Elasticity and Oscillations Lecture 3

Goals Review Kinetic and Potential Energy Gain an understanding of Simple Harmonic Motion (SHM) Understand Period and Frequency as related to SHM

Review of Energy A good equation to start with when working energy problems is the following: If there are no outside forces (gravity and spring forces are not outside forces), set Wnc equal to zero

Lets try an Example Consider a sled at the top of a frictionless hill 20 meters high. Once it reaches the horizontal bottom friction is present, how far does it slide if the coefficient of kinetic friction is .1?

Lets look at a Spring Example Consider a 5 kg block moving horizontally with speed 2 m/s. It collides directly with a spring that has a 100 N/m constant. How far does the spring compress?

Introduction to Simple Harmonic Motion (SHM) SHM is a special type of motion that occurs whenever the restoring force is proportional to the displacement from equilibrium Many types of natural phenomena exhibit this motion This includes springs and pendulums to a reasonable order

Still Confused? If you are still confused it may be helpful to look at the result of SHM The displacement from equilibrium will be a sine wave, the offset is determined by the starting location

Lets look at the Result of the Motion

Period and Frequency as Related to SHM Looking at the previous drawing we can see that the period is the time for the object to leave and then return to the start location The frequency is the inverse of the period and tells us how many cycles occur in a second, it has the units of inverse seconds or Hertz (HZ)

SHM for a Spring For a spring vibrating starting from equilibrium we know the following holds Angular frequency is 2pi divided by the period, or multiplied by the frequency

Frequency of Motion for the a Spring in SHM Due to the periodic nature of simple harmonic motion, circular motion can be used to produce theoretical results For SHM observed for spring motion, this can give the following result

Velocity and Acceleration in SHM

Another SHM Oscillator is the Vertical Spring A spring that is hung vertically rather than horizontally also exhibits SHM, the only difference is the equilibrium position

Lets See the Difference

Pendulums are can show SHM To a degree, pendulums also act as Simple Harmonic Oscillators Both the Simple and Physical Pendulum can be modeled this way if the angle is small Review the Physical Pendulum in your book

Lets see why we can Approximate the Pendulum’s Motion as SHM

Lets Finally Derive the Period for a Simple Pendulum

Simple Harmonic Motion Derivation Frestore = Fparallel = - m g sin θ sin θ ≈ θ The angle is also related to the displacement by θ = y/L

Simple Pendulum Derivation Continued Applying Newton’s Second Law: This is analogous to the restoring force for a spring, we can therefore replace the spring constant for SHM observed in springs to give the frequency of a simple pendulum

Additional Notes