simple harmonic motion


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simple harmonic motion

  1. 1. Simple HarmonicMotionPhysics1/23/2013
  2. 2. Simple Harmonic Motion simple harmonic motion (SHM) –vibration about an equilibrium positionin which a restoring force isproportional to the displacement fromequilibrium two common types of SHM are avibrating spring and an oscillatingpendulum springs can vibrate horizontally (on africtionless surface) or vertically
  3. 3. Oscillating Spring
  4. 4. SHM and Oscillating Springs in an oscillating spring, maximumvelocity (with Felastic = 0) is experiencedat the equilibrium point; as the springmoves away from the equilibriumpoint, the spring begins to exert aforce that causes the velocity todecrease the force exerted is maximum whenthe spring is at maximumdisplacement (either compressed orstretched)
  5. 5. SHM and Oscillating Springs at maximum displacement, thevelocity is zero; since the spring iseither stretched or compressed at thispoint, a force is again exerted to startthe motion over again in an ideal system, the mass-springsystem would oscillate indefinitely
  6. 6. SHM and Oscillating Springs damping occurs when friction slowsthe motion of the vibrating mass,which causes the system to come torest after a period of time if we observe a mass-spring systemover a short period of time, damping isminimal and we can assume an idealmass-spring system
  7. 7. SHM and Oscillating Springs in a mass-spring system, the springforce is always trying to pull or pushthe mass back toward equilibrium;because of this, we call this force arestoring force in SHM, the restoring force isproportional to the mass’displacement; this results in all SHMto be a simple back-and-forth motionover the same path
  8. 8. Hooke’s Law in 1678, Robert Hooke proposed thissimple relationship between force anddisplacement; Hooke’s Law isdescribed as:Felastic = -kx where Felastic is the spring force, k is the spring constant x is the maximum displacement fromequilibrium
  9. 9. Hooke’s Law the negative sign shows us that the force isa restoring force, always moving the objectback to its equilibrium position the spring constant has units ofNewtons/meter the spring constant tells us how resistant aspring is to being compressed or stretched(how many Newtons of force are required tostretch or compress the spring 1 meter) when stretched or compressed, a springhas potential energy
  10. 10. Simple Pendulum simple pendulum – consists of a mass(called a bob) that is attached to afixed string; we assume that the massof the bob is concentrated at a point atthe center of mass of the bob and themass of the string is negligible; wealso disregard friction and airresistance
  11. 11. Simple Pendulum
  12. 12. Simple Pendulum for small amplitude angles (less than15°), a pendulum exhibits SHM at maximum displacement fromequilibrium, a pendulum bob hasmaximum potential energy; atequilibrium, this PE has beenconverted to KE amplitude – the displacement fromequilibrium
  13. 13. Period and Frequency period (T) – the time, in seconds, toexecute one complete cycle of motion;units are seconds per 1 cycle frequency (f) – the number ofcomplete cycles of motion that occurin one second; units are cycles per 1second (also called hertz)
  14. 14. Period and Frequency frequency is the reciprocal of period, so the period of a simple pendulum dependson the length of the string and the value forfree-fall acceleration (in most cases, gravity)
  15. 15. Period of a SimplePendulum notice that only length of the string and thevalue for free-fall acceleration affect theperiod of the pendulum; period isindependent of the mass of the bob or theamplitude
  16. 16. Period of a Mass-SpringSystem period of a mass-spring system depends onmass and the spring constant notice that only the mass and the springconstant affect the period of a spring; periodis independent of amplitude (only forsprings that obey Hooke’s Law)
  17. 17. Comparison of a Pendulumand an Oscillating Spring