Question #43738

A uniform cylindrical telescope of mass $M = 10 \text{ kg}$ and length $L = 0.6 \text{ m}$ is suspended vertically from a frictionless pivot at its upper end in a uniform gravitational field $g = 10 \text{ m/s}^2$. The telescope has an objective lens of diameter $D = 6.1 \text{ cm}$ and observes light of wavelength $\lambda = 500 \text{ nm}$. A horizontal impulse $J = 1.0 \times 10^{-4} \text{ N s}$ is applied to the lower end of the telescope, causing it to oscillate as a physical pendulum. A long-exposure photograph of a distant binary star system is taken. Due to the oscillation, the time-averaged image of each star is broadened into a line segment. Assume the effective angular width of a single star's image is $W = \Delta \theta_{\text{motion}} + \Delta \theta_R$, where $\Delta \theta_{\text{motion}}$ is the peak-to-peak angular sweep of the telescope's optical axis, and $\Delta \theta_R = 1.22 \frac{\lambda}{D}$ is the static Rayleigh resolution limit. The binary stars are considered just resolved when their actual angular separation $\Delta \phi$ equals $W$. Calculate the minimum angular separation $\Delta \phi$ required to resolve the binary star system.