\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 ${K}$. The balance
is at temperature ${T}$ and gravity acceleration is ${g}$ in the ${x}$
direction. A small mass ${\mass}$ hangs at the end of the spring. There
is an option to apply an external force $F(t)$, to which $x$ is
conjugate or apply an external vector potential $A(t)$.
\begin{itemize}
\item [{{(a)}}] Find the partition function $Z$.
\item [{{(b)}}] Find $\langle x \rangle$ and $\langle x^{2}\rangle$ and $\mbox{Var}(x)$.
\item [{{(c)}}] Write a Langevin equation for $x(t)$, with friction $\gamma$, and a random force $f(t)$.
\item [{{(d)}}] Assuming $\langle f(t)f(0)\rangle=C\delta(t)$, find $\mbox{Var}(x)$,
and deduce what is $C$ by comparing with the canonical result.
\item [{{(e)}}]
Assuming $x$ is measured in the lab by averaging over time period $t_0$,
what is the minimal mass that can be meaningfully measured?
\item [{{(f)}}] Describe the external force $F(t)$ by a scalar potential and demonstrate FDT.
\item [{{(g)}}] Describe the external force $F(t)$ by a vector potential and demonstrate FDT.
\end{itemize}
\Dn
Note: $\int \frac{d\omega}{(\omega^2-\omega_0^2)^2+\gamma^2\omega^2}=\frac{\pi}{\gamma \omega_0^2}$.
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