Frictionless, rotating circular hoop

In summary, we have a small bead sliding without friction on a circular hoop with a radius of 0.100m in a vertical plane. The hoop rotates at a constant rate of 4.00\frac{rev}{s} about a vertical diameter. We need to find the angle \theta at which the bead is in vertical equilibrium and determine if it is possible for the bead to "ride" at the same elevation as the center of the hoop. Additionally, we need to consider what would happen if the hoop rotates at a slower rate of 1.00\frac{rev}{s}. To solve this problem, we can use the equations F=m\frac{v^2}{r} and s=r\theta,
  • #1
courtrigrad
1,236
2
A small bead can slide without friction on a circular hoop that is in a vertical plane and has a radius of [i tex]0.100 m[/itex]. The hoop rotates at a constant rate of [itex] 4.00 \frac{rev}{s} [/itex] about a vertical diameter. (a) Find the angle [itex] \theta [/itex] at which the bead is in vertical equilibrium. (It has a radial acceleration toward the axis) (b) Is it possible for the bead to "ride" at the same elevation as the center of the hoop? (c) What will happen if the hoop rotates at [itex]1.00 \frac{rev}{s} [/itex]?

All I really know is that you have to find the velocity [itex] \frac{2\pi r}{T} [/itex]. You have to use the equation [itex] F = m\frac{v^{2}}{r} [/itex]. Is it possible to use the arc length formula [itex] s = r\theta [/itex]?

Any help would be appreciated!

Thanks :smile:
 
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  • #2
HINT: The hoop can only provide the radial component of the centripetal force - i.e. gravity must supply the tangential component. :)
 
  • #3
So the hoop is rotating about a vertical circle, while the bead is rotating in its own circle within the vertical circle? The direction of the acceleration is toward the center.

Any help would be appreciated

Thanks Tide for your hint
 
  • #4
can someone please help me out
 
  • #5
If you measure the angle from the vertical line connecting the center of the hoop to the lowest point on the hoop, then the radius of the "orbit" is [itex]R \sin \theta[/itex]. This makes the component of centripetal force on the bead tangential to the hoop [itex]m \omega^2 R \sin \theta \cos \theta[/itex]. The component of the gravitational force tangential to the hoop is [itex]m g\sin \theta[/itex].

Draw some pictures to verify the above and then proceed.
 

1. What is a frictionless, rotating circular hoop?

A frictionless, rotating circular hoop is a theoretical object that is used in physics to study the effects of rotational motion without the presence of friction. It is a hoop-shaped object that is able to rotate freely without any resistance or slowing down due to friction.

2. How is a frictionless, rotating circular hoop different from a regular hoop?

A regular hoop experiences friction as it rotates, which can cause it to slow down and eventually stop. A frictionless, rotating circular hoop does not experience this resistance, allowing it to maintain its rotational motion indefinitely.

3. What are the applications of studying a frictionless, rotating circular hoop?

The study of a frictionless, rotating circular hoop can help scientists understand the principles of rotational motion, such as angular momentum, torque, and centripetal force. It can also be used in theoretical models and simulations to understand the behavior of objects in a frictionless environment.

4. Is a frictionless, rotating circular hoop possible in real life?

No, a frictionless, rotating circular hoop is a theoretical concept and cannot exist in real life. In order for an object to rotate, there must be some form of friction present, even if it is minimal.

5. How can the concept of a frictionless, rotating circular hoop be applied to real-world situations?

The principles and equations used to study a frictionless, rotating circular hoop can be applied to real-world situations, such as the motion of planets and galaxies in space or the spinning of a gyroscope. They can also be used in engineering and design to minimize the effects of friction in rotating machinery.

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