The pendulum equation A pendulum consisting of a bob of

Chapter , Problem 47

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The pendulum equation A pendulum consisting of a bob of mass m swinging on a massless rod of length / can be modeled as an oscillator (see figure). Let u1t2 be the angular displacement of the pendulum t seconds after it is released (measured in radians). Assuming that the only force acting on the bob is the gravitational force, we write Newtons second law in the direction of motion (perpendicular to the rod). Notice that the distance along the arc of the swing is s1t2 = /u1t2, so the velocity of the bob is s_1t2 = /u_1t2 and the acceleration is s_1t2 = /u_1t2. _ _ Circular arc mg a. Considering only the component of the force in the direction of motion, explain why Newtons second law is mlu_1t2 = -mg sin u1t2, where g = 9.8 m>s2 is the acceleration due to gravity. b. Write this equation as u_ + v0 2 sin u = 0, where v0 2 = g / . c. Notice that this equation is nonlinear. It can be linearized by assuming that the angular displacements are small 1_u_ V 12 and using the approximation sin u _ u. Show that the resulting linear pendulum equation is u_ + v0 2 u = 0. d. Express the period of the pendulum in terms of g and /. If the length of the pendulum is increased by a factor of 2, by what factor does the period change?