Equation to model dripping faucet or slowing roulette wheel?

[azpat]

I've noticed when I take a shower that after I turn off the water, the water that's left in the pipe runs out in a regular pattern. At first it streams out continuously, and then as the amount of water decreases, the stream turns to a trickle, and then a drip, and the drip is the most interesting part. At first the drip is several times a second,

drip,drip,drip,drip

then it slows to about once a second
drip..drip..drip..

and then it slows to where there are several seconds between drips so that the entire pattern is

dripdripdripdrip..drip..drip...drip...drip...drip...drip.....drip
you get the idea.

I've noticed a similar pattern when watching wheel of fortune and seeing the wheel spin around, the ticking sound makes the same pattern as the wheel slows.

The question is whether there is an equation that describes this pattern. In trying to develop one, I've come to a few other questions specific to my attempt.

So, I'll leave the first question as "is there an equation that models the dripping of water (not from a leaking pipe) but from a pipe with a finite amount of water, or similarly, is there an equation that models the spinning of a slowing roulette wheel such as the wheel of fortune wheel or the price is right wheel."

part of me thinks that this is simply a constant deceleration problem and the (1/2)at^2 nature is messing with my head, but I'm not familiar enough with physics to work out whether this hunch has merit.

For my faucet, I've approximated the time between drips as being 3/2 times the previous time between drips. For example, if the first drip and the second drip were a second apart, the second and the third drip would be a second and a half apart. So, the time between drips 1 and 2 is 1, and between 2 and 3 is 3/2*1 and between 3 and 4 is 3/2*3/2 and between 4 and 5 is 3/2(3/2*3/2) = (3/2)^3 which is the geometric series for (3/2), which diverges for values greater than 1.

I know this isn't an exact model, because the bathtub drips come to be approximately 6 seconds apart steadily, but by this point I was more interested in the question of finding an equation for the drips if the rate of dripping was given by the above geometric series.

So I said that the rate of dripping versus the rate of drops (time as a function of drops) was equal to (3/2)^n, or
dt/dn = (3/2)^n, and I tried to solve this differential equation, which I presumed for a given value of n, would be equal to the finite value of the geometric series for r!=1.
Which is ( 1 - r^n ) / (1 - r) according to
http://www.cartage.org.lb/en/themes/Sciences/Mathematics/calculus/seriesexpan/GeometricSeries.htm .

Unfortunately, I couldn't get this to work out. So my second questions is can anyone get this model (albeit not the correct model for the motivating problem) to work out so that the function, total time t(n) for the nth drop is equal to (1-r^n)/(1-r).

Thanks. I hope these problems are as appealing to some of you as they have been to me.

 

[AndyW]

Thank you for bringing up this interesting observation about the regular pattern in the dripping of water from a pipe and the spinning of a slowing roulette wheel. I am always fascinated by these types of natural phenomena and enjoy exploring the underlying physics behind them.

Firstly, to answer your initial question, yes, there is indeed an equation that can model the dripping of water from a pipe or the spinning of a slowing roulette wheel. This equation is known as the exponential decay equation, which is commonly used to describe the decrease in a quantity over time. In this case, the quantity is the amount of water in the pipe or the speed of the spinning wheel, and time is the time elapsed since the initial event (turning off the water or spinning the wheel).

The exponential decay equation is given by:

N(t) = N0 * e^(-λt)

Where N(t) is the quantity at time t, N0 is the initial quantity, and λ is the decay constant. This equation is commonly used in various fields such as radioactive decay, population growth, and chemical reactions.

Now, to apply this equation to your specific situations, we can consider the dripping of water from a pipe. In this case, the quantity N(t) represents the amount of water left in the pipe at time t, N0 represents the initial amount of water, and λ represents the rate at which the water is dripping. The rate of dripping can be determined by counting the number of drips in a given time interval, for example, one second. This is known as the frequency of dripping, and it is given by:

f = 1/t

Where t is the time between two consecutive drips. Therefore, we can rewrite the exponential decay equation as:

N(t) = N0 * e^(-ft)

This equation describes the decrease in the amount of water in the pipe over time, and it follows a similar pattern as the one you described in your forum post. As the amount of water decreases, the frequency of dripping also decreases, resulting in a slower and more spaced out pattern.

Similarly, we can apply the exponential decay equation to the spinning of a slowing roulette wheel. In this case, the quantity N(t) represents the speed of the wheel at time t, N0 represents the initial speed, and λ represents the deceleration rate. The rate of deceleration can be determined by measuring the change in speed over a given time