bead to be in equilibrium the component of force acting along on the bead in the direction of the wire due to that acceleration must be equal in magnitude and opposite in direction to the component of the bead's weight acting along the wire.
I have difficulty visualizing what that means ..are you saying that the vertical component and the horizontal component of force acting on the bead are equal and opposite ?
If so then I have to work out what each of these component values are in terms of the weight of the bead and the sin and cos of the angle a ?
Or am I way off the mark ?

Regards
JacktheHat
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Old Aug 12th 2015, 11:39 AM   #4
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Maybe I should have used the phrase "acting tangentially to the hoop" instead of "in the direction of the wire." See the attached sketch. The force due to gravity is shown in green - what is the magnitude of the component of that force acting on the bead in the direction tangential to the hoop (shown as a dashed green line)? The force due to angular velocity is shown in red - the component of that force acting tangentially up the hoop is shown as a dashed red line - what is its magnitude?
Attached Thumbnails
Circular motion in two planes problem-bead-wire.jpg  
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Old Aug 12th 2015, 12:44 PM   #5
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Hi ChipB,
Thanks for that .. I will try that. Just one small thing .. The term mw(squared)r
is that the force that is equal in magnitude but opposite in direction to the force produced by the mass and radial acceleration (inwards towards the axis) ?
Regards,
JacktheHat
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Old Aug 12th 2015, 02:18 PM   #6
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Originally Posted by jackthehat View Post
.. The term mw(squared)r
is that the force that is equal in magnitude but opposite in direction to the force produced by the mass and radial acceleration (inwards towards the axis) ?
Yes - it's due to the centripetal acceleration of the bead as it travels in a horizontal circle around the vertical axis of rotation.
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