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Unformatted text preview: EXPERIMENT SIMPLE HARMONIC MOTION THE VARIABLE-g PENDULUM Introduction: Simple Harmonic Motion (SHM) is one of the most common types of motion found in nature: the elastic properties and tension of a plucked guitar string combine to produce SHM; the forces of gravity and buoyancy combine to produce SHM when a float is dropped into water; the tension in the supporting cord combined with the force of gravity produce SHM when a suspended mass is displaced from equilibrium and released. In all of the above examples there exists a restoring force F that is directed towards the position of equilibrium and is proportional to the displacement x of the object from equilibrium: F x . If we insert a constant of proportionality k (often referred to as the spring constant ) into this relation we have the basic force equation governing elastic behavior: F = k x (1) If the force F is acting on a body with mass m and acceleration a, then we must have ma = k x or (2) 2 + d where . Eq.(2) is a second order differential equation describing periodic motion with angular frequency of oscillation and period of oscillation T where T=2 / . It has a general solution of the form x = x o cos t, where x o represents the maximum displacement of the object from equilibrium. Substitution into Eq.(2) demonstrates that it is indeed a solution; however, note that it is not the only solution. 2 2 = x dt x m k / = A simple pendulum consists of a massive bob suspended from a fixed point by a cord of negligible mass with respect to the bob. If the bob is displaced by a small amount from its position of equilibrium, the subsequent motion will be SHM. If the bob has mass m then the only external force (neglecting friction) on the bob is m g . This force may be separated into a component F // parallel and a component F perpendicular to the direction of motion . The F component is the force the cord exerts on the bob to maintain the trajectory and the F // component supplies the restoring force necessary for SHM (see Fig.1)....
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