Finding the effect of air resistance on period of oscilation

[Muizz]

Homework Statement

I'm currently working on the "cavendish" experiment and wish to use/develop a method separate from the casus we've been provided. Now I've nicely calculated and derived everything I need to know, including all the corrections that have to be made for the mass of the rod, the finite size of the masses, etc.
However: I'm running into a brick wall when it comes to finding a way to bring air resistance into the picture.

For those of you don't know: The cavendish experiment is a torsion balance that is used to measure gravity. The way it works is by suspending a rod with some small masses on either end by a wire with low torsion spring coefficient. These small masses will start to move towards an equilibrium where the force supplied by the torsion in the wire and the force of gravity supplied by two larger masses brought in close proximity of the smaller masses cancel out.

The problem is that these small masses have a certain moment of inertia, so they'll end up oscillating about the position of equilibrium. This oscillation is of course dampened by air resistance. The most common way to go about finding the effect of air resistance is by setting up and solving a differential equation, which I prefer not to do. Because of this, I'm now looking into alternative methods of (numerically) finding the way air resistance affects the period of the oscillation.

My work so far
So far, I've determined that air resistance is irrelevant when determining the equilibrium position of the bar (which is used to determine $G$). Where air resistance does become relevant is in finding the value for $\kappa$ (torsion spring coefficient). Most of you will know that one can describe kappa with $\kappa=2m(\frac{\pi L}{T})^2$ where m is the point mass's mass and L the length of the rod.
Because $T$ gets changed by friction, $\kappa$ can not be accurately determined without finding a way to go from $T_{measured}$ to $T_{frictionless}$. Herein lies my struggle, as I haven't been able to do so.

 

[Dr.D]

What is your definition of kappa? (I'm not familiar with your notation.)

Are you willing to accept an assumed form for the air resistance, such as either linear or quadratic with the velocity?
Why are you so adverse to using a differential equation of motion? This is the long accepted way to describe mechanical vibrations, the class of problems that this falls into.

 

[haruspex]

adverse

Averse.

 

[Muizz]

What is your definition of kappa? (I'm not familiar with your notation.)

Are you willing to accept an assumed form for the air resistance, such as either linear or quadratic with the velocity?
Why are you so adverse to using a differential equation of motion? This is the long accepted way to describe mechanical vibrations, the class of problems that this falls into.

"Where air resistance does become relevant is in finding the value for $\kappa$ (torsion spring coefficient)." ^^

Well, mainly because I just want to explore the science behind it. I can solve the differential equation, or I can try to find a different way which should also work ^^
I'm only willing to accept results that can either be derived or have a strong argument attached to them.

 

[haruspex]

The most common way to go about finding the effect of air resistance is by setting up and solving a differential equation, which I prefer not to do.

For such low speeds, you can take it to be linear with speed. Solutions are available online. http://farside.ph.utexas.edu/teaching/315/Waves/node10.html provides the equation for the affect on the period.
Alternatively, if you are setting up the experiment, can you measure the period in a vacuum chamber?

 

[Muizz]

For such low speeds, you can take it to be linear with speed. Solutions are available online. http://farside.ph.utexas.edu/teaching/315/Waves/node10.html provides the equation for the affect on the period.
Alternatively, if you are setting up the experiment, can you measure the period in a vacuum chamber?

Yeah, I agree.

The source you've given is still using differential equations of motion though. Is that really something you're fundamentally stuck to?
Edit: No, I'm unable to measure the period in a vacuum chamber, the anti-vibration desks just wouldn't be able to be moved.

 

[haruspex]

The source you've given is still using differential equations of motion though.

I thought you were just trying to avoid dealing with such equations yourself. It sounds like you would prefer a direct measurement instead of trusting the calculus. But to use the period of oscillation to infer the torsion constant is also trusting the calculus, so I'm unsure what your objection is.

 

[Muizz]

I thought you were just trying to avoid dealing with such equations yourself. It sounds like you would prefer a direct measurement instead of trusting the calculus. But to use the period of oscillation to infer the torsion constant is also trusting the calculus, so I'm unsure what your objection is.

I have no objection against the calculus. I'm just trying to explore the science: to find out if there are other avenues that would still work. That includes a different way of measuring the torsion constant if such a way exists (provided that you don't directly measure it but use the measurements of the apparatus to determine it)

 

[Muizz]

I ended up doing this:

 

[haruspex]

I don't understand your thinking in equations 15 and 16.
In 15, T is the dampened period.
How is ξ defined?
Working backwards, you seem to be claiming that the frequency ratio is δ² : δ²+4π². I don't see why that would be.

 

[Muizz]

https://en.wikipedia.org/wiki/Logarithmic_decrement
It's mainly based on that article and a message I got from someone:

For small amounts of damping, as in this experiment, it doesn't really matter if the damping force is proportional to the velocity or not. The important quantity is the amount of energy taken out of the system per cycle of oscillation. You can measure that from the rate of decay of the amplitude and convert that to an equivalent linearized damping constant, and estimate the shift in resonant frequency. In practice, you will only measure a significant amplitude decay over several days, and the frequency correction will be most likely be negligible compared with other sources of error. Google "logarithmic decrement" or "log dec" for the way to convert the amplitude decay into a damping parameter.

Another resource:

 

[haruspex]

https://en.wikipedia.org/wiki/Logarithmic_decrement
It's mainly based on that article and a message I got from someone:

Another resource:

If you follow the link through to https://en.m.wikipedia.org/wiki/Damping_ratio, you will see that the derivation depends on the differential equation you were at pains not to depend on.

 

[Muizz]

If you follow the link through to https://en.m.wikipedia.org/wiki/Damping_ratio, you will see that the derivation depends on the differential equation you were at pains not to depend on.

Yeah, I found that too. But I like this because it shows a different angle of the situation, which is what I was attempting to do. Doing mechanics without differential equations is like doing special relativity without lorentz transformations, it's just not happening. But that doesn't mean you can't find another way in the system to broaden your knowledge and find another connection of which you didn't know before ^^

 

[haruspex]

Yeah, I found that too. But I like this because it shows a different angle of the situation, which is what I was attempting to do. Doing mechanics without differential equations is like doing special relativity without lorentz transformations, it's just not happening. But that doesn't mean you can't find another way in the system to broaden your knowledge and find another connection of which you didn't know before ^^

Fair enough, but personally I find solving the differential equation to get the exponential decay gives insight into the process, whereas quoting equations as in post #9 conceals it.

 

[Muizz]

Fair enough, but personally I find solving the differential equation to get the exponential decay gives insight into the process, whereas quoting equations as in post #9 conceals it.

I agree. The derivation gets a little hairy though, and already got 5 pages of those, so I decided to keep it out for now. I can always add it, but that's a lot of work which is time I'll have to pay for somewhere else. Usually I derive everything I use, even little things such as the inertial moment of a sphere or bar.

So in short: I agree, but things are getting a little too long in the assignment :P