Bead on a Rotating Hoop - Constraints (Holonomic / Nonholonomic)

In summary, we have a bead sliding on a vertically oriented hoop that is rotating about its center and a vertical axis. The constraints in both cases are time-dependent, with the first case being holonomic and the second case being non-holonomic due to the involvement of friction.
  • #1
Mistro116
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1. Homework Statement
We have a bead sliding with friction on a hoop oriented vertically. First the hoop rotates about its center with rotation axis perpendicular to its plane.

Second, the hoop rotates about a vertical axis as well.
In both of these cases, are the constraints holonomic or nonholonomic? Are the time dependent or independent?

I think in both cases the constraints will be time-dependent. I am confused as to whether or not they will be holonomic / non-holonomic. I thought they would both be non-holonomic since friction is involved. Is this right?
 
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  • #2
2. Homework Equations N/A3. The Attempt at a Solution In the first case, the constraint is holonomic since it is independent of time. In the second case, the constraint is non-holonomic since friction is involved and it is time dependent.
 

FAQ: Bead on a Rotating Hoop - Constraints (Holonomic / Nonholonomic)

What is a holonomic constraint in the context of a bead on a rotating hoop?

A holonomic constraint is a constraint that can be expressed as an equation between the position, velocity, and time of the bead on a rotating hoop. This means that the constraint does not depend on the direction of motion, only on the position and time.

Can you give an example of a holonomic constraint in the context of a bead on a rotating hoop?

An example of a holonomic constraint in this context would be the constraint that the distance between the bead and the center of the hoop remains constant. This can be expressed as an equation: r = constant, where r is the distance and constant is a fixed value.

How is a nonholonomic constraint different from a holonomic constraint in the context of a bead on a rotating hoop?

A nonholonomic constraint is a constraint that cannot be expressed as an equation between the position, velocity, and time of the bead on a rotating hoop. This means that the constraint depends on the direction of motion. An example of a nonholonomic constraint in this context would be the constraint that the bead can only move in a certain direction along the hoop.

What is the significance of constraints in the study of the bead on a rotating hoop?

Constraints play a crucial role in the study of the bead on a rotating hoop as they restrict the possible motions of the bead, making the problem easier to solve. They also allow us to make predictions about the behavior of the bead and understand the underlying principles that govern its motion.

Can constraints be applied to other systems besides a bead on a rotating hoop?

Yes, constraints can be applied to many different systems in physics and engineering. They are used to simplify problems and make them more manageable, while still providing valuable insights and predictions about the behavior of the system. Constraints are also a key concept in the field of mechanics and are widely used in the study of various mechanical systems.

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