The Rolling Observatory · AI Generator #GEN-4DCCCBBD

Physics · mechanics

Problem

An astronomical observatory dome consists of a uniform hemispherical shell of mass

M = 2400\;\text{kg}

and radius

R = 8.0\;\text{m}

. The dome can rotate about a vertical axis through its center. To open the observation slit, the dome must be rotated by

90°

in exactly

T = 45\;\text{s}

to avoid tracking errors with the telescope. The rotation motor applies a constant torque for the first half of the rotation time, then applies an equal and opposite torque for the second half to bring the dome to rest at the correct position. (a) Find the moment of inertia of the hemispherical dome about the vertical rotation axis. (b) What constant torque magnitude

\tau

must the motor apply during each phase of the motion? Note: The moment of inertia of a uniform solid sphere of mass

m

and radius

r

about a diameter is

I = \frac{2}{5}mr^2

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