Consider a hollow sphere (I = 2/3 M R2 when rotated about its center) of radius 0.49 m.  The sphere is pinned at its north pole (this is not its center) at allowed to undergo small oscillations about this point.  Calculate the period of the oscillation, is s, using g = 10 m/s2

Sine and cosine have period 2π.

A uniform annular ring of mass m and inner and outer radil a and b, respectively, is pivoted around an axis perpendicular to the plane of the ng at point P (see figurm bet). Detemine ts penod sf cscillation. (Use the following as necessary: m, a, b, and g for acceleration due to gravity) T-

Consider the solution tothe harmonic oscillator given above by x(t)=Ccos(wt−v) Prove tha tx(t0)=x(t0+2piw) In other words, the solution has the same value at time:t0 and at time:t0+2piw regardless of what value we have for ?0. The value 2piw is then the period T of the harmonic oscillator.

(a) What does the quadrupole formula (P) = = = (Qij Q³ ³) compute? Reason the answer. (b) A point mass m undergoes a harmonic motion along the z-axis with frequency w and amplitude L, x(t) = y(t) = 0, z(t) = L cos(wt). Show that the only non-vanishing component of the quadrupole moment tensor is = Im L² cos² (wt). (c) Use the quadrupole formula to compute the power radiated by the emission of gravitational waves. (Hint: recall that (cos(t)) = (sin(t)) = 0 and (cos² (t)) = (sin² (t)) = ½½ for a given frequency 2.)