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I. The same three point masses of mass m are connected with massless rigid beam as shown in the figure. In the figure, all the three point masses are on the ()-yz plane. Set the Cartesian coordinate system (-ryz so that the z axis is vertically upward, the y axis is horizontally right direction, and the r axis is perpendicular to the paper surface (with the front side of the paper being positive). Consider the small vibration around a pivot point of this object. The gravitational acceleration is g as indicated by an arrow in the figure. Answer the following questions. (25 points) (1) Find the center of the gravity GIGYG, 2G) of the object. (2) Determine the potential energy U and the kinetic energy T when the object rotates & radian in yz plane around the r axis. Let @ be a generalized coordinate, and the reference plane of the potential energy be m 9 2 = 0 m m (3) Derive an equation of motion of the object for the angle 0, and linearize the equation when the vibration is very small (0 <1,0 <1). (4) Find a natural period T of this small vibration around the x axis. (5) Considering the case of the same small vibration but around the y axis not around the r axis, derive a linearized equation of motion for a rotation angle o around the y axis. And also find a natural period T2 of this vibration.