Unit 6 Part 2: Simple Harmonic Motion Book Section 10.2

Simple Harmonic Motion  Also called vibrational motion, oscillation, or periodic motion.  Think springs, pendulums, atoms carrying sound, or buildings or trees swaying in the wind.  Key difference:  Translational motion – object is permanently moved.  Vibrational motion – object periodically returns to a “resting” position.

Simple Harmonic Motion

The “Resting” Position  Also known as the “equilibrium” position.  All forces are balanced.  Place where an object is “happy” in absence of any forces.

The “Resting” Position  Where would the resting position be for a pendulum?  At the bottom.  For a building, tree, or bobble head?  Standing straight up.  For a hanging spring & mass?  Where the system comes to rest after released.

Vibration  Vibration is displacement from the resting position that follows a periodic pattern. System begins in resting position. 1.Force causes displacement from resting position. 2.Restoring force causes the system to move back towards resting position. 3.Overshoots resting position, so displaced to the other direction. 4.Restoring force works (in the other direction now) moves back toward the resting position again. 5.Overshoots again, and so on…

Restoring Force  Restoring force is the force that causes the spring, pendulum, etc. to move back toward the resting position.  Can be provided by…  Gravity  Tension/Compression  NOT friction  NOT Normal Force  CAN be electromagnetic force

Damping  In ideal motion (with no friction, air resistance, etc.) vibrations go on forever.  Just as we learned in our unit about Energy!  In real life, vibrations “die out” due to damping.  Damping is caused by nonconservative forces such as air resistance and friction, which work against the restoring forces in vibrational motion, causing the displacement from resting position to be less and less over time.  If there is any damping, the system will eventually return to resting position.

Periodic Motion Quantified

Period for a Mass on a Spring

Frequency for a Mass on a Spring

Since f is proportional to the square root of k/m, compare k/m for each system. The ranking should be A>B>D>C.

Displacement vs. Time  A displacement (from resting position) vs. time graph looks like a sine wave.

Velocity vs. Time  At maximum displacement, v=0, and at displacement=0, v=max.  A velocity vs. time graph looks like a cosine wave

Acceleration vs. Time  Similarly, at maximum velocity, a=0, and at velocity=0, a=max.  An acceleration vs. time graph looks like the opposite of the displacement vs time. (a negative sine wave)

Amplitude  Amplitude (A) = maximum displacement from rest position.  NOT peak to trough!  Measure of energy, independent of period or frequency.

Answer: T = 8 s, the amount of time that it takes for one complete cycle. Measure peak to peak or similarly.

Answer: f = 1/period = 1/8 Hz. Meaning it goes through 1/8 th of a cycle in 1 second.

Answer: Since velocity is in the positive at the beginning, the cart must have been displaced to the left.