Two spinning disks, did I mess up?

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It seems like the mistake was in assuming that the larger gear would spin at the same speed as the smaller gear after they were meshed together. In reality, the larger gear will spin twice as fast as the smaller gear, resulting in a decrease in angular speed of 20% for the larger gear. In summary, the mistake was in assuming equal angular speeds after meshing, when in fact the larger gear will spin twice as fast as the smaller gear.
  • #1
etotheipi
Homework Statement
See below
Relevant Equations
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Hey guys, I need to check that I'm not doing something stupid. You have two uniform disc-shaped gears, one of which has twice the radius of the other [and 16 times the moment of inertia]. Initially, the larger gear is spinning at ##\omega##, and then they're suddenly meshed together. We need to show that the angular speed of the larger disk drops by 20%. [Edit. To clarify, the disks are given to have equal uniform area densities ##\sigma##].

Since the torque of the tangential force on the larger disk about its centre is twice in magnitude compared to the torque of the tangential force on the smaller disk about its centre, I reckoned that the change in angular momentum of the larger one would be twice that of the smaller one, i.e. that ##16(\omega - \omega_f) = 2\omega_f##. But that gives me that ##\omega_f = (8/9)\omega##, which clearly isn't a 20% decrease.

What did I miss?! It's almost completely improbable that the question contains a mistake, so the error must be on my end. Thanks!
 
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  • #2
16 times the moment of inertia? Its quadratic in the radius.
 
  • #3
hutchphd said:
16 times the moment of inertia? Its quadratic in the radius.

But the mass of the uniform disk is also quadratic in the radius, so the product ##I = (1/2)mr^2 = (1/2) \pi r^2 \sigma r^2 = (1/2) \pi \sigma r^4## is quartic in the radius. I should have specified that both are given to have equal area densities ##\sigma##, and not equal masses. This part is in agreement with the given solutions.
 
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  • #4
I see it. The larger gear will get twice the torque impulse as you notice. But it will also be spinning twice as fast as the big gear ...so that should be ##2(2\omega_f)## on the RHS of the equation.
I believe I have violated the "homework rule".
 
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  • #5
hutchphd said:
I see it. The larger gear will get twice the torque impulse as you notice. But it will also be spinning twice as fast as the big gear ...so that should be ##4\omega_f## on the RHS of the equation.
I believe I have violated the "homework rule".

That’s it, thanks :wink:
 
  • #6
This was a really interesting problem.
 
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1. What are two spinning disks?

Two spinning disks refer to two circular objects that are rotating around a central axis. These objects can be any shape, size, or material, as long as they are spinning in a circular motion.

2. How do I know if I messed up with two spinning disks?

If you are referring to a scientific experiment involving two spinning disks, you can determine if you messed up by analyzing your results. If the data does not match your hypothesis or if there are inconsistencies in your methods, it is possible that you made a mistake. However, making mistakes is a normal part of the scientific process and can lead to valuable insights and discoveries.

3. What factors can affect the spinning of two disks?

There are several factors that can affect the spinning of two disks, including the shape and size of the disks, the speed and direction of rotation, the surface they are spinning on, and external forces such as friction or air resistance. It is important to control these variables in a scientific experiment to ensure accurate results.

4. Can two spinning disks be used in practical applications?

Yes, two spinning disks can be used in various practical applications such as power generation, gyroscopes, and data storage devices. The rotational motion of the disks can be harnessed to produce energy, stabilize the orientation of objects, and store and retrieve information.

5. How can I conduct a successful experiment with two spinning disks?

To conduct a successful experiment with two spinning disks, it is important to clearly define your research question or hypothesis, carefully control variables, and accurately measure and record data. It is also helpful to consult with other scientists and conduct multiple trials to ensure the reliability of your results.

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