\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)}}] What is the minimal mass that can be meaningfully measured? \item [{{(d)}}] Write a Langevin equation for $x(t)$, with friction $\gamma$, and a random force $f(t)$. \item [{{(e)}}] 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 [{{(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: $\int \frac{d\omega}{(\omega^2-\omega_0^2)^2+\gamma^2\omega^2}=\frac{\pi}{\gamma \omega_0^2}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%