\sect{Mass on a spring}

A balance for measuring weight consists of a sensitive spring
which hangs from a fixed point. The spring constant is is ${K}$,
i.e. the force opposing a length change ${x}$ is  ${-Kx}$. The
balance is at a temperature ${T}$ and gravity acceleration is
${g}$. A small mass m hangs at the end of the spring.

\begin{itemize}

\item [(a)]
Write the partition function and evaluate the average ${\langle
x\rangle}$ and the fluctuation ${\langle\left(x - \langle
x\rangle\right)^{2}\rangle}$. What is the minimal ${m}$ which can be
meaningfully measured?

\item [(b)]
Write a Langevin equation for ${x\left(t\right)}$ with friction
$\gamma$ and a random force ${A\left(t\right)}$. Assuming
${\langle A\left(t\right)A\left(0\right)\rangle = C
\delta\left(t\right)}$ evaluate the spectrum ${ |\tilde{x}(\omega)
|^{2}}$ where ${\tilde{x}\left(\omega\right )}$  is the Fourier
transform of ${\tilde{x}  = x - \langle x\rangle}$. Evaluate
${\langle \tilde{x} ^{2}\left(t\right)\rangle}$ and from (a) find
the coefficient ${C}$. [You may use $\int d\omega
/[(\omega^2-K/m)^2+\gamma^2\omega^2]=m\pi/\gamma K$.]

\item [(c)]
Consider response to a force that couples to the velocity, i.e.
the Langevin equation acquires a term $-\partial F/\partial t$.
Evaluate the dissipation function ${Im\alpha_v \left(\omega\right
)}$ and the power spectrum of the velocity $\phi_v(\omega)$.

\end{itemize}

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