Frequency of Circular Motion in Ring Rolling Inside a Cone

Thread starter Saitama
Start date Apr 3, 2014
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Cone Ring Rolling

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The discussion centers on calculating the frequency of circular motion for a small ring rolling inside a cone with friction. Participants analyze the forces and torques acting on the ring, particularly the torque due to friction along the cone's slant height, which is necessary for the ring's precessional motion. Confusion arises regarding the relationship between the angular velocities of the ring and its center of mass, leading to a deeper exploration of the dynamics involved. The final conclusion reached is that the frequency of the ring's motion can be expressed as ω = (1/tanθ)√(g/2h), confirming the correct relationship between the variables involved. The discussion emphasizes the importance of understanding the dynamics of rolling motion and the role of torque in maintaining circular motion.

Apr 24, 2014

Saitama

haruspex said:

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!