Motion of Sphere Rolling Down Rotating Cone

In summary, the motion of a sphere rolling down a rotating cone involves the interaction between gravitational forces and the cone's angular momentum. As the sphere descends, it experiences both translational and rotational motion due to the cone's rotation. The dynamics of the system can be analyzed using principles of classical mechanics, including the conservation of energy and angular momentum. The resulting motion is characterized by complex trajectories influenced by the cone's geometric properties and the sphere's initial conditions, leading to rich behavior in both the sphere's path and speed as it rolls down the sloped surface.
  • #1
qianqian07
4
0
Homework Statement
See image below
Relevant Equations
.
1703283666298.png

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?
 
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  • #2
Is this problem supposed to be solved through Newtonian or Langrangian methods?
 
  • #3
Newtonian, if possible.
 
  • #4
qianqian07 said:
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?
 
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  • #5
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.
 
  • #6
  • #7
qianqian07 said:
... 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.
 
  • #8
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?
 
  • #9
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.
 
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  • #10
qianqian07 said:
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.
 
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FAQ: Motion of Sphere Rolling Down Rotating Cone

What factors influence the motion of a sphere rolling down a rotating cone?

The motion of a sphere rolling down a rotating cone is influenced by several factors, including the angular velocity of the cone, the angle of the cone, the radius and mass of the sphere, the coefficient of friction between the sphere and the cone, and gravitational acceleration. These factors determine the sphere's trajectory, speed, and stability as it rolls.

How does the angular velocity of the cone affect the sphere's motion?

The angular velocity of the cone introduces a centrifugal force that acts outward from the axis of rotation. This force affects the sphere's trajectory, potentially causing it to move radially outward or inward depending on the balance between the centrifugal force and gravitational pull. Higher angular velocities can lead to more complex motion patterns or even cause the sphere to lose contact with the cone surface if the centrifugal force exceeds gravitational force.

What role does friction play in the motion of the sphere?

Friction between the sphere and the cone is crucial for rolling motion. It prevents the sphere from simply sliding down the cone and ensures that it rolls without slipping. The coefficient of friction determines the maximum angular velocity at which rolling without slipping can occur. Insufficient friction can lead to slipping, altering the expected trajectory and dynamics of the sphere.

Can the motion of the sphere be described using classical mechanics?

Yes, the motion of the sphere can be described using classical mechanics principles, including Newton's laws of motion, rotational dynamics, and energy conservation. Equations of motion can be derived to predict the sphere's velocity, acceleration, and position as functions of time, taking into account the forces acting on the sphere, such as gravitational, normal, frictional, and centrifugal forces.

What are the potential applications of studying the motion of a sphere on a rotating cone?

Understanding the motion of a sphere on a rotating cone has applications in various fields such as mechanical engineering, robotics, and geophysics. For instance, it can help in designing stable rotating machinery, improving ball-bearing systems, or understanding natural phenomena like the motion of debris on rotating planetary surfaces. Additionally, it provides insights into more complex dynamical systems and can be used in educational settings to illustrate principles of rotational dynamics and friction.

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