Rolling Cone - (Rotating Coordinate systems)

In summary, to describe the motion of a cone rolling on a flat surface, we can use the equations Vcm = r*OMEGAcm + r*OMEGA and OMEGAcmlab = OMEGA.
  • #1
m0nk3y
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Homework Statement


A cone rolls on a flat surface. The instantaneous axis of rotation lies parallel to the point where the cone touches the surface and the angular velocity OMEGA. The motion of the center of mass (Vcm) plus a rotation OMEGAcm about the center if mass. Describe this motion by finding Vcm and OMEGAcmin the laboratory (space) system.

PLEASE HELP!


Homework Equations





The Attempt at a Solution


- i made the origin at the end of the cone (close side)
- rotated about the x-axis.
- got the matrix of the rotation about the x, so x'
now, i don't know where to go OR if I did anything right

Thanks!
 
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  • #2


Thank you for your question. It seems like you are on the right track with setting up the coordinate system and finding the rotation matrix. However, in order to fully describe the motion of the cone, you will need to use some additional equations.

First, let's define some variables:
- r is the radius of the cone
- h is the height of the cone
- m is the mass of the cone
- g is the acceleration due to gravity

To find the velocity of the center of mass in the laboratory system, we can use the equation:
Vcm = Vcminstantaneous + Vcmrotation

Vcminstantaneous is the instantaneous velocity of the point where the cone touches the surface. Since the cone is rolling without slipping, we know that this point is not moving in the x-direction, so Vcminstantaneous = 0. In the y-direction, this point is moving with the same velocity as the center of mass, which we can find using the equation:
Vcm = r*OMEGAcm

Vcmrotation is the velocity due to the rotation of the cone about its center of mass. This can be found using the equation:
Vcmrotation = r*OMEGA

Now, to find the angular velocity OMEGAcmin the laboratory system, we can use the equation:
OMEGAcmlab = OMEGAcminstantaneous + OMEGAcmmotion

OMEGAcminstantaneous is the instantaneous angular velocity of the cone about the point where it touches the surface. This can be found using the equation:
OMEGAcminstantaneous = Vcminstantaneous/r = 0

OMEGAcmmotion is the angular velocity due to the rotation of the cone about its center of mass. This can be found using the equation:
OMEGAcmmotion = OMEGA

Therefore, the final equations for Vcm and OMEGAcmin the laboratory system are:
Vcm = r*OMEGAcm + r*OMEGA
OMEGAcmlab = OMEGA

I hope this helps. Good luck with your calculations!
 

1. What is a rolling cone?

A rolling cone is a type of rotating coordinate system used to describe the motion of a rigid body. It is composed of three mutually perpendicular axes, with the center of the cone moving along a circular path and the cone itself rotating around its axis.

2. How is a rolling cone different from other coordinate systems?

A rolling cone is unique because it is a non-inertial frame of reference that takes into account both translation and rotation. This allows for a more accurate description of the motion of a rigid body compared to other coordinate systems, such as the fixed or moving coordinate systems.

3. What are the advantages of using a rolling cone?

One of the main advantages of using a rolling cone is that it simplifies the equations of motion for a rigid body in rotational motion. It also allows for a more intuitive understanding of the dynamics of rotating objects, making it easier to analyze and predict their behavior.

4. How is a rolling cone used in real-world applications?

Rolling cones are commonly used in fields such as physics, engineering, and robotics to describe the motion of rotating objects. They are also used in computer graphics and animation to simulate the movement of objects in three-dimensional space.

5. Can a rolling cone be used for non-rigid bodies?

No, a rolling cone is specifically designed for rigid bodies that maintain their shape and size during motion. For non-rigid bodies, other coordinate systems may be more suitable, such as the deforming coordinate system which takes into account the changing shape of the object.

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