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  two common types of SHM are a vibrating spring and an oscillating pendulum  springs can vibrate horizontally (on a frictionless surface) or vertically

Oscillating Spring

SHM and Oscillating Springs  in an oscillating spring, maximum velocity (with F elastic = 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 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 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 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  in 1678, Robert Hooke proposed this simple relationship between force and displacement; Hooke’s Law is described as: F elastic = -kx  where F elastic is the spring force,  k is the spring constant  x is the maximum displacement from equilibrium

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 – 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 Pendulum

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 (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 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  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 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)

Comparison of a Pendulum and an Oscillating Spring