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Periodic Motion

Periodic Motion. Definition of Terms. Periodic Motion: Motion that repeats itself in a regular pattern. Cycle: One complete vibration or oscillation. Equilibrium position: Position of the object when it is at rest. No energy is stored, and no net force acts on the object.

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Periodic Motion

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  1. Periodic Motion

  2. Definition of Terms • Periodic Motion: Motion that repeats itself in a regular pattern. • Cycle: One complete vibration or oscillation. • Equilibrium position: Position of the object when it is at rest. No energy is stored, and no net force acts on the object. • Restoring Force: Force that brings an oscillating object back to its equilibrium position.

  3. Robert Hooke (1635-1703) • In 1678, he determined that the deformation of an elastic object is directly proportional to the force causing the deformation. (HOOKE’S LAW)

  4. Hooke’s Law F  x or F = -kx • F = applied force (N) • x = amount of deformation (m) • k = spring (force) constant • Units = N/m

  5. Given: m = 0.25 kg F = mg = -2.4 N k = 48 N/m Find: x = ? F = -kx F/k = x (2.4 N) / (48 N/m) = x 0.050 m = x Ex: A 0.25 kg mass is suspended from a spring with a force constant of 48 N/m. How far does the spring stretch?

  6. Elastic Potential Energy (PEe) • Energy stored in an elastic object (usually a spring) by deforming it (doing work on it). PEe = ½ kx2 • x = distance spring is deformed (stretched or compressed) • k = spring constant: How resistant an elastic object is to being stretched or compressed (stiffness). Units = N/m • Units = N/m (m2) = N*m = Joules

  7. Given: k = 160 N/m x = 0.080 m Find: PEe = ? PEe = ½ kx2 = ½ (160 N/m)(.080 m)2 PEe = 0.51 J Ex: A spring with a spring constant of 160 N/m is compressed by 8.0 cm. How much energy is stored in the spring?

  8. Simple Harmonic Motion (SHM) • Any periodic motion in which the restoring force is proportional to the displacement from equilibrium. • Mass on a Spring • Simple Pendulum (for small angles)

  9. Measures of Simple Harmonic Motion • Amplitude (A): Maximum displacement from equilibrium position (meters) • Period (T): Time it takes to execute one complete cycle of motion (seconds) • Frequency (f): Number of cycles or vibrations per unit of time • Units = Hertz (Hz) or s-1 or 1/s

  10. Calculating Period • Simple Pendulum: T = period (s) L = length of pendulum (m) g = free fall acceleration (m/s2)

  11. Given: L = 1.20 m g = 9.81 m/s2 Find: T =? T = 2π√(L/g) = 2π√(1.20 m / 9.81 m/s2) = 2.20 s Ex: What is the period on Earth of a simple pendulum that has a length of 1.20 m?

  12. Resonance • Forced Vibrations: One vibrating objectcauses another object to vibrate at the same frequency. • Natural Frequency: Frequency at which minimum energy is required to produce forced vibrations. An object “prefers” to vibrate at this frequency. • Resonance occurs when the frequency of a forced vibration matches the natural frequency of a system.

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