Oscillatory motion (chapter twelve) Motion of a particle on a spring Simple harmonic motion Energy in SHM Simple pendulum Physical pendulum Damped oscillations Forced oscillations

Particle attached to a spring We can model oscillatory motion as a mass attached to a spring (linear restoring force) Causes displaced mass to to be restored to the equilibrium position. Potential energy  Kinetic energy. At equilibrium – large KE but force is now zero. Newton’s first law - keeps moving.

Particle attached to a spring We can use Newton’s 2nd law to quantitatively describe the motion Acceleration proportional to displacement. Opposite direction.

Simple Harmonic Motion Defining the ratio k/m2, the equation of motion becomes (in one dimension) This equation has the solution

SHM http://webphysics.davidson.edu/Applets/java10_Archive.html A: amplitude of the motion (maximum displacement) : =(k/m)½ – angular frequency of the motion : phase – where the motion starts A and  are set by the initial conditions,  is fixed by the mass and spring constant http://webphysics.davidson.edu/Applets/java10_Archive.html

SHM Period of one full cycle of motion: Maximum velocity and acceleration:

Energy in SHM Kinetic energy: Potential energy: Total energy of the system: Total energy is constant!

Energy in SHM Oscillation is repeated conversion of kinetic to potential energy and back. Using the expression for the total energy, we can find the velocity as a function of position http://webphysics.davidson.edu/physletprob/ch7_in_class/in_class7_1/default.html

The simple pendulum  L T Fg Small angle approximation - sin

The simple pendulum This equation has the same form as that for the motion of the mass attached to a spring. If we define we get the exact same differential equation, and so the system will undergo the same oscillatory motion as we saw earlier. Note – the frequency (and period) of the pendulum are independent of the mass!

The Physical Pendulum An object hanging from a point other than its COM d  COM

Damped Oscillations If we add in a velocity dependent resistive force The solution to this DE when the resistive force is weak This describes an underdamped oscillator

Damped Oscillations

Damped Oscillations The frequency of oscillation is In other words, some natural frequency plus a change due to the damping When b=2m, the system is critically damped (returns to equilibrium) For b>2m, the system is overdamped – also returns to equilibrium (slower rate).

Forced Oscillations If we try to drive an oscillator with a sinusoidally varying force: The steady-state solution is where 0=(k/m)½ is the natural frequency of the system. The amplitude has a large increase near 0 - resonance

Forced Oscillations