Swing mass overhead - forces involved

[Mancer]

I was considering the simple scenario: A mass is attached to a string, which is then twirled around by a scientist overhead. Assuming the scientist is spinning the mass fast enough, it will eventually be rotating in plane above the scientist.

I'm having some trouble working out the dynamics of this system. Obviously the mass is acted on by gravity, and in order to 'rise' from it's initial position will need a force opposing gravity.

I've reasoned that the tension on the string, while the system is gaining angular momentum from the scientist, will have a component upwards and, provided an appropriate tension, should be enough to get the mass up.

My problem comes when the string eventually is rotating in the plane parallel to the ground; obviously, there is no component of the string tension to oppose gravity. What keeps the mass from falling in this situation?

Best guess is that the mass does indeed fall downward, giving rise to an upward component of the force from the tension in the string, which then returns the mass back to the parallel plane, and oscillates like this while there is still tension in the string.

I considered conservation of angular momentum (neglecting the scientist and frictional forces) but I can't figure how this would stop the mass from falling due to gravity; only that it would have the rotational speed increase as the mass is pulled downward.

 

[Borek]

Assuming the scientist is spinning the mass fast enough, it will eventually be rotating in plane above the scientist.

No, it will be not plane, but surface of the cone. Plane is a limiting case for infinite rotation speed.

 

[Doc Al]

Assuming the scientist is spinning the mass fast enough, it will eventually be rotating in plane above the scientist.

The mass will rotate in a plane, but that plane will always be below the support point. As you yourself have pointed out, you need a component of string tension acting vertically to counteract the force of gravity.

My problem comes when the string eventually is rotating in the plane parallel to the ground; obviously, there is no component of the string tension to oppose gravity. What keeps the mass from falling in this situation?

As Borek points out, the string will never become parallel to the ground.

 

[Borek]

Oops, mass is in a plane, string is on the cone surface, lack of precision on my side.

 

[PJC01]

I can provide some insights into the forces involved in this scenario. Firstly, you are correct in identifying that gravity and tension are the two main forces at play here. The tension in the string is what provides the centripetal force necessary for the mass to maintain a circular path. Without this tension, the mass would simply fly off in a straight line due to its inertia.

Now, when the string is rotating in the plane parallel to the ground, the tension in the string is still present, but it is now acting perpendicular to the direction of the mass's motion. This means that the tension is no longer able to provide the necessary force to keep the mass at a constant height. As a result, the mass will begin to fall downward.

However, as the mass falls, the tension in the string will also decrease, as the string becomes more slack due to the decrease in distance between the scientist and the mass. This decrease in tension will cause the mass to slow down in its descent and eventually come to a stop at the bottom of its path.

At this point, the tension in the string will be at its minimum, and the mass will start to rise again due to the tension providing an upward force. This process will continue, and the mass will oscillate up and down while the string is still rotating in the parallel plane.

In terms of conservation of angular momentum, this principle does play a role in this scenario. As the mass falls, its distance from the axis of rotation decreases, causing its angular velocity to increase. This increase in angular velocity is offset by the decrease in the mass's mass, resulting in the conservation of angular momentum.

Overall, the forces involved in this scenario are complex, but they can be explained by the interplay between gravity, tension, and conservation of angular momentum. Further analysis and calculations may be necessary to fully understand the dynamics of this system.