Oscillations and Harmonic Motion Notes Ch.15 Oscillations and Harmonic Motion

Periodic Motion Regular vibrations or oscillations repeat the same movement on either side of the equilibrium position f times per second (f is the frequency) Displacement is the distance from the equilibrium position Amplitude is the maximum displacement Period (T) is the time for one cycle or or 1 complete oscillation

Producing time traces 2 ways of producing a voltage analogue of the motion of an oscillating system

The conical pendulum The vertical component of the tension (Tcosθ) supports the weight (mg) The horizontal component of tension (Tsinθ) provides the centripetal force

Time traces

15.1 Simple Harmonic Motion Simple harmonic motion: Motion that follows a repetitive pattern, caused by a restoring force that is proportional to displacement from the equilibrium position. Simple harmonic motion is linear motion in which the acceleration is proportional to the displacement from an equilibrium position and directed toward that position Period is independent of amplitude Same time for a large swing and a small swing For a pendulum this only works for angles of deflection up to about 20º Simple harmonic motion

15.2 Simple harmonic motion: graph and equation Gradient of displacement v. time graph gives a velocity v. time graph Max veloc at x = 0 Zero veloc at x = max Graphing simple harmonic motion

Simple Harmonic Motion cont… Acceleration v. time graph is produced from the gradient of a velocity v. time graph Max a at V = zero Zero a at v = max

Simple Harmonic Motion cont… Displacement and acceleration are out of phase a is proportional to – x Hence the minus

Simple Harmonic Motion cont… a = -ω²x equation defines SHM T = 2π/ω F = -kx eg. a trolley tethered between two springs

Circular Motion and SHM The peg following a circular path casts a shadow which follows SHM This gives a mathematical connection between the period T and the angular velocity of the rotating peg

The Pendulum Pendulum – an object suspended so that it can swing back and forth about an axis. Rules of simple pendulums Period is independent of mass Period is independent of amplitude as long as the arc is small (<20º) Period is directly proportional to the square root of the length. Period is inversely proportional square root of the acceleration of gravity