Question #43913

A straight bimetallic strip of length $L$, width $w$, and total thickness $2d$ (composed of two metals of equal thickness $d$, identical Young's modulus, and thermal expansion coefficients $\alpha_1$ and $\alpha_2$ with $\alpha_1 > \alpha_2$) is clamped at one end in a vacuum chamber at absolute zero. One of its large faces is uniformly bombarded by an effusive beam of monatomic ideal gas molecules (mass $m$) escaping from a small hole of an oven. The gas inside the oven has a number density $n$ and the variance of the kinetic energy of an individual molecule in the oven is $V$. The molecules strike the strip normally, stick to it, and transfer all their energy. The strip reaches a steady-state uniform temperature $T_s$ by dissipating heat to the vacuum exclusively via a linearized heat loss mechanism, where the total power lost is $P_{\text{loss}} = \beta L w T_s$ ($\beta$ is a known constant). Assuming the mechanical pressure of the gas beam is negligible compared to the thermal stress, what is the steady-state radius of curvature $R$ of the strip?