Motion of Sphere Rolling Down Rotating Cone

[qianqian07]

Homework Statement

See image below

Relevant Equations

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I am trying to understand the motion of the sphere in the image above, and I am a bit confused about the motion. How does the ball move down the cone? Will the rotation of the cone cause the ball to rotate with it, and which direction would the static friction be in? What does the path the ball take look like? From my understanding, if there is no friction, then the ball will just roll down the side of the cone in a straight line. However, when the friction is nonzero, how does it affect the motion, given that the cone is rotating?

 

[PhDeezNutz]

Is this problem supposed to be solved through Newtonian or Langrangian methods?

 

[qianqian07]

Newtonian, if possible.

 

[kuruman]

Newtonian, if possible.

What are we solving for? There is no question posed by the problem. How about posting the entire statement of the problem?

 

[qianqian07]

This wasn't from a full problem, it is just a scenario that I thought of. I would just like to understand conceptually how the sphere will move and how the rotation of the cone affects it.

 

[haruspex]

This wasn't from a full problem, it is just a scenario that I thought of. I would just like to understand conceptually how the sphere will move and how the rotation of the cone affects it.

Apart from the contribution from gravity, it is much like the motion of a ball rolling on a rotating disc.
Try https://phys.libretexts.org/Bookshe...g_Plane/30.05:_Ball_Rolling_on_Rotating_Plane

 

[Lnewqban]

... From my understanding, if there is no friction, then the ball will just roll down the side of the cone in a straight line. However, when the friction is nonzero, how does it affect the motion, given that the cone is rotating?

It seems to me that the ball would not start rolling at all, if there is no friction.
It would slide down the side of the cone, ignoring its rotation.

When the friction is nonzero, the rotational inertia of the ball needs to be considered in order to compute the acceleration, reason for which the radius and the mass of the ball are provided.

 

[qianqian07]

Okay, thanks everyone for the help. When there is friction between the two surfaces, by Newton's third law, there should be a force exerted on the cone as well, so it seems that there should be a torque exerted on the cone. The rotational inertia of the ball about the axis of the cone's rotation at a given height can be found using the parallel axis theorem (using the fact that the rotational inertia of the ball is $\frac{2}{5}mr^2$). When this is known, how could I calculate the angular acceleration of the cone? If I understand correctly, the ball's acceleration depends on the cone's acceleration, and the cone's acceleration depends on the friction, which depends on the ball's acceleration. Would finding expressions for the ball or the cone's acceleration then involve some kind of system of differential equations?

 

[Lnewqban]

The rotation of the cone is happening at constant angular speed, according to the problem.

It seems that ω will remain constant during the sphere movement, regardless of how much resistance that movement may put against that rotation.

The sphere is the only one increasing its velocity from zero, when at the apex of the cone.

 

[haruspex]

Would finding expressions for the ball or the cone's acceleration then involve some kind of system of differential equations?

Yes. As @Lnewqban points out, the cone is given as having constant angular velocity, but the interplay between the acceleration of the ball, its position and its velocity will lead to ODEs.
It may become airborne at some point.