Periodic Motion Periodic Motion When a vibration or

Periodic Motion

Periodic Motion • When a vibration or oscillation repeats itself over the same time period • Simple Harmonic Motion (SHM) - any vibrating system where the restoring force is directly proportional to the negative displacement

Oscillating Spring System • Equilibrium positionwhere spring exerts no force on the mass (m) • Natural state

Oscillating Spring System • Spring stretched • Exerts a force to move it back to equilibrium

Oscillating Spring System • Spring compressed • Force exerted pushing it back to equilibrium

Restoring Force (F) • Force exerted when away from equilibrium • Directly proportional to the displacement (x) F = - kx • k – spring constant • Greater the value of k the more force is needed to stretch spring - stiffer

• F is not constant, depends on x • F = ma • Therefore acceleration is not constant, depends on x

Question • At what position(s) is the acceleration the greatest? • Where is it the least? • Where is velocity greatest? And least?

Example • A. 10 kg object is suspended from a spring with a spring constant of 10 N/m. The object is pulled 15 cm from its equilibrium position and released, what is its maximum acceleration and when does it experience this?

Other Definitions • Amplitude (A) – point of greatest displacement • Cycle – complete oscillation • Period (T) – time for one complete oscillation • Frequency (f) – cycles per second T = 1/f

Period of Vibration •

Pendulum • Small object (the bob) suspended from the end of a lightweight cord • Motion of pendulum very close to SHM if the amplitude of oscillation is fairly small • Restoring force is the component of the bobs weight – depends on the weight and the angle

Period of Pendulum •

Example • A pendulum with a length of 36. 9 cm has a period of 1. 22 s. What is the gravitational field at the pendulum’s location?

Resonance • Every vibrating system has its own natural frequency • If you try to vibrate the system at other than its natural frequency, it will bounce around but never reach any great amplitude • For example, if you push on a swing at random frequency

• If you push on the object at the same frequency as its natural frequency, the amplitude will increase dramatically • This effect is called resonance • The natural vibrating frequency of an object is its resonant frequency