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Simple Harmonic Motion. Pre-AP Physics Pearland High School Mr. Dunk. Simple Harmonic Motion. simple harmonic motion (SHM) – vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium

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Simple Harmonic Motion


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simple harmonic motion

Simple Harmonic Motion

Pre-AP Physics

Pearland High School

Mr. Dunk

simple harmonic motion1
Simple Harmonic Motion
  • simple harmonic motion (SHM) – vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium
  • two common types of SHM are a vibrating spring and an oscillating pendulum
  • springs can vibrate horizontally (on a frictionless surface) or vertically
shm and oscillating springs
SHM and Oscillating Springs
  • in an oscillating spring, maximum velocity (with Felastic = 0) is experienced at the equilibrium point; as the spring moves away from the equilibrium point, the spring begins to exert a force that causes the velocity to decrease
  • the force exerted is maximum when the spring is at maximum displacement (either compressed or stretched)
shm and oscillating springs1
SHM and Oscillating Springs
  • at maximum displacement, the velocity is zero; since the spring is either stretched or compressed at this point, a force is again exerted to start the motion over again
  • in an ideal system, the mass-spring system would oscillate indefinitely
shm and oscillating springs2
SHM and Oscillating Springs
  • damping occurs when friction slows the motion of the vibrating mass, which causes the system to come to rest after a period of time
  • if we observe a mass-spring system over a short period of time, damping is minimal and we can assume an ideal mass-spring system
shm and oscillating springs3
SHM and Oscillating Springs
  • in a mass-spring system, the spring force is always trying to pull or push the mass back toward equilibrium; because of this, we call this force a restoring force
  • in SHM, the restoring force is proportional to the mass’ displacement; this results in all SHM to be a simple back-and-forth motion over the same path
hooke s law
Hooke’s Law
  • in 1678, Robert Hooke proposed this simple relationship between force and displacement; Hooke’s Law is described as:

Felastic = -kx

  • where Felastic is the spring force,
  • k is the spring constant
  • x is the maximum displacement from equilibrium
hooke s law1
Hooke’s Law
  • the negative sign shows us that the force is a restoring force, always moving the object back to its equilibrium position
  • the spring constant has units of Newtons/meter
  • the spring constant tells us how resistant a spring is to being compressed or stretched (how many Newtons of force are required to stretch or compress the spring 1 meter)
  • when stretched or compressed, a spring has potential energy
simple pendulum
Simple Pendulum
  • simple pendulum – consists of a mass (called a bob) that is attached to a fixed string; we assume that the mass of the bob is concentrated at a point at the center of mass of the bob and the mass of the string is negligible; we also disregard friction and air resistance
simple pendulum2
Simple Pendulum
  • for small amplitude angles (less than 15°), a pendulum exhibits SHM
  • at maximum displacement from equilibrium, a pendulum bob has maximum potential energy; at equilibrium, this PE has been converted to KE
  • amplitude – the maximum displacement from equilibrium
period and frequency
Period and Frequency
  • period (T) – the time, in seconds, to execute one complete cycle of motion; units are seconds per 1 cycle
  • frequency (f) – the number of complete cycles of motion that occur in one second; units are cycles per 1 second (also called hertz)
period and frequency1
Period and Frequency
  • frequency is the reciprocal of period, so
  • the period of a simple pendulum depends on the length of the string and the value for free-fall acceleration (in most cases, gravity)
period of a simple pendulum
Period of a Simple Pendulum
  • notice that only length of the string and the value for free-fall acceleration affect the period of the pendulum; period is independent of the mass of the bob or the amplitude
period of a mass spring system
Period of a Mass-Spring System
  • period of a mass-spring system depends on mass and the spring constant
  • notice that only the mass and the spring constant affect the period of a spring; period is independent of amplitude (only for springs that obey Hooke’s Law)