Mass dropped onto rotating disk

In summary: Summary: In summary, the mass is captured by a rotating spiral that spins the axis and allows the mass to slide, scoop and redirect.
  • #1
Jazzjohn
1
2
TL;DR Summary
How would a mass fly off a rotating disk/vane system?
Picture a flat disk of radius r with a radial vane. The disk is rotating at angular velocity w. Assume the vane is straight, starts at the center and ends at the perimeter of the disk.
A very small round mass ( of m grams) is dropped onto the disk very near the center. The vane contacts it and pushes it. The mass slides along the vane and travels toward the disk rim.
Assume no friction along the vane and disk surfaces.
At the end of the vane (at the edge of the disk), the mass is spun off the disk.

1. What direction does it travel off the disk? Intuition tells me there will be a radial force component in addition to the tangential force component.

2. If the vane is not restricted to a straight line, what shape will increase the tangential direction while reducing the radial direction? Is there a theoretical optimal shape for a given mass size?
 
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  • #2
Jazzjohn said:
1. What direction does it travel off the disk? Intuition tells me there will be a radial force component in addition to the tangential force component.
There are both radial and tangential velocity components relative to the fixed surface below the disk.
Jazzjohn said:
2. If the vane is not restricted to a straight line, what shape will increase the tangential direction while reducing the radial direction? Is there a theoretical optimal shape for a given mass size?
If the groove curves towards the edge, then the mass may come out with minimal radial velocity component.
 
  • #3
Jazzjohn said:
Summary: How would a mass fly off a rotating disk/vane system?

Intuition tells me there will be a radial force component in addition to the tangential force component.
Your intuition needs to be tempered by logic. All contacts are assumed frictionless. A frictionless surface can only a force perpendicular and away from it. The vane extends along a radius and can only exert a force tangent perpendicular to the radius. The disk is horizontal and can only exert a force "up", perpendicular to the radius.
 
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  • #4
Conservation of energy arguments work, even when made using the rotating frame. There is a centrifugal potential associated with a uniformly rotating frame.
 
  • #5
Jazzjohn said:
Summary: How would a mass fly off a rotating disk/vane system?

1. What direction does it travel off the disk? Intuition tells me there will be a radial force component in addition to the tangential force component.
I would do this problem using the Lagrange multiplier approach. The KE in polar coordinates is: ##T = \frac{1}{2}m \dot r^2+\frac{1}{2}m r^2 \dot \theta^2##, and this problem has no potential so ##V=0##. Then the rotating vane can be represented by the constraint ##\theta=\omega t##. This gives us the constrained Lagrangian $$L=\frac{1}{2}m \dot r^2+\frac{1}{2}m r^2 \dot \theta^2+\lambda(\theta-\omega t)$$

The Euler Lagrange equations then give us $$ m(r \dot \theta^2-\ddot r)=0$$ $$\lambda-mr(2\dot r \dot \theta+r \ddot \theta)=0$$ $$ -t \omega + \theta=0$$ which, assuming ##r(0)=r_0## and ##\dot r(0)=0## we can solve to obtain $$r(t)=r_0 \cosh(\omega t)$$

Jazzjohn said:
Summary: How would a mass fly off a rotating disk/vane system?

2. If the vane is not restricted to a straight line, what shape will increase the tangential direction while reducing the radial direction? Is there a theoretical optimal shape for a given mass size?
You could do this by changing the constraint equation, but I will leave that as an exercise for the interested reader
 
  • #6
@Jazzjohn You ought to make some attempt next time when posting a problem.

To improve reaction acceleration times & direction backwards in the axial direction rather than radial and tangential, one needs to capture the particle mass in a curved spiral that rotates the axis and captures the mass by spinning fast enough to have the rotational duration between vanes less than the mass linear velocity so as not to miss it.

But we think that the spiral will allow the mass to slide, scoop and redirect , we would be making an invalid assumption on the stiffness and elastic properties of both materials combined. Until all assumptions are stated, the question is actually invalid. (remember this in future as it can lead to false positive conclusions)
 

Related to Mass dropped onto rotating disk

1. What is the purpose of dropping mass onto a rotating disk in a scientific experiment?

The purpose of this experiment is to study the effects of angular momentum and conservation of energy on a rotating system. By dropping mass onto a rotating disk, we can observe how the system's angular velocity and energy change.

2. How does the mass affect the rotation of the disk?

The mass dropped onto the rotating disk will cause a change in the disk's angular velocity. This change is due to the conservation of angular momentum, where the initial angular momentum of the system is equal to the final angular momentum after the mass is dropped.

3. What factors can affect the results of this experiment?

The results of this experiment can be affected by various factors such as the mass and size of the disk, the mass and velocity of the dropped object, and the initial angular velocity of the disk. Other external factors such as air resistance and friction can also play a role in the results.

4. How is energy conserved in this experiment?

The total energy of the system, which includes the kinetic energy of the rotating disk and the potential energy of the dropped mass, remains constant throughout the experiment. This is known as the principle of conservation of energy.

5. What real-life applications can be derived from this experiment?

The principles of angular momentum and conservation of energy observed in this experiment have various real-life applications. These include understanding the behavior of objects in space, designing efficient energy systems, and developing technologies such as gyroscopes and flywheels.

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