Chapter 14 OutlinePeriodic Motion

• Oscillations• Amplitude, period, frequency

• Simple harmonic motion• Displacement, velocity, and acceleration

• Energy in simple harmonic motion

• Pendulums• Simple

• Physical

• Damped oscillations

• Resonance

Periodic Motion

• Many types of motion repeat again and again.• Plucked string on a guitar, a child in a swing, sound waves in a

flute…

• This is called periodic motion, or oscillation.• Stable equilibrium point

• Displacement from equilibrium leads to a force (or torque) to return it to equilibrium.

• Kinetic energy leads to overshoot, causing repeat

Periodic Motion Notation

• Amplitude – The maximum magnitude of displacement from equilibrium

• Period, – The time for one cycle ()

• Frequency, – Number of cycle per unit time ()

• Angular frequency, – Equal to ()

• From these definitions,

, and

Simple Harmonic Motion

• If the restoring force is directly proportional to the displacement, the resulting oscillation is simple harmonic motion.

• One example is a mass on spring that obeys Hooke’s law.

Mass on a Spring

• Using Newton’s second law, in one dimension,

• Since acceleration is the second time derivative of position,

Mass on a Spring

• Sinusoidal functions satisfy this differential equation.

• Plugging this into the equation, we can solve for ,

• Comparing to ,

Simple Harmonic Motion of a Mass on a Spring

• Angular frequency

• Frequency

• Period

• Amplitude does not enter the equations as long as the spring obeys Hooke’s law. (Generally for small amplitudes.)

Displacement, Velocity and Acceleration

• As we saw earlier, sinusoidal functions satisfy the differential equation for simple harmonic motion.

• is the amplitude, and is the phase angle (starting position)

• We find the velocity at any time by taking the time derivative,

• Likewise, for the acceleration,

SHM Example

Energy in Periodic Motion

• If there are no dissipative forces (friction), the total energy is constant.

Energy Plots in Periodic Motion

𝐸=𝐾 +𝑈=12𝑚𝑣2+

12𝑘 𝑥2

Vertical SHM

• We looked at a mass oscillating horizontally on a frictionless table.

• How does this change if the mass is hanging vertically?

• The equilibrium point will not be at the same extension of the spring. ( instead of )

• This the same as before, but is measured from the new equilibrium position.

Molecular Vibrations

• Interactions between neutral atoms can be described by the Lennard-Jones potential.

• The positive term is due to the Pauli repulsion.• The negative, is due to long range attractive forces (LDF).

• This does not look like the simple parabolic potential well from a mass on a spring, but we can still look at the oscillations.

Molecular Vibrations

• The restoring force is:

• Using, the binomial theorem,

• For small oscillations, the restoring force reduces to:

• Where

• This is now Hooke’s law.

Simple Pendulum

• The simplest pendulum is a point mass on a massless string.

• What provides the restoring force?

• How can we describe the motion?

• Mass , length .

Simple Pendulum

• For small amplitudes,

Physical Pendulum

• A real, or physical, pendulum is an extended object.

• Now, we need to know the moment of inertia, , as well as the mass , and distance from the pivot point to the center of mass, .

Physical Pendulum

• Again, for small amplitudes,

Physical Pendulum Example

Damped Oscillations

• We have assumed there was no friction, and therefore the amplitude never decreases, but in reality, energy will be lost to friction.

• This decrease in amplitude is damping.

• Consider the simplest case where the frictional damping is proportional to the velocity of the object.

• This force always opposes the motion.

Underdamped Oscillations

• Solving this differential equation for small ,

• This is very similar to simple harmonic motion but with an exponentially decreasing amplitude.

• Also, the angular frequency is decreased.

Critical and Over-damped Oscillations

• When the equation for ,

• The system is critically damped, and it no longer oscillates, but returns exponentially to equilibrium.

• If , the system is overdamped, and it decays to zero with a double exponential.

Forced Oscillations and Resonance

• If the system is driven at some arbitrary frequency, there will not be much of a response.

• But, if it is driven near the natural resonance frequency, the response can be quite large.

Chapter 14 SummaryPeriodic Motion

• Oscillations• Amplitude,

• Period,

• Frequency, , and angular frequency,

• Simple harmonic motion – Spring:

• Energy in simple harmonic motion

Chapter 14 SummaryPeriodic Motion

• Pendulums• Simple:

• Physical:

• Damped oscillations

• Resonance – driven near