Vibrations and Waves (II)

• Conservation of Energy
• applying the idea of conservation of energy allows us an alternate description of the motion.
• energy can be stored in the spring
• the spring force is conservative
• we can define an elastic potential energy as
• U_s = 1/2kx²
• Total Mechanical Energy E = U + KE
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• Thus, for an object undergoing SHM, total mechanical energy is conserved.



• At the turning points (max displacement from equilibrium) the speed is zero.
• KE = 0, PE = 1/2kA² = total energy.



• at equilibrium point, speed is a maximum



• at any other point,
• 1/2kx² + 1/2mv² = 1/2kA²
• we can use this to find the velocity as a function of position



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• with the + or - sign dependant on the direction of motion




• Simple Pendulum

• The motion of a pendulum is periodic but is it SHM
• The criteria for SHM is a restoring force proportional to the displacement.


• The equilibrium position of the mass is directly below the point of support.
• When the mass is displaced from the equilibrium position, we are interested in the force that will return the mass to its equilibrium position.
• That restoring force is the component of the weight tangent to the circular path.
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• In order to have SHM we need a restoring force proportional to the displacement which is s, the length of the circular arc. But if is measured in radians, then s = ql, and the restoring force can be proportional to q and result in SHM. Unfortunately, the restoring force is not proportional to q but rather to sinq. However, if q is small,
• and the restoring force can be written as
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• By comparing to SHM for a mass on a spring
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• not all objects can be considered as point masses supported by weightless strings.
• for rotational motion



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• If the angle is small, then we can make the same approximation as with the simple pendulum and by comparison



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