The pendulum seems to demonstrate simple harmonic motion. In previous sections you've worked your way through "Simple Harmonic Motion" and "The Mass-Spring System." Can you answer this question? What is it about the restoring force that defines a harmonic motion? ... Inccorect. Gravity always pulls the mass toward the center of the swing. ... Simple harmonic motion occurs when the restoring force is always directed toward the equilibrium position and is proportional to the distance from the equilibrium position. This is expressed in Hooke's Law, F equals minus k x. Our next step is to prove that this swinging pendulum exhibits harmonic motion. Consider the force that returns it to the rest position. Is the force proportional to the distance from the equilibrium position? Let's calculate s, the displacement of the mass along the arc. It is equal to the length of the pendulum L times the angle theta, expressed in radians. Recall Hooke's Law: F equals minus k times x. Since, the displacement of the pendulum is s equals L times theta, and L is constant, the restoring force should be proportional to the angle for the pendulum to exhibit harmonic motion. Take a minute to analyze this sketch. The forces acting on the mass are the gravitational force mg and the tension T exerted by the string. The gravitational force can be resolved into a component along the string, mg cosine theta, which is opposite to the string's tension and a component perpendicular to the string, mg sine theta that is tangential to the mass' displacement and is always directed toward the equilibrium position. Only the tangential force, F equals minus mg sine theta, is a restoring force. Since m and g are constant, they can be combined and replaced by the single constant k. Accordingly, F equals minus k times sine theta. The restoring force is not proportional to minus k times theta, but to minus k times sine theta. The restoring force does not satisfy Hooke's law, therefore, the pendulum does not exhibit harmonic motion.

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