# What Causes the Loss of Mechanical Energy in a Frictionless Spinning Cylinder?

• guv
guv
Homework Statement
Imagine a frictionless horizontal table. On the table a uniform cylinder is spinning with one flat face touching the table at angular velocity ##\omega_0##. Through the flat faces of the cylinder, a tiny hole is drilled through. As the cylinder is spinning in place, a pin is inserted through the cylinder very quickly and the cylinder starts to spin around the pin. The horizontal position of the pin does not change (imagine the insertion happens very quickly). Can we apply conservation of energy to find the final angular velocity?
Relevant Equations
N/A
We all know we need to apply conservation of angular momentum here. This necessarily leads to a difference in mechanical energy. Since initial rotational inertial is less than final rotational inertia, there is a loss of mechanical energy. However, I have not been able to convince myself what's doing work to take away the mechanical energy of the disk? Or is it because the problem is set up in a too idealistic way? Thanks,

Delta2
It doesn't say, but I assume the hole is off-centre.

There is an inelastic impact as the part of the cylinder adjacent to the pin is suddenly brought to a stop.

guv
What if the disk and the pin are perfectly rigid and the diameter of the pin and the diameter of the hole are identically infinitesimally small?

I think the problem is the assumption, that the pin can move downward infinitely fast. It cannot, therefore there is a negative work done on the disk as pin slides down the hole while the disk is moving.

guv said:
What if the disk and the pin are perfectly rigid and the diameter of the pin and the diameter of the hole are identically infinitesimally small?
So we would have an infinite pressure and an infinite force acting over an infinitesimal area for an infinitesimal time.

As @haruspex has remarked elsewhere, the correct way to deal with ideal infinities and infinitesimals in physics is by taking the limit of finite quantities as those are increased without bound or decreased toward zero.

If you carry out the limiting process, you will find that every realistic collision with finite holes, finite pins, finite rigidity and finite insertion speeds is either inelastic or leaves some undamped vibrational energy.

guv said:
I think the problem is the assumption, that the pin can move downward infinitely fast. It cannot, therefore there is a negative work done on the disk as pin slides down the hole while the disk is moving.
Not necessarily. There would, in the real world, be some slack between pin and hole. The insertion could be fast enough to use that. But then you get the collision.

This is in the same spirit as a number of other electrical (charging a perfect capacitor at fixed V) and mechanical (putting a mass on a spring into a gravity field) examples: the energy goes into (usually macroscopic) oscillations via an elastic interaction which then vibrationally "leaks" the energy into other (usually microscopic) degrees of freedom while we aren't paying attention.

## 1. Can a spinning cylinder actually be frictionless?

In reality, achieving a completely frictionless environment is practically impossible due to microscopic imperfections and interactions at the molecular level. However, in theoretical physics and certain idealized situations, we can approximate a frictionless condition to simplify the analysis of mechanical systems.

## 2. What are the primary causes of mechanical energy loss in a spinning cylinder?

In a truly frictionless spinning cylinder, the primary causes of mechanical energy loss would be due to non-mechanical factors such as air resistance (if not in a vacuum), internal material deformations (if the material is not perfectly rigid), and electromagnetic interactions (if the cylinder is made of a conductive material and moving through a magnetic field).

## 3. How does air resistance affect a spinning cylinder?

Air resistance, or drag, acts against the motion of the spinning cylinder, causing it to lose kinetic energy over time. This resistance converts some of the mechanical energy into thermal energy, leading to a gradual decrease in the cylinder's rotational speed.

## 4. Can internal material deformations cause energy loss in a spinning cylinder?

Yes, internal material deformations can cause energy loss. If the material of the cylinder is not perfectly rigid, internal stresses and strains can lead to the conversion of mechanical energy into heat within the material, resulting in a loss of mechanical energy.

## 5. How do electromagnetic interactions contribute to energy loss in a spinning cylinder?

If the spinning cylinder is made of a conductive material and is in the presence of a magnetic field, electromagnetic interactions can induce eddy currents within the material. These currents create resistive heating, which dissipates mechanical energy as heat, leading to a reduction in the cylinder's rotational energy.

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