- #1

Twigg

Science Advisor

Gold Member

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Trying this out for fun, and seeing if people find this stimulating or not. Feedback appreciated! There's only 3 problems, but I hope you'll get a kick out of them. Have fun!

Two damped, unforced springs are weakly coupled and obey the following equations of motion: $$\ddot{x}_a +\gamma \dot{x}_a + \omega_{0,a}^2 x_a + \beta^2 (x_b - x_a) = 0$$ $$\ddot{x}_b +\gamma \dot{x}_b + \omega_{0,b}^2 x_b - \beta^2 (x_b - x_a) = 0$$ You wish to measure the difference between the two springs' natural (undamped) resonant frequencies: ##\Delta = \omega_{0,b} - \omega_{0,a}##. Your measurement will be complicated by the coupling coefficient β and the damping coefficient γ. Design a simple procedure for measuring ##\Delta##.

Assume for simplicity that ##\omega_0 = \frac{1}{2}\left(\omega_{0,b} + \omega_{0,a}\right) = 1\mathrm{kHz}## and ##\gamma = 125\mathrm{s^{-1}}## are known exactly. You are also given that Δ is of order ##2\pi\times 1\mathrm{mHz}## and β is of order ##2\pi\times100\mathrm{mHz}##. The uncertainty in either spring's measured position is determined by the spring's initial conditions by $$\sigma_x = \left(2.6\times10^{-7}\right)\sqrt{x(0)^2 + \frac{\dot{x}(0)^2}{\omega_0^2 - \gamma^2 / 4}}$$

Your answer should include a set of times at which to measure the spring positions ##x_a## and ##x_b##, a formula for ##\Delta## in terms of these measurements, and a standard deviation on the value of ##\Delta##. Optimal answers should have uncertainty ##\sigma_\Delta \approx 2\pi \times 10\mathrm{\mu Hz}## with as little as two position measurements. Numerical and analytical methods are accepted so long as the results are valid!

You and your coworker Bob are studying a chemical reaction ##A + B \leftrightarrow C##. For this study, you vary the temperature of the mixture and record the concentration of species C: $$x = \frac{N_C}{N_A + N_B + N_C}$$ where ##N_A##, ##N_B##, and ##N_C## refer to the total number of each species A, B, and C respectively. For each temperature setting, you record a number (M) of measurements of ##N_A##, ##N_B##, ##N_C## (M measurements of each). Furthermore, you know that ##N_A##, ##N_B##, and ##N_C## are all Poisson distributed. You then calculate an average ##\mathrm{E}[x]## and standard error ##\sigma_{\mathrm{E}[x]}## for each set of measurements. Up to this point, everything makes sense.

Your coworker Bob comes up with a wacky idea. Bob re-defines the concentration (now called ##x'##) within a set of M measurements as follows: $$x'_i = \frac{N_{C,i}}{\mathrm{E}[N_A] + \mathrm{E}[N_B] + N_{C,i}} \; \; \mathrm{for} \; i=1,2,...,M$$ Bob argues that taking expectation values over ##N_A## and ##N_B## in the denominator eliminates extraneous noise. What's more, Bob has a mathematical proof that shows that ##\mathrm{Var}[x']\leq\mathrm{Var}[x]##. You make a bet with Bob: you collect 100 data sets, each consisting of M measurements, and compare the

Consider the following bridge circuit, where the variable resistor sees "pink" noise (aka 1/f noise):

All 4 resistors have identical resistance on average, but the top right resistor fluctuates with a pink spectrum: $$P_{\delta R}(\omega) = \frac{A}{\omega}$$ where ##P_x(\omega)## is the power spectral density (PSD) of the function ##x(t)##. Each resistor also puts out thermal noise (Johnson-Nyquist noise). Find an expression in terms of the measured voltages ##V_A##, ##V_B##, and ##V_S## that is proportional to the fluctuating resistance ##\delta R## but is independent of thermal noise.

Some sample data is attached (filename is “pinkdatafinalfinal.csv”), where each voltage (##V_A##,##V_B##, and ##V_S##) is reported versus time in a CSV format. Extract the constant A as defined above and state your uncertainty on A, given ##R = 1\mathrm{\Omega}##. My solution has uncertainty on the order of ##1 \times 10^{-8} \mathrm{\mathrm{\Omega^2}}##. There are many methods for tackling this problem and some give higher precision than others.

**1. Springey Thingies:**Two damped, unforced springs are weakly coupled and obey the following equations of motion: $$\ddot{x}_a +\gamma \dot{x}_a + \omega_{0,a}^2 x_a + \beta^2 (x_b - x_a) = 0$$ $$\ddot{x}_b +\gamma \dot{x}_b + \omega_{0,b}^2 x_b - \beta^2 (x_b - x_a) = 0$$ You wish to measure the difference between the two springs' natural (undamped) resonant frequencies: ##\Delta = \omega_{0,b} - \omega_{0,a}##. Your measurement will be complicated by the coupling coefficient β and the damping coefficient γ. Design a simple procedure for measuring ##\Delta##.

Assume for simplicity that ##\omega_0 = \frac{1}{2}\left(\omega_{0,b} + \omega_{0,a}\right) = 1\mathrm{kHz}## and ##\gamma = 125\mathrm{s^{-1}}## are known exactly. You are also given that Δ is of order ##2\pi\times 1\mathrm{mHz}## and β is of order ##2\pi\times100\mathrm{mHz}##. The uncertainty in either spring's measured position is determined by the spring's initial conditions by $$\sigma_x = \left(2.6\times10^{-7}\right)\sqrt{x(0)^2 + \frac{\dot{x}(0)^2}{\omega_0^2 - \gamma^2 / 4}}$$

Your answer should include a set of times at which to measure the spring positions ##x_a## and ##x_b##, a formula for ##\Delta## in terms of these measurements, and a standard deviation on the value of ##\Delta##. Optimal answers should have uncertainty ##\sigma_\Delta \approx 2\pi \times 10\mathrm{\mu Hz}## with as little as two position measurements. Numerical and analytical methods are accepted so long as the results are valid!

**2. "Honey, I shrunk the error bars!"**You and your coworker Bob are studying a chemical reaction ##A + B \leftrightarrow C##. For this study, you vary the temperature of the mixture and record the concentration of species C: $$x = \frac{N_C}{N_A + N_B + N_C}$$ where ##N_A##, ##N_B##, and ##N_C## refer to the total number of each species A, B, and C respectively. For each temperature setting, you record a number (M) of measurements of ##N_A##, ##N_B##, ##N_C## (M measurements of each). Furthermore, you know that ##N_A##, ##N_B##, and ##N_C## are all Poisson distributed. You then calculate an average ##\mathrm{E}[x]## and standard error ##\sigma_{\mathrm{E}[x]}## for each set of measurements. Up to this point, everything makes sense.

Your coworker Bob comes up with a wacky idea. Bob re-defines the concentration (now called ##x'##) within a set of M measurements as follows: $$x'_i = \frac{N_{C,i}}{\mathrm{E}[N_A] + \mathrm{E}[N_B] + N_{C,i}} \; \; \mathrm{for} \; i=1,2,...,M$$ Bob argues that taking expectation values over ##N_A## and ##N_B## in the denominator eliminates extraneous noise. What's more, Bob has a mathematical proof that shows that ##\mathrm{Var}[x']\leq\mathrm{Var}[x]##. You make a bet with Bob: you collect 100 data sets, each consisting of M measurements, and compare the

*estimated*standard error on the mean ##\sqrt{\frac{1}{M}\mathrm{Var}[x']}## (aka the "error bars" on the mean of each set of M measurements) with the*observed*standard deviation on the means of each of the 100 sets of measurements ##\sigma_{\mathrm{E}[x']}##. The data shows that ##\sigma_{\mathrm{E}[x']} > \sqrt{\frac{1}{M}\mathrm{Var}[x']}##, and more specifically that $${\sigma_{\mathrm{E}[x']}}=\sigma_{\mathrm{E}[x]}$$ This last result can be interpreted to mean there is no free lunch for Bob. Reproduce Bob's proof that ##\mathrm{Var}[x']\leq\mathrm{Var}[x]## and prove the "no free lunch" result ##\sigma_{\mathrm{E}[x']}=\sigma_{\mathrm{E}[x]}##.**3. Pink, pink, you stink!**Consider the following bridge circuit, where the variable resistor sees "pink" noise (aka 1/f noise):

All 4 resistors have identical resistance on average, but the top right resistor fluctuates with a pink spectrum: $$P_{\delta R}(\omega) = \frac{A}{\omega}$$ where ##P_x(\omega)## is the power spectral density (PSD) of the function ##x(t)##. Each resistor also puts out thermal noise (Johnson-Nyquist noise). Find an expression in terms of the measured voltages ##V_A##, ##V_B##, and ##V_S## that is proportional to the fluctuating resistance ##\delta R## but is independent of thermal noise.

Some sample data is attached (filename is “pinkdatafinalfinal.csv”), where each voltage (##V_A##,##V_B##, and ##V_S##) is reported versus time in a CSV format. Extract the constant A as defined above and state your uncertainty on A, given ##R = 1\mathrm{\Omega}##. My solution has uncertainty on the order of ##1 \times 10^{-8} \mathrm{\mathrm{\Omega^2}}##. There are many methods for tackling this problem and some give higher precision than others.