Consider a homogeneous hemispherical shell (hollow bowl) with mass and radius , as shown in the figure on the left.

(a) In the configuration in the figure on the left, obtain the position of the center of mass of the hemispherical shell taking the origin at the point .

(b) Get the moments of inertia of the hemispherical shell for rotations around axes parallel to the direction that pass through the origin , through the center of mass and by the pole . Pay attention when applying the parallel axes theorem.

(c) As illustrated in the figure on the right, consider a physical pendulum composed of a hemispherical shell with a flat circular edge facing downwards and set to oscillate around a fixed point , located at a distance above its pole . The rigid rod that connect the points and has negligible mass and remains always perpendicular to the spherical shell at its pole . From the conservation of mechanical energy or the resulting torque (in relation to ) acting on the system, obtain the equation differential satisfied by the angle which describes the position of the pendulum's center of mass in the presence of the constant gravitational field . Note that the movement of the center of mass occurs along a single plane, perpendicular to the direction. What is the frequency of small oscillations of the system around the vertical equilibrium position ?

12 Ay bi , M R = (X,Y,Z) 0 = (0, 0, 0) G 1.14.1. -2 G= CM = R P = (0, 0, -a 0 = arccos(R) w=0 12 Ay bi , M R = (X,Y,Z) 0 = (0, 0, 0) G 1.14.1. -2 G= CM = R P = (0, 0, -a 0 = arccos(R) w=0