Ring rolling inside a cone

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haruspex
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But how does it explain that ##\omega h\tan\theta## doesn't come in play. I mean, if the point of contact is at rest, its net velocity must be zero i.e
$$v_{cm}+\omega h\tan\theta=\omega' r$$
I understand that the above is supposed to be incorrect but I just don't see why.
Need to distinguish between the point of contact and a point X on the ring that's currently in contact.
The velocity of X is the velocity of the centre of the ring plus the velocity of X relative to that. The velocity of the centre of the ring is ωre; the velocity of X relative to that is -ω'r. But X is instantaneously stationary, so the sum of these is zero.
The point of contact, on the other hand, travels faster than the centre of the ring, in the ratio h tan θ to re.
 
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Need to distinguish between the point of contact and a point X on the ring that's currently in contact.
Sorry if this sounds silly but aren't you talking about the same point? I don't see the difference between "point of contact" and "point X on the ring that's currently in contact". :confused:
 
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haruspex
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Sorry if this sounds silly but aren't you talking about the same point? I don't see the difference between "point of contact" and "point X on the ring that's currently in contact". :confused:
If a wheel rolls along level ground, the point of contact is always below the centre of the wheel, so moves at the same speed. The point on the wheel which is in contact with the ground at some instant is stationary at that instant; it is the instantaneous centre of rotation.
 
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If a wheel rolls along level ground, the point of contact is always below the centre of the wheel, so moves at the same speed. The point on the wheel which is in contact with the ground at some instant is stationary at that instant; it is the instantaneous centre of rotation.
I honestly don't see the difference yet. I have attached a sketch.

Is the blue part "the point of contact" or "the point X"? :confused:
 

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haruspex
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I honestly don't see the difference yet. I have attached a sketch.

Is the blue part "the point of contact" or "the point X"? :confused:
The "point of contact" of one object on another does not refer to a fixed piece of the first object; it refers dynamically to that part of the first object which is contact with the second at any given instant. When a wheel rolls along a road, the point of contact is always on the road directly under the centre of the wheel. Thus, it moves along at the same speed as the wheel.
If the blue part you have marked is intended as a mark on the wheel, that will descriibe a cycloid. When it makes contact with the road (becoming, transiently, the point of contact) it is instantaneaously at rest.
The equation you wrote taking the velocity of the centre of the ring, then adding to that the relative velocity of a point on the ring, gave you the velocity of that piece of the ring which was instantaneously in contact with the cone. That velocity was therefore zero.
 
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The "point of contact" of one object on another does not refer to a fixed piece of the first object; it refers dynamically to that part of the first object which is contact with the second at any given instant. When a wheel rolls along a road, the point of contact is always on the road directly under the centre of the wheel. Thus, it moves along at the same speed as the wheel.
If the blue part you have marked is intended as a mark on the wheel, that will descriibe a cycloid. When it makes contact with the road (becoming, transiently, the point of contact) it is instantaneaously at rest.
The equation you wrote taking the velocity of the centre of the ring, then adding to that the relative velocity of a point on the ring, gave you the velocity of that piece of the ring which was instantaneously in contact with the cone. That velocity was therefore zero.
Ok, I see it now, thanks haruspex for your patience! :smile:
 

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