A cylinder of side L meters lies one quarter submerged and upright in a certain fluid

Chapter 8, Problem 17

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A cylinder of side L meters lies one quarter submerged and upright in a certain fluid. At t = 0, the cylinder is pushed down a distance of L/2 meters and released from rest. Show that the resulting motion is simple harmonic, and determine the circular frequency and period of the motion.A simple pendulum consists of a mass, m kilograms, attachedto the end of a light rod of length L meters, whose other endis fixed. (See Figure 8.5.11.)If we let radians denote the angle the rod is displacedfrom the vertical at time t, then the component of the velocityin the direction of motion is v = L ddt , so thatmg mg sin uu LFigure 8.5.11: The simple pendulumthe component of the acceleration in the direction of motionis L d2dt2 . Further, the tangential component of the forceis FT = mg sin , so that, from Newtons second law, theequation of motion of the pendulum ism Ld2dt2 = mg sin .That is,d2dt2 + gLsin = 0. (8.5.27)This is a nonlinear differential equation. However, if we recallthe Maclaurin expansion for sin , namely,sin = 13! 3 +15! 5 ,it follows that for small oscillations, we can approximatesin by . Then Equation (8.5.27) can be replaced to reasonableaccuracy by the simple linear differential equationd2dt2 + gL = 0. (8.5.28)