Lecture Notes

$\newcommand{\vect}[1]{{\mathbf{#1}} }$ $\renewcommand{\vec}[1]{{\overrightarrow{#1}} }$ $\newcommand{\norm}[1]{{\left|\left|#1\right|\right|} }$ $\newcommand{\D}{{{\mathcal D}} }$ $\newcommand{\C}{{{\mathcal C}} }$ $\newcommand{\E}{{{\mathcal E}} }$ $\newcommand{\R}{{\mathbb{R}} }$ $\newcommand{\Rtwo}{{\mathbb{R}^2} }$ $\newcommand{\Rthree}{{\mathbb{R}^3} }$ $\newcommand{\V}{{\mathbb{V}} }$ $\newcommand{\T}{{\mathbb{T}} }$ $\newcommand{\nhat}{{\vect{\hat{n}}} }$ $\newcommand{\vi}{{\vect{i}} }$ $\newcommand{\vj}{{\vect{j}} }$ $\newcommand{\vk}{{\vect{k}} }$ $\newcommand{\f}{{\vect{f}} }$ $\newcommand{\g}{{\vect{g}} }$ $\newcommand{\vz}{{\vect{0}} }$ $\newcommand{\Du}{{D_{\vect{u}}} }$ $\DeclareMathOperator{\sgn}{sgn}$ $\DeclareMathOperator{\dist}{dist}$ $\DeclareMathOperator{\area}{Area}$ $\DeclareMathOperator{\vol}{Volume}$ $\newcommand{\xcm}{{x_\mathrm{CM}} }$ $\newcommand{\ycm}{{y_\mathrm{CM}} }$ $\DeclareMathOperator{\grad}{\bf grad}$ $\DeclareMathOperator{\ave}{ave}$ $\DeclareMathOperator{\std}{std}$

Notes on Experiment 9: Maxwell's Wheel

Sujeet Akula

Moment of Inertia

In the first investigation, you have to measure the mass and dimensions of the wheel and compute the moments of inertia of the disk and axle, including the uncertainties. To simplify the computation drastically, we will use the quoted density of aluminum, \(\rho=2700\,\mathrm{kg}/\mathrm{m}^3\). Remember that $r$ is the radius of the axle, and $\ell$ is its length. Meanwhile, $R$ is radius of the disk and $d$ is its depth. We do not need the uncertainty in volumes or masses. We only want the uncertainty in the moments of inertia. We can start by calculating the volumes and masses. Note: I consider the center of the wheel as being a part of the axle. \begin{align} V_a &= \pi r^2 \ell \\ m_a &= \rho \pi r^2 \ell \\ V_d &= \pi\left(R^2 - r^2\right)d \\ m_d &= \rho \pi\left(R^2 - r^2\right)d \end{align} Now let's calculate the moments of inertia. \begin{align} I_a &= \frac{1}{2}\pi \rho r^4 \ell \\ \delta(I_a) &= I_a \sqrt{ \left(\frac{\delta \ell}{\ell}\right)^2 + \left(4 \frac{\delta r}{r}\right)^2} \\ I_d &= \frac{1}{2}\pi \rho d \left( R^4 - r^4 \right) \end{align} \begin{equation} \delta(I_d) = \sqrt{ \left(I_d \frac{\delta d}{d}\right)^2 + \left(2\pi \rho d\right)^2 \left[ \left(R^3\delta R\right)^2 + \left(r^3\delta r\right)^2 \right] } \end{equation} Finally, the total moment of inertia $I$ is obtained by the simple sum \begin{equation} I = I_d + I_a , \end{equation} and the uncertainty is given by \begin{equation} \delta I = \sqrt{\left(\delta I_d\right)^2 + \left(\delta I_a\right)^2} . \end{equation}

Dynamics Analysis

You have taken data on the time it takes for the wheel to fully unwind given $n=8,6,4$ initial windings, and you have taken several measurements for each case. For a given choice of $n$, you will have a set of time measurements \(\left\{t_i\right\}\), and the choice of $n$. First you will calculate the vertical distance that the wheel's center of mass traveled (ignoring the string's radius): \begin{align} y_n &= 2\pi rn \\ \Rightarrow \delta(y_n) &= 2\pi n\left(\delta r\right) . \end{align} Now, we need to deal with the timing data. Start with the average and standard deviation: \begin{equation} t = \ave\left\{t_i\right\} \text{ and } \delta t = \sqrt{\left(\std\left\{t_i\right\}\right)^2 + \Delta_t^2} , \end{equation} where $\Delta_t$ is the systematic uncertainty, and we can estimate that \( \Delta_t = 0.1\,\mathrm{s} \). Since we wish to make a plot of $y$ vs $t^2$, you must next compute $t^2$, and use \(\delta\left(t^2\right) = 2t\delta t\). You can now make a plot of $y$ vs $t^2$ using $\delta y$ as the vertical error bars and $\delta\left(t^2\right)$ as the horizontal error bars. Use the website to obtain the slope and slope-uncertainty, $\mu$ and $\delta\mu$. The acceleration $a$ is very directly related to the slope: \(a = 2\mu\), and \(\delta a = 2\delta\mu\). We can now obtain the total moment of inertia according to the equations of motion. Using \begin{align} a &= \cfrac{g}{1 + \cfrac{I}{Mr^2}} \\ \Rightarrow I &= Mr^2\left(\frac{g}{a} - 1\right) . \end{align} The uncertainty in this calculation of the total moment of inertia is \begin{equation} \delta I = \sqrt{ I^2\left[ \left(\frac{\delta M}{M}\right)^2 + \left(2\frac{\delta r}{r}\right)^2 \right] + \left(\frac{Mr^2g}{a^2}\right)^2\left(\delta a\right)^2 } \end{equation} We can now compare this result to what we measured in Investigation 1.

---