Consider a pendulum in the basement consisting of a point mass m fixed to the en

Question

Consider a pendulum in the basement consisting of a point mass m fixed to the end of a massless rod with length 1. The other end is pivoted from the ceiling of the basement and the rod swings freely in the vertical plane. During the earthquake. when the transverse seismic waves arrive, the basement is oscillating horizontally along the x-direction as xp(1) = A sin omega t in the inertial Lab frame. A is the oscillation amplitude. We assume that the seismic wave fequency is much higher than the intrinsic frequency of the pendulum. and the oscillation angle of the pendulum is small. I) Consider the non-inertial frame of the basement Write down the equation of motion for the other end of the pendulum in this frame. 2) What is the oscillation amplitude of the other end of the pendulum in the frame of the basement. (Hint: consider the pendulum as a forced oscillator whose motion is induced by oscillation of the basement.)

Explanation / Answer

Rather than tracking the change of momentum, the precession of the oscillation plane can efficiently be described as a case of parallel transport. For that, it can be demonstrated, by composing the infinitesimal rotations, that the precession rate is proportional to theprojection of the angular velocity of Earth onto the normal direction to Earth, which implies that the trace of the plane of oscillation will undergo parallel transport. After 24 hours, the difference between initial and final orientations of the trace in the Earth frame is? = ?2?sin(?), which corresponds to the value given by the Gauss

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