Finding the effect of air resistance on period of oscilation

In summary: I am with the methods that I'm looking into.In summary, the Cavendish experiment is a torsion balance used to measure gravity. The problem is that the small masses have a certain moment of inertia, so they'll oscillate about the equilibrium position. However, air resistance becomes relevant in finding the value for $\kappa$ (torsion spring coefficient). Air resistance is not an issue in determining the equilibrium position, but is in finding the value for $\kappa$ when trying to find the torsion constant.
  • #1
Muizz
11
0

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.
 
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  • #2
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.
 
  • #3
Dr.D said:
adverse
Averse.
 
  • #4
Dr.D said:
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.
 
  • #5
Muizz said:
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?
 
  • #6
haruspex said:
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.
 
  • #7
Muizz said:
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.
 
  • #8
haruspex said:
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)
 
  • #9
I ended up doing this:
4b952c9384.png
 
  • #10
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.
 
  • #11
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:
 
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  • #13
haruspex said:
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 ^^
 
  • #14
Muizz said:
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.
 
  • #15
haruspex said:
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
 

FAQ: Finding the effect of air resistance on period of oscilation

1. What is air resistance?

Air resistance, also known as drag, is the force that opposes the motion of an object through air. It is caused by the interaction between the object and the air molecules surrounding it.

2. How does air resistance affect the period of oscillation?

Air resistance can affect the period of oscillation by slowing down the motion of an object, resulting in a longer period. This is because the force of air resistance acts in the opposite direction of the object's motion, causing it to lose energy and decrease its amplitude over time.

3. Can air resistance be ignored in the calculation of period of oscillation?

No, air resistance cannot be ignored in the calculation of period of oscillation. While it may be negligible for some objects, it can have a significant impact on the period for others, especially those with larger surface areas or slower velocities.

4. How can the effect of air resistance on period of oscillation be measured?

The effect of air resistance on period of oscillation can be measured by conducting experiments with varying air resistance, such as using different objects with different surface areas or adding materials like feathers or paper to alter the drag force.

5. How can the data collected from the experiment be analyzed to determine the relationship between air resistance and period of oscillation?

The data collected from the experiment can be analyzed by plotting the period of oscillation against the air resistance and observing the trend in the data. This can also be done mathematically by using equations that relate air resistance and period of oscillation, such as the drag force equation or the equation for damped oscillations.

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