Coursework help - Stokes Law equation + graphs

[astro111]

Homework Statement

Hi there.
For my A2 physics coursework I have been doing an experiment into stokes law, in which I dropped ball bearings of various diameters into a tube filled with a liquid, worked out their terminal velocities and then used Stokes Law to calculate the viscosity of the liquid. My results have been consistent and in line with my expectations. I then did another experiment, the reverse in a way, in which I used my calculated values for the viscosity of the liquids to find the diameter of ball bearing before measuring it to ensure I was gaining accurate results.

However now I am at the stage of writing up my experiment report and I am a little unsure of what graphs to plot. I've been reading and it sounds as though I should plot a graph of radius² over velocity, however I am unsure as to what this would show. For example, would the gradient show viscosity?

Homework Equations

The equations I've been using are μ=(d²g(Pρ-Ps))/18ν, where (Pρ-Ps) is the density of the ball bearing minus the density of the fluid, d is the diameter of the ball bearing, g is equal to 9.81 and v is the terminal velocity of the ball bearing. I believe this equation can also be written as μ=(2/9)*((Pρ-Ps)/v)*gr², where r is the radius of the ball bearing.

The Attempt at a Solution

I have tried to rearrange the second equation in order to get r²/v, as the graph would be I think? leaving me with 9μ/(2(Pρ-Ps)g) = r²/v, which I think would mean that r² ∝ 1/v?
However I am confused as to what my graph would show (i.e what the gradient would be equal to etc) and whether I have rearranged the equation correctly.

As a side note, I am unable to ask my teacher presently as I am currently off school ill and ideally I would like to have finished my report when I go back in.
Any help would be much appreciated, thank you!
Alex

 

[BvU]

Hello Alex, welcome to PF :) Hope you get well soon.

It's good practice to plot experimental results by using the y-axis for what you measure and the x-axis for what you vary. I suppose you measured the time it took the balls to fall from one level to a lower one. And you varied the diameter and the material of the balls.

Another good practice is to try to plot results in such a way that a linear relationship is achieved (at least according to the prediction of the theory). Sometimes that means you need to square, to take a square root or a logarithm.

If you look at your expression, you see you can write $\displaystyle v = {d^2 \; \Delta\rho g \over 18\mu}$ which looks like $v = C\; d^2$, so you expect a straight line if you plot something that is proportional to v as a function of something like d².

Here, v = distance / time means that 1/time is a good one to plot vertically. The distance $\Delta y$ is the same in all cases (right?), so you divide that out in a later stage (*).

For the horizontal axis, asw we saw, diameter d doesn't yield a linear relationship in the theory, but d² does.

With three materials you hope to get get three more or less straight lines, each with slope $\displaystyle\Delta\rho g \over 18\mu \Delta y$. Draw the best possible straight lines to determine best values for these slopes.

As a next step, you can claim you measured three slopes as a function of $\Delta \rho$, so you can plot these slopes vertically and $\Delta \rho$ horizontally. That line is expected to have a slope $\displaystyle g \over 18\mu \Delta y$. Again, draw the best possible straight line to determine its values. Finally, calculate $\mu$.

(*) because distance is the same for all measurements, the error in the slopes doesn't get smaller if you use more different ball sizes to get a better line. (The error in the distance is 'systematic' and not statistical)

 

[Chestermiller]

You have the right idea. Consider plotting v/r² vs Δρ/μ (so that the left hand side is all geometric and kinematic, and the right hand side is all physical properties). The data should all lie on a straight line.

Chet

 

[astro111]

Thank you to both of you, you've helped me greatly!

 

[HalfLostAndSearching]

Hi Alex,

It sounds like you have conducted a thorough and well-designed experiment for your A2 physics coursework. Stokes Law is a fundamental equation in fluid mechanics and your application of it to calculate the viscosity of a fluid is a great way to demonstrate your understanding of the concept.

In terms of plotting graphs, there are a few options you could consider. One option is to plot a graph of terminal velocity (v) against ball bearing radius (r). This graph would show the relationship between these two variables and could help you to determine whether there is a linear relationship between them. If there is, then you could use the equation v=2/9 * (d^2g(Pρ-Ps))/μ to calculate the viscosity of the fluid.

Another option is to plot a graph of terminal velocity (v) against the inverse of ball bearing radius (1/r). This graph would also show the relationship between these two variables and could help you determine if there is a linear relationship. If there is, then you could use the equation v=2/9 * (d^2g(Pρ-Ps))/μ to calculate the viscosity of the fluid, with the added benefit of having a straight line on your graph.

As for your proposed graph of r^2/v, it is not a commonly used graph in this context and may not provide the most informative results. However, if you are interested in exploring this relationship, you could plot a graph of r^2 against 1/v and see if there is a linear relationship. The gradient of this graph would then be equal to 2/9 * (d^2g(Pρ-Ps))/μ.

I hope this helps and good luck with your report! If you have any further questions, feel free to ask your teacher or seek help from a knowledgeable peer. As a scientist, it's important to seek guidance and collaborate with others in order to improve your understanding and results. Keep up the good work!