Thin hoop suspended by string and rotating

In summary, a thin hoop of mass M and radius R suspended on a string at a high angular velocity ω will spin with its plane nearly horizontal and its center nearly on the axis of support. The string makes an angle α with the vertical. The small angle β between the plane of the hoop and the horizontal can be found approximately, as well as the radius of the small circle traced out by the center of mass about the vertical axis. The equation for the moment of inertia may be useful in solving this problem.
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Homework Statement



A thin hoop of mass M and radius R is suspended from a string through a point on the rim of the hoop. If the support is turned with a high angular velocity ω, the hoop will spin as shown in the diagram above (see the top figure in the attachment), with its plane nearly horizontal and its center nearly on the axis of support. The string makes an angle α with the vertical.

(a) Find, approximately, the small angle β between the plane of the hoop and the horizontal (see the bottom figure in the attachment).

(b) Find, approximately, the radius of the small circle traced out by the center of mass about the vertical axis.

Homework Equations


The Attempt at a Solution



I am really struggling with these rotational dynamics problems. My biggest issue with this particular problem is that I don't even know how to start the solution.
 

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I think that I am supposed to use the equation for the moment of inertia, but I'm not sure how to apply it. Any help would be greatly appreciated!
 

FAQ: Thin hoop suspended by string and rotating

How does a thin hoop suspended by string rotate?

A thin hoop suspended by string rotates due to the principle of conservation of angular momentum. As the hoop is set into motion, the string exerts a force on the hoop in the direction perpendicular to the plane of rotation. This force causes the hoop to accelerate and rotate around the string.

What factors affect the rotation of a thin hoop suspended by string?

The rotation of a thin hoop suspended by string can be affected by several factors, such as the length and tension of the string, the mass and shape of the hoop, and any external forces acting on the hoop.

Is there a limit to how fast a thin hoop suspended by string can rotate?

Yes, there is a limit to how fast a thin hoop suspended by string can rotate. This limit is determined by the tension and strength of the string, as well as the mass and shape of the hoop. If these factors are exceeded, the string may break or the hoop may deform or fly off in a different direction.

How does the rotation of a thin hoop suspended by string demonstrate angular momentum?

The rotation of a thin hoop suspended by string demonstrates angular momentum by showing the conservation of angular momentum. As the hoop rotates, its angular momentum remains constant unless acted upon by an external torque. This is similar to how a spinning top will continue to spin unless an external force is applied to change its direction or speed.

What real-life applications are there for a thin hoop suspended by string rotating?

There are several real-life applications for a thin hoop suspended by string rotating, such as in gyroscope technology and in the design of satellite stabilization systems. The principles of conservation of angular momentum and rotational motion are also important in fields like robotics, aerodynamics, and engineering.

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