Dropping the Ball: paper experiments in Dynamics and Inference (part 2)

2013 April 18

Attached data: drops and balls which contain the timing data for 30 drops, and the mass and size data for 10 balls.

From my previous article proposing a simple experiment of dropping a ball that will experience drag and then doing inference about the unknown quantities based on a mechanical model that will induce a likelihood: Here’s an update on the model and experimental protocol. I ran the experiment today and the data is posted as text files above.

We started with Newton’s Laws for a falling sphere with drag:

\[ma = -mg + c(\rho v r / \mu) \rho A(r) \frac{v^2}{2}\]

\[v = \frac{dh}{dt}\]

and the initial conditions $$v(0)=0, h(0)=h_0+\epsilon_h$$ where $$h_0$$ is the measured initial height and $$\epsilon_h$$ accounts for the difference between measured and actual.

It makes sense to nondimensionalize every physical equation. We will define new variables: $$a’=a/g_n, v’=v/(h_0/\sqrt{2h_0/g_n}), h’ = h/h_0, t’=t/\sqrt{2h_0/g_n}$$ where $$g_n$$ is the nominal “standard” value for g, defined as $$g_n=9.80665 {\rm m/s^2}$$. Substituting for the variables $$a,v,h,t$$ and solving for the new equations in $$a’,v’,h’,t’$$ gives us:

\[a’ = -\frac{g}{g_n} + c\left (\frac{v’r\rho \sqrt{g_nh_0/2}}{\mu}\right ) \frac{h_0A(r)\rho}{4m} {v’}^2\]

\[v’ = \frac{dh’}{dt’}\]

These are dimensionless equations (choose a system of units, such as SI, and calculate the units of any term, and you will see they all cancel) and it emphasizes that we don’t need to know exactly each constant. Ultimately we need 3 dimensionless ratios, some of which are exact and for a frequentist are not random variables at all (they are fixed in every experiment): $$g/g_n, \rho r \sqrt{g_n h_0/2}/\mu, \rho Ah_0/(4m)$$. We are modeling $$A$$ as $$\pi r^2$$ and we have that $$v’$$ is a dynamic variable that our deterministic model will predict. The uncertainty comes in the variables $$r, m, g, \rho, \mu, \epsilon_h$$ which is 6 unknown variables, however we only need to infer 4 quantities thanks to our nondimensionalization $$g, r/\nu, \rho/m, \epsilon_h/h_0$$ where $$\nu = \mu/\rho$$ is called the “kinematic viscosity”.

Next, we need to consider the differences between our predictions from the ODE model and the measured fall times. From the ODE model we have that the fall time is $$t:h(t)=0$$ and given values for the constants, we can estimate this essentially exactly (ie. 8 or 10 decimal places, which is enormously precise compared to other aspects of our problem). However our measured time will definitely be different from this predicted time. The dimensionless version of the measured time $$t_m’ = t_m/\sqrt{2h_0/g_n}$$ will differ from the predicted amount by a “reaction time” error $$t_r’$$. If an experimenter flubs the measurement entirely (forgets to press start, misses the end time and lets the timer run for several extra seconds, etc), they will redo the experiment rather than recording it, so they will only accept values that are reasonably tightly clustered around something “true” and hence a normal random variable is a reasonable model for $$t_m$$. Alternatively if they might record several minutes of fall time due to letting the stopwatch go on accidentally, we might prefer to use a complicated mixture model or something with fat tails, like a t distribution. However even if we use the normal model, the bias and variance in this procedure are unknown. Having approximated the experiment a few times we know that the fall time will be on the order of 0.5 seconds, and that the fastest we can double-click the stopwatch is on the order of 0.1 seconds, so we might expect that the bias is some number whose size is on the order of 0.1 seconds and might be either positive or negative, and the variance in the measurement error should be on the order of 0.2^2 or less. For a Bayesian, we can define hyperparameters $$\mu$$ and $$\sigma$$, and a reasonable prior for them is something like $$N(0,0.1^2)$$ and $$\Gamma(2,0.1)$$ where here the 0.1 in the Gamma distribution is the scale parameter of the Gamma, (sometimes people parameterize in terms of a rate = 1/scale).

Now for more specifics on the data collection: create two sheets of paper, one with headings:

Ball Mass D1 D2 D3

Where you record info about the balls, namely the ball number, the mass in g based on weighing on a scale, and three approximate diameters (lay it on top of your tape measure and read off something that seems reasonable, record it in cm).  On the other sheet of paper we have:

Ball h0 t

Where we record which ball we dropped, the nominal height we tried to drop it from (generated randomly in a range that you can easily read without standing on something or stretching too much). Record these values in cm and seconds. For analysis we’ll convert distances to meters to get consistent units.

At the beginning of this post I have links to data files that I generated thanks to Rama and Evangelia for helping me collect the data. BTW the temperature in the room was about 72 C 72 F plus or minus a degree or so, that’s useful for looking up the kinematic viscosity of air to get a prior for that parameter.

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