Calculating Fractional Change in Angular Velocity of a Spinning Cone of Sand

In summary, the conversation discusses the calculation of the fractional change in angular velocity of a light, hollow cone filled with sand as it drains through a hole in the apex. The conservation of angular momentum and the moment of inertia of a solid cone are considered, and it is determined that the shape of the sand will form a conical ring. The final conclusion is that the change in angular velocity can be calculated by finding the new height of the cone with half the mass.
  • #1
zerakith
7
0
1. A light, hollow cone is filled with sand set spinning about a vertical axis through its apex on a frictionless bearing. Sand is allowed to drain slowly through a hole in the apex. Calculate the fractional change in angular velocity when the sand level has fallen to half its original value. You may neglect the contribution of the hollow cone to the moment of inertia.
2. Moment of Inertia of Solid Cone: [tex]I=\frac{\rho\pi R^4h}{10} [/tex]
Conservation of Angular Momentum: [tex]I_0\omega_0=I_1\omega_1 [/tex]
Dimensions of the cone: Length: h. Radius of circle at end of cone: R3. So its clear to me that the way to proceed is to consider the conservation of angular momentum. At the start the cone is full of sand and thus the system is just a solid cone so:
[tex]I_0=\frac{\rho\pi R^4h}{10} [/tex]
For [tex]I_1 [/tex] i need to calculate the moment of inertia, the sticking point for me is the shape the sand will take within the cone. Intuitivly i think that the sand will form a conical ring around the cone (i.e so there is a smaller cone of empty space inside the cone and the rest is filled with sand). I'm not really happy with jumping to that conclusion however if I do that and work it through I get:
[tex]\frac{\omega_1}{\omega_0}=\frac{h^3}{3}[/tex]
I've no real way to determine whether this is correct. Am I right in the assumption about the shape of the sand (and ideally why?), and does the fractional change I have ended up with make sense?
There is no rush to this, it's not examined and term is over, it is however, bugging me.

Thanks in advance

Zerakith
 
Last edited:
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  • #2
I have now figure this out, the cone was vertical with the apex pointing downwards, hence it was simply matter of finding the new height of the cone, given that there was half the mass.
 

1. How do you calculate the fractional change in angular velocity of a spinning cone of sand?

To calculate the fractional change in angular velocity, you will need to know the initial and final angular velocities of the spinning cone of sand. Then, you can use the formula (final angular velocity - initial angular velocity) / initial angular velocity to calculate the fractional change.

2. What are the units of measurement for the fractional change in angular velocity?

The fractional change in angular velocity is a dimensionless quantity, meaning it does not have any specific units of measurement. It is simply a ratio of two angular velocities.

3. Can the fractional change in angular velocity be negative?

Yes, the fractional change in angular velocity can be negative if the final angular velocity is less than the initial angular velocity. This would indicate a decrease in the speed of the spinning cone of sand.

4. What factors can affect the fractional change in angular velocity of a spinning cone of sand?

The primary factor that can affect the fractional change in angular velocity is the external torque applied to the spinning cone of sand. Other factors such as the shape and size of the cone, the surface it is spinning on, and air resistance can also play a role.

5. How is the fractional change in angular velocity of a spinning cone of sand related to its rotational kinetic energy?

The fractional change in angular velocity is directly proportional to the change in the rotational kinetic energy of the spinning cone of sand. This means that a larger fractional change in angular velocity will result in a larger change in rotational kinetic energy.

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